Limit trees for free group automorphisms: universality

Abstract

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism’s dominating exponential dynamics.

As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn–Handel’s recognition theorem for atoroidal outer automorphisms.

Introduction

We previously constructed a limit pretree that detects the exponential dynamics for an arbitrary free group automorphism [22]. In this sequel, we prove the construction is canonical. This completes the existence and uniqueness theorem for a free group automorphism’s limit pretree. Recall that if we record all the compact geodesics in an $\mathbb R$ -tree but forget their lengths, then the resulting structure is a pretree; briefly, a pretree is a set with a structure that encodes the notion of closed intervals satisfying certain axioms. Pretrees are the baseline of our constructions; for instance, ( $\mathbb R$ -)trees will be defined as pretrees with convex metrics, and pseudotrees as pretrees with a certain hierarchy of convex pseudometrics.

In [22], we motivated the existence and uniqueness theorem of a limit pretree by describing it as a free group analogue to the Nielsen–Thurston theory for surface homeomorphisms, which in turn can be seen as the surface analogue to the Jordan canonical form for linear maps. We now give our own motivation for this result.

Universal representation of an endomorphism

It feels rather odd to discuss my personal motivation while using the communal ‘we’; excuse me as I break this convention a bit for this section. In my doctoral thesis, I extended Brinkmann’s hyperbolization theorem to mapping tori of free group endomorphisms. This required studying the dynamics of endomorphisms. Along the way, I proved that injective endomorphisms have canonical representatives. More precisely, suppose $\phi \colon F \to F$ is an injective endomorphism of a finitely generated free group; then there is

1. 1. a minimal simplicial F-action on a simplicial tree T with trivial edge stabilizers;

2. 2. a $\phi $ -equivariant expanding embedding $f\colon T \to T$ (unique up to isotopy); and

3. 3. an element in F is T-elliptic if and only if one of its forward $\phi $ -iterates is conjugate to an element in a $[\phi ]$ -periodic free factor of F.

Existence of the limit free splitting (i.e., T with its F-action) for the outer class $[\phi ]$ was the core of my thesis (see also [21, Theorem 3.4.5]). Universality follows from bounded cancellation: any other simplicial tree $T'$ satisfying these three conditions will be uniquely equivariantly isomorphic to T [21, Proposition 3.4.6].

In a way, the limit free splitting detects and filters the ‘nonsurjective dynamics’ of the (outer) endomorphism. When $\phi \colon F \to F$ is an automorphism, then T is a singleton and the free splitting provides no new information. On the other extreme, the F-action on T can be free; in this case, let be the quotient graph. Then the outer endomorphism $[\phi ]$ is represented by a unique expanding immersion $[f]\colon \Gamma \to \Gamma $ and $[\phi ]$ is expansive – such outer endomorphisms are characterized by the absence of $[\phi ]$ -periodic (conjugacy classes of) nontrivial free factors [21, Corollary 3.4.8]. The most important thing is that the expanding immersion $[f]$ has nice dynamics and greatly simplifies the study of expansive outer endomorphisms.

After completing my thesis, I found myself in a paradoxical situation: I had a better ‘understanding’ of nonsurjective endomorphisms than automorphisms – the main obstacle to studying the dynamics of nonsurjective endomorphisms was understanding the dynamics of automorphisms. The naïve expectation (when I started my thesis) had been that nonsurjective endomorphisms have more complicated dynamics as they are not invertible. The current project was born out of an obligation to correct this imbalance.

Universal representation of an automorphism

What follows is a direct analogue of the above discussion in the setting of automorphisms. The main theorem of [22] produces an action that detects and filters the ‘exponential’ dynamics of an automorphism. Specifically, suppose $\phi \colon F \to F$ is an automorphism of a finitely generated free group. Then there is

1. 1. a minimal rigid F-action on a real pretree T with trivial arc stabilizers;

2. 2. a $\phi $ -equivariant ‘expanding’ pretree-automorphism $f \colon T \to T$ ; and

3. 3. an element in F is T-elliptic if and only if it grows polynomially with respect to $[\phi ]$ .

The pair of the pretree T and its rigid F-action is called a (forward) limit pretree for the outer automorphism $[\phi ]$ . The theorem is stated properly in Chapter 3 as Theorem 3.1. When $[\phi ]$ is polynomially growing, then the limit pretree is a singleton (and hence unique) but provides no new information. We are mainly interested in exponentially growing $[\phi ]$ as their limit pretrees are not singletons. On the other hand, the F-action on a limit pretree is free if and only if $[\phi ]$ is atoroidal, (i.e., there are no $[\phi ]$ -periodic (conjugacy classes of) nontrivial elements) [22, Corollary III.5]. As with expanding immersions and expansive outer endomorphisms, the expanding ‘homeomorphism’ $[f]$ (on the quotient space $F \backslash T$ ) has dynamics that could facilitate the study of atoroidal outer automorphisms.

Unlike the endomorphism case, uniqueness of limit pretrees requires a more involved argument. It was remarked in the epilogue of [22] that the only source of indeterminacy in the existence proof was [22, Proposition III.2]; this proposition is restated in Section 1.4 as Proposition 1.2, and a proof is sketched in Sections 2.1 and 2.4. The main result of this paper is a universal version of the proposition. It can also be thought of as an existence and uniqueness theorem for an action that detects and filters the ‘dominating’ exponential dynamics of an outer automorphism:

Main Theorem (Theorems 3.10–3.11).

Let $\phi \colon F \to F$ be an automorphism of a finitely generated free group and $\{\mathcal A_j^{dom}[\phi ]\}_{j=1}^k$ a (possibly empty) subset of $[\phi ]$ -orbits of dominating attracting laminations for $[\phi ]$ .

Then there is

1. 1. a minimal factored F-tree $(Y,\Sigma _{j=1}^k \delta _j)$ with trivial arc stabilizers;

2. 2. a unique $\phi $ -equivariant expanding dilation $f \colon (Y, \Sigma _{j=1}^k \delta _j) \to (Y, \Sigma _{j=1}^k \delta _j)$ ; and

3. 3. for $1 \le j \le k$ , a nontrivial element in F is $\delta _j$ -loxodromic if and only if its forward $\phi $ -iterates have axes that weakly limit to $\mathcal A_j^{dom}[\phi ]$ ;

moreover, the factored F-tree $(Y,\Sigma _{j=1}^k \delta _j)$ is unique up to a unique equivariant dilation.

Thus, the factored tree (up to rescaling of its factors $\delta _j$ ) is a universal construction for outer automorphisms of free groups, and we call it the complete dominating (resp. topmost) tree if we consider the whole set of orbits of dominating (resp. topmost) attracting laminations. As a corollary, the previously constructed limit pretrees are independent of the choices made in the proof of Theorem 3.1 (i.e., the limit pretree is canonical) (Corollary 3.9). Let us now briefly define the emphasized terms in the theorem’s statement.

An F-tree is an ( $\mathbb R$ -)tree with an isometric F-action. Informally, an F-tree is factored if its metric has been equivariantly decomposed as a sum $\sum _{j=1}^k \delta _j$ of pseudometrics. For a factored F-tree $(Y,\Sigma _{j=1}^k \delta _j)$ , an element in F is $\delta _i$ -loxodromic if it is Y-loxodromic and its axis has positive $\delta _i$ -diameter. An equivariant homeomorphism $(T, \Sigma _{j=1}^k d_j) \to (Y,\Sigma _{j=1}^k \delta _j)$ of factored F-trees is a dilation if it is a homothety of each pair of factors $d_j$ and $\delta _j$ ; a dilation is expanding if each factor-homothety is expanding.

A lamination in F is a nonempty closed subset in the space of lines in F. A sequence of lines (e.g., axes) weakly limits to a lamination if some subsequence converges to the lamination. Any $[\phi ]$ has a finite set of attracting laminations which is empty if and only if $[\phi ]$ is polynomially growing; this set is partially ordered by inclusion and has an order-preserving $[\phi ]$ -action. The maximal elements of the partial order are called topmost. An attracting lamination A for $[\phi ]$ has an associated stretch factor $\lambda (A)$ ; it is dominating if any distinct attracting lamination $A'$ for $[\phi ]$ containing A has a strictly smaller stretch factor $\lambda (A') < \lambda (A)$ . Topmost attracting laminations are vacuously dominating; moreover, the $[\phi ]$ -action permutes the dominating attracting laminations.

Remark. If one considers a subset $\{\mathcal A_j^{top}[\phi ]\}_{j=1}^k$ of $[\phi ]$ -orbits of topmost attracting laminations, then we prove the topmost tree has the additional property that its factor-pseudometrics are pairwise mutually singular: for each i, there is an element that is $\delta _i$ -loxodromic but $\delta _j$ -elliptic for $j \neq i$ (see Section 3.4). We highlight this feature by using the notation $(Y, \oplus _{j=1}^k \delta _j)$ for topmost trees.

Some applications of universal representations. Fix an automorphism $\phi \colon F \to F$ ; since $[\phi ]$ has a unique equivariant dilation class $[Y, \Sigma _{j=1}^k \delta _j]$ of complete dominating limit trees, any invariant of the class is automatically an invariant of $[\phi ]$ . For instance, the Gaboriau–Levitt index $i(Y)$ (as defined in [11, Chapter III]) is the dominating forward index for $[\phi ]$ . In fact, since the limit pretree T for $[\phi ]$ is canonical, its index $i(T)$ (defined in [22, Appendix A]) is the exponential (forward) index for $[\phi ]$ ; when $[\phi ]$ is atoroidal, the index $i(T)$ is closely related to the Gaboriau–Jaeger–Levitt–Lustig index for $[\phi ]$ defined in [10, Section 6]. Each factor $\delta _j$ has an associated F-tree $(Y_j^{dom}, \delta _j)$ ; the pairing of $\delta _j$ with the orbit of dominating attracting lamination $\mathcal A_j^{dom}[\phi ]$ means $i(Y_j^{dom})$ is an index for $\mathcal A_j^{dom}[\phi ]$ , respectively.

Our main application is a characterization of minimal F-trees with $\phi $ -equivariant expanding homotheties:

Main Corollary (Theorem 5.3).

Let $\phi \colon F \to F$ be an automorphism and $(Y, \delta )$ a minimal very small F-tree. The F-tree $(Y, \delta )$ admits a $\phi $ -equivariant expanding homothety if and only if it is equivariantly isometric to the dominating tree for $[\phi ]$ with respect to a subset of $[\phi ]$ -orbits of dominating attracting laminations with the same stretch factor.

In the appendix, we prove a variation of Feighn–Handel’s recognition theorem for atoroidal outer automorphisms.

Some historical context

This paper continues Gaboriau–Levitt–Lustig’s philosophy of prioritizing limit trees in their alternative proof of the Scott conjecture [12]. In particular, our paper relies only on the existence of irreducible train tracks [4, Section 1] but none of the typical splitting paths analysis of relative train tracks [3, 9]. Zlil Sela gave another dendrogical proof the conjecture (now Bestvina–Handel’s theorem) that used Rips’s theorem in place of train track technology [25]. Frédéric Paulin gave yet another dendrological proof that avoids both train tracks and Rips’s theorem [23].

About the same time, Bestvina–Fieghn–Handel used train tracks and trees to prove fully irreducible (outer) automorphisms have universal limit trees [2]. They used this to give a short dendrological proof of a special case of the Tits Alternative for $\operatorname {Out}(F)$ ; their later proof of the general case was much more involved due to the lack of such a universal limit construction [3]. Universal limit trees have been indispensable for studying fully irreducible automorphisms. In principle, a universal construction of limit trees for all automorphisms would lead to a dendrological proof of the Tits alternative and extend much of the theory for fully irreducible automorphisms to arbitrary automorphisms. Speaking of dendrological proofs of the Tits alternative, we mention that Camille Horbez gave such a proof with a very different approach [15].

Continuing the work started in [3], Feighn–Handel defined and proved the existence of completely split relative train tracks (CTs) in [9, Section 4]; they use CTs to characterize abelian subgroup of $\operatorname {Out}(F)$ [8]. The main obstacle when working with topological representatives is that they are not canonical, which can make defining invariants of the outer automorphism quite technical. This is the difficulty that we had to deal with in this paper; however, now that it is done, we can use our new universal representatives to define other invariants rather easily. A minor inconvenience when working with CTs is that they are only proven to exist for some (uniform) iterate of the outer automorphism; we were very careful (perhaps to a fault) in this paper to ensure our universal representatives exist for all outer automorphisms. Finally, a subtle advantage to our approach is that we find universal representatives for automorphisms and not just outer automorphisms!

In a sequel to [25], Sela used limit trees and Rips’s theorem to give a canonical hierarchical decomposition of the free group F that is invariant under a given atoroidal automorphism [24]. This second paper was never published, and a third announced paper that extends the canonical decomposition to arbitrary automorphisms was never released even as a preprint (as far as we know). We remark that the limit trees used in that paper were not (or rather, were never proven to be) canonical/universal. Perhaps, one could combine Sela’s canonical decomposition with Bestvina–Feighn–Handel’s work to give a universal construction of limit trees for atoroidal automorphisms – our approach is independent of Sela’s work and applies more generally to exponentially growing automorphisms. Conversely, we suspect that a careful study of the structure of our topmost trees might recover Sela’s canonical hierarchical decomposition.

Morgan–Shalen introduced the term ‘ $\mathbb R$ -trees’ in [20]. They also defined ‘ $\Lambda $ -trees’ for an ordered abelian group $\Lambda $ . At first glance, the hierarchy of pseudometrics on a real pretree (defined in Section 1.2) looks like a $\Lambda $ -tree. But paths in our constructed hierarchies ‘exit’ infinitesimal trees through metric completion points, whereas paths in a $\Lambda $ -tree exit at infinity. Hierarchies appear to be a new construction to the best of our knowledge.

Proof outline for existence of topmost tree (Theorem 3.7)

One method for constructing limit trees is iterating expanding irreducible train tracks. This is carried out in Section 2.1, but it has two drawbacks: exponentially growing automorphisms do not always have expanding irreducible train tracks; and even when they do, the point stabilizers of the corresponding limit tree are not canonical as they can change with the choice of train tracks. We handle the first obstacle in Section 2.4 by constructing a limit tree $(Y_1, \delta _1)$ using a descending sequence of irreducible train tracks, where only the last train track is expanding. Such descending sequences always exist for exponentially growing automorphisms.

Next, we construct in Section 3.1 a pretree with an F-action whose point stabilizers are canonical. Set , and let ${\mathcal {G}}_2$ be the $[\phi ]$ -invariant subgroup system determined by the point stabilizers of $G_1$ acting on $Y_1$ . By restricting $[\phi ]$ to ${\mathcal {G}}_2$ and inductively repeating the construction, we get a descending sequence of limit forests $(\mathcal Y_i, \delta _i)_{i=1}^n$ . Each limit forest $(\mathcal Y_i, \delta _i)$ has ( $[\phi ]$ -orbits of) attracting laminations $\mathcal A_i[\phi ]$ for $[\phi ]$ that are forward limits of $\mathcal Y_i$ -loxodromic elements in ${\mathcal {G}}_i$ . Starting with $X^{(1)} = Y_1$ , equivariantly replace the points in $X^{i}$ fixed by ${\mathcal {G}}_{i+1}$ with the pretrees $\mathcal Y_{i+1}$ to produce $X^{(i+1)}$ for $i < n$ . The limit pretree $T = X^{(n)}$ has canonical point stabilizers: the maximal polynomially growing subgroups.

Everything we have mentioned so far is a rehash of [22]. From the blow-up construction, the limit pretree T inherits an F-invariant hierarchy $(\delta _i)_{i=1}^n$ of convex pseudometrics – the pseudometric $\delta _i$ is defined on maximal ${\mathcal {G}}_i$ -invariant convex subsets of T of $\delta _{i-1}$ -diameter $0$ . The theorem is finally proven in Section 3.4. The new insight for this proof: if attracting laminations $\mathcal A_i[\phi ]$ are topmost, then the ${\mathcal {G}}_i$ -invariant pseudometric $\delta _i$ can be extended to an F-invariant convex pseudometric, still denoted $\delta _i$ , on T. Let $\{\mathcal A_{\iota (j)}[\phi ]\}_{j=1}^k$ be a subset of topmost attracting laminations. The sum of the corresponding F-invariant pseudometrics on T, denoted $\oplus _{j=1}^k \delta _{\iota (j)}$ , is an F-invariant convex pseudometric on T. Let Y be the partition of T into its maximal subsets of $\oplus _{j=1}^k \delta _{\iota (j)}$ -diameter $0$ ; as these subsets are convex, Y inherits a pretree structure from T. The pseudometric $\oplus _{j=1}^k \delta _{\iota (j)}$ on T induces a convex metric, also denoted $\oplus _{j=1}^k \delta _{\iota (j)}$ , on Y. The metric space $(Y, \oplus _{j=1}^k \delta _{\iota (j)})$ is our topmost tree. This concludes the outline.

At the end of Section 3.5, we prove universality. The proof relies on Chapter 4: variations of Bestvina–Feighn–Handel’s convergence criterion [2]; it boils down to bounded cancellation and Perron–Frobenius theory.

We use the results of [22] as black boxes, and the two papers can be read in any order.

1 Preliminaries

In this paper, F denotes a free group with $2 \le \operatorname {rank}(F) < \infty $ . Subscripts never indicate the rank but instead are used as indices. For inductive arguments, we also work with a free group system of finite type: disjoint union $\bigsqcup _{j \in J} F_j$ of nontrivial finitely generated free groups $F_j$ indexed by a possibly empty finite set J. In this paper, $\mathcal F$ is always a free group system of finite type with some component $F_j$ that is not cyclic.

1.1 Group systems and actions

Nearly all statements and results about groups and connected spaces that we are interested in still hold when ‘connectivity’ is relaxed and we work with ‘systems’ componentwise. In general (almost categorical) terms, a system of [?-objects] is a disjoint union $\mathcal O = \bigsqcup _{j \in J} O_j$ of [?-objects] $O_j$ indexed by some set J. An [?-isomorphism] of systems $\psi \colon \mathcal O \to \mathcal O'$ is a bijection $\sigma \colon J \to J'$ of the corresponding indexing sets and a union of [?-isomorphisms] $\psi _j \colon O_j \to O_{\sigma \cdot j}'$ . The calligraphic font is reserved for systems.

In more concrete terms, here are some basic concepts that will show up in the paper:

1. 1. an isomorphism of group systems $\psi \colon {\mathcal {G}} \to {\mathcal {G}}'$ is a bijection whose restriction to any component ${G_j \subset {\mathcal {G}}}$ is a group isomorphism of components; for group systems, we always assume (for convenience) components are nontrivial if the system is nonempty.

2. 2. two isomorphisms of group systems $\psi ,\psi '\colon {\mathcal {G}} \to {\mathcal {G}}'$ are in the same outer class $[\psi ]$ if the component isomorphisms $\psi _j,\psi _j' \colon G_j \to G_{\sigma \cdot j}'$ differ only by post-composition with an inner automorphism of $G_{\sigma \cdot j}'$ for all $j \in J$ .

3. 3. a metric on a set system $\mathcal X$ is a disjoint union of metrics $d_j\colon X_j \times X_j \to \mathbb R_{\ge 0}$ on the components $X_j \subset \mathcal X$ .

4. 4. for a group system ${\mathcal {G}}$ indexed by J and object system $\mathcal O$ indexed by $J'$ , a $\underline{\mathcal{G}-\mathrm{action}}$ on $\mathcal O$ (or $\underline{\mathcal{G}-\text{object} \text{system} \mathcal{O}}$ ) is a union of component $G_j$ -actions on $O_{\beta \cdot j}$ for some bijection $\beta \colon J \to J'$ .

5. 5. for an automorphism of a group system $\psi \colon {\mathcal {G}} \to {\mathcal {G}}$ and a ${\mathcal {G}}$ -object system $\mathcal O$ , the ψ-twisted ${\mathcal {G}}$ -object system $\mathcal O \psi $ is given by precomposing the component $G_{\sigma \cdot j}$ -action on $O_{\beta \sigma \cdot j}$ with the component isomorphism $\psi _j\colon G_j \to G_{\sigma \cdot j}$ to get a $G_j$ -object $O_{\beta \sigma \cdot j}$ .

1.2 Pretrees, trees and hierarchies

Pretrees are what arises when one wants to discuss ‘treelike’ objects without reference to a metric or topology. In this paper, the pretrees are the ‘primitive’ objects, and metrics/topologies are additional structures on the pretree – think of it the same way a Riemannian metric is a compatible addition to a manifold’s smooth structure.

Fix a set T; an interval function on T is a function $[\cdot , \cdot ]\colon T \times T \to \mathcal P(T)$ , where $\mathcal P(T)$ is the power set of T, that satisfies the following axioms: for all $p,q,r \in T$ ,

1. 1. (symmetric) $[p,q] = [q,p]$ contains $\{p, q\}$ ;

2. 2. (thin) $[p,r] \subset [p,q] \cup [q,r]$ ; and

3. 3. (linear) if $r \in [p,q]$ and $q \in [p,r]$ , then $q = r$ .

A pretree is a pair $(T, [\cdot , \cdot ])$ of a nonempty set T and an interval function $[\cdot , \cdot ]$ on T.

The subsets $[p,q] \subset T$ are called closed intervals, and they should be thought of as the points between p and q (inclusive). We can similarly define open (resp. half-open) intervals by excluding both (resp. exactly one) of $\{p,q\}$ . Generally, ‘interval’ (with no qualifier) refers to any of the three types of intervals we have defined. An interval is degenerate if it is empty or a singleton. We usually omit the interval function and denote a pretree by T. Note that the real line $\mathbb R$ is a pretree.

Any subset $S \subset T$ of a pretree inherits an interval function: for all $u,v \in S$ . A subset $C \subset T$ is convex if $[p,q] \subset C$ for all $p,q \in C$ ; or equivalently, $[\cdot , \cdot ]_C$ is the restriction of $[\cdot , \cdot ]$ to $C \times C \subset T \times T$ . A system of pretrees is a set system $\mathcal T = \bigsqcup _{j \in J} T_j$ and a disjoint union of interval functions on $T_j$ ; we call these systems pretrees for short.

Let $(T, [\cdot , \cdot ])$ and $(T', [\cdot , \cdot ]')$ be pretrees. A pretree-isomorphism is a bijection $f\colon T \to T'$ satisfying $f([p,q]) = [f(p), f(q)]'$ for all $p,q \in T$ . Similarly, a pretree-automorphism of $(T, [\cdot , \cdot ])$ is a pretree-isomorphism $g \colon (T, [\cdot , \cdot ]) \to (T, [\cdot , \cdot ])$ . A pretree is real if its closed intervals are pretree-isomorphic to closed intervals of $\mathbb R$ . By definition, the real line $\mathbb R$ is a real pretree. Note that being real is a property of a pretree, not an added structure like a metric! An arc of a real pretree T is a nonempty union of an ascending chain of nondegenerate intervals. A real pretree is degenerate if it is a singleton; and a system of real pretrees is degenerate if all components are degenerate.

Fix a real pretree T; a convex pseudometric on T is a function $d\colon T \times T \to \mathbb R_{\ge 0}$ satisfying the following axioms: for all $p,q,r \in T$ ,

1. 1. (symmetric) $d(p,q) = d(q,p)$ ;

2. 2. (convex) $d(p,r) = d(p,q) + d(q,r)$ if $q \in [p,r]$ ; and

3. 3. (continuous) $d(p,q) = 2\, d(p,q')$ for some $q' \in [p,q]$ .

For any given convex pseudometric d on T, the preimage $d^{-1}(0) \subset T \times T$ is an equivalence relation on the real pretree T such that each equivalence class is convex and the set $T_d$ of equivalence classes inherits a real pretree structure. A convex metric on T is a convex pseudometric whose equivalence relation $d^{-1}(0)$ is the equality relation on T. A (metric) tree (or $\mathbb R$ -tree) is a real pretree with a convex metric; a forest is a system of trees. For example, the real line $\mathbb R$ is a tree with the standard metric . Note that a convex pseudometric d on a real pretree T induces a convex metric, still denoted d, on the real pretree $T_d$ ; we refer to the tree $(T_d, d)$ as the associated tree.

A λ-homothety of trees $h \colon (T, d) \to (Y, \delta )$ is a pretree-isomorphism $h \colon T \to Y$ that uniformly scales the metric d by $\lambda $ :

$$\begin{align*}\delta(h(p),h(q)) = \lambda \, d(p,q) ~ \text{for all} ~ (p,q) \in \operatorname{dom}(d)= T \times T; \end{align*}$$

equivalently, $h^*\delta = \lambda d$ , where $h^*\delta $ is the pullback of $\delta $ via h. A homothety is a $\lambda $ -homothety for some $\lambda> 0$ ; it is expanding (resp. an isometry) if $\lambda> 1$ (resp. $\lambda = 1$ ). An isometry $\iota \colon (T,d) \to (T, d)$ is elliptic if it fixes a point of T; otherwise, it is loxodromic and acts by a nontrivial translation on its axis, the unique $\iota $ -invariant arc of $(T, d)$ isometric to $(\mathbb R,d_{\mathrm {std}})$ ; the translation distance $\|\iota \|_d \in \mathbb R_{\ge 0}$ is $0$ if $\iota $ is elliptic and equal to the displacement of points in $\iota $ ’s axis if $\iota $ is loxodromic. These definitions extend componentwise to forests.

Let $d_1$ be a nonconstant convex pseudometric on T and $d_{i+1} \colon d_i^{-1}(0) \to \mathbb R_{\ge 0}$ a nonconstant disjoint union of convex pseudometrics for $1 \le i < n$ . The sequence $(d_i)_{i=1}^n$ will be known as an n-level hierarchy of convex pseudometrics on T; we will say just hierarchies for short. A hierarchy $(d_i)_{i=1}^n$ has full support if $d_n$ is a disjoint union of convex metrics. A pseudotree is a pair $(T, (d_i)_{i=1}^n)$ of a real pretree and a hierarchy with full support; a pseudoforest is a system of pseudotrees. A (λ _i ) _i=1 ⁿ -homothety of n-level pseudoforests $h \colon (\mathcal T, (d_i)_{i=1}^n) \to (\mathcal Y, (\delta _i)_{i=1}^n)$ is a system of pretree-isomorphisms $h \colon \mathcal T \to \mathcal Y$ that scales each pseudometric $d_i$ by $\lambda _i$ :

$$\begin{align*}\delta_i(h(p),h(q)) = \lambda_i \, d_i(p,q) ~ \text{for all} ~ i \ge 1 ~ \text{and} ~ (p,q) \in \operatorname{dom}(d_i);\end{align*}$$

a homothety is a $(\lambda _i)_{i=1}^n$ -homothety for some $\lambda _i> 0$ ; it is expanding (resp. isometry) if each $\lambda _i> 1$ (resp. each $\lambda _i = 1$ ). As with trees, an isometry of a pseudotree is either elliptic (fixes a point) or loxodromic (translates a ‘pseudoaxes’). Hierarchies and pseudoforests are the fundamental (perhaps novel) tool in this paper. They are first used in Chapter 3.

1.3 Simplicial actions and train tracks

For a pretree T, a direction at $p \in T$ is a maximal subset $D_p \subset T \setminus \{ p \}$ not separated by p (i.e., $p \notin [q,r]$ for all $q,r \in D_p$ ). A branch point is a point with at least three directions, and a branch is a direction at a branch point. An endpoint is a point with at most one direction. A simple pretree is a pretree whose closed intervals are finite subsets. A pretree T is simplicial if it is real, its subset V of branch points and endpoints is a simple pretree, and no convex proper subset contains V; a vertex is a point in V. An (open) edge in a simplicial pretree T is a maximal convex subset $e \subset T$ that contains no vertex. By construction, edges are open intervals; the corresponding closed intervals in T are called closed edges.

Remark. Being simplicial is a property of a pretree, not an added structure! Besides that, our definition of a simplicial pretree is more general (with one exception) than the standard definition of a simplicial tree and has the advantage that it is independent of any choice of metric/topology. See [22, Interlude] for a discussion on this distinction. The one exception: the real line ℝ is not a simplicial pretree!

An F-pretree is a pretree with an F-action by pretree-automorphisms. An F-pseudotree is a pair of a real F-pretree and an F-invariant hierarchy with full support; equivalently, an F-pseudotree (resp. F-tree) is a pseudotree with an isometric F-action. An F-pseudotree or F-tree is minimal if the underlying F-pretree has no proper nonempty F-invariant convex subset; in this case, the underlying F-pretree has no endpoints. We mostly consider minimal F-pseudotrees with trivial arc (pointwise) stabilizers.

Suppose an F-pseudotree $(T, (d_i)_{i=1}^n)$ has trivial arc stabilizers. For any nontrivial subgroup $G \le F$ , the characteristic convex subset (of T) for G is the unique minimal nonempty G-invariant convex subset $T(G) \subset T$ . In an F-tree $(T, d)$ with trivial arc stabilizers, the restriction of d to $T(G)$ is a G-invariant convex metric, still denoted d; the minimal G-tree $(T(G), d)$ is the characteristic subtree (of $(T, d)$ ) for G.

Remark. We do not really need an isometric action to define characteristic convex subsets and minimality. All we need is the F-action on the real pretree T to be rigid/non-nesting: no closed interval is sent properly into itself by the F-action [22, Section II.2]. While rigid actions are central to [22], they are superseded by isometric actions in this paper.

An F-pretree T is simplicial if T is simplicial and admits an F-invariant convex metric d; equivalently, a simplicial F-pretree is a simplicial pretree with a rigid F-action. Any simplicial F-pretree has an open cone (over a finite dimensional open simplex) worth of F-invariant convex metrics (up to an equivariant isometry isotopic to the identity map). The definitions given so far extend componentwise to systems.

Let $\mathcal T$ and $\mathcal T'$ be simplicial pretrees and $f\colon \mathcal T \to \mathcal T'$ a tight cellular map (i.e., a function that maps vertices to vertices and the restriction to any closed edge is a pretree-embedding – that is, a pretree-isomorphism onto its image). For any choice of convex metrics $d, d'$ on $\mathcal T, \mathcal T'$ , respectively, there is a unique map $(\mathcal T, d) \to (\mathcal T', d')$ that is linear on edges and isotopic to f; whenever a choice of convex metrics is made, we implicitly replace f with this map.

Let $\mathcal T$ be a free splitting of $\mathcal F$ – that is, minimal simplicial $\mathcal F$ -pretrees with trivial edge stabilizers, and suppose $\psi \colon \mathcal F \to \mathcal F$ is an automorphism of a free group system. The $\psi $ -twisted free splitting $\mathcal T\psi $ is the same real pretrees $\mathcal T$ , but the original simplicial $\mathcal F$ -action is precomposed with $\psi $ . A (relative) topological representative for $\psi $ is a $\psi $ -equivariant tight cellular map $f\colon \mathcal T \to \mathcal T$ on a nondegenerate free splitting $\mathcal T$ of $\mathcal F$ : $\psi $ -equivariance means $f(x \cdot p) = \psi (x) \cdot f(p)$ for all $x \in \mathcal F$ and $p \in \mathcal T$ , or equivalently, $f\colon \mathcal T \to \mathcal T \psi $ is equivariant. Given a topological representative $f \colon \mathcal T \to \mathcal T$ for $\psi $ , we let $[f]$ denote the induced map on the quotient $\mathcal F \backslash \mathcal T$ ; we say $[f]$ is a topological representative for the outer class $[\psi ]$ . A (relative) train track for $\psi $ is a topological representative $\tau \colon \mathcal T \to \mathcal T$ for $\psi $ whose iterates $\tau ^m~(m \ge 1)$ are topological representatives for $\phi ^m$ – or equivalently, whose iterates $\tau ^m$ restrict to pretree-embeddings on closed edges.

For any free splitting $\mathcal T$ of $\mathcal F$ , Bass-Serre theory gives a uniform bound on the number of $\mathcal F$ -orbits of edges (linear in $\operatorname {rank}(\mathcal F)$ ) and relates the vertices with nontrivial stabilizers in a (componentwise) connected fundamental domain to a (possibly empty) free factor system $\mathcal F[\mathcal T]$ of $\mathcal F$ – take this as the working definition of free factor systems. The theory also gives a uniform bound on the complexity (e.g., ranks) of free factor systems. A free factor system $\mathcal F[\mathcal T]$ is proper if $\mathcal F[\mathcal T] \neq \mathcal F$ ; equivalently, $\mathcal F[\mathcal T]$ is proper if and only if $\mathcal T$ is not degenerate. Any proper free factor system of $\mathcal F$ has strictly lower complexity than $\mathcal F$ . The trivial free factor system of $\mathcal F$ is the (possibly empty) free factor system consisting of the cyclic $\mathcal F$ -components; it is always proper since we assume $\mathcal F$ has a noncyclic component.

Remark. We will abuse notation and write $\underline{\mathcal{F}[\mathcal{T}] = \mathcal{F}[\mathcal{T}^{\prime}]}$ for two free splittings $\mathcal T, \mathcal T'$ of $\mathcal F$ when we mean an element of $\mathcal F$ is $\mathcal T$ -elliptic if and only if it is $\mathcal T'$ -elliptic.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ and a topological representative $f\colon \mathcal T \to \mathcal T$ for $\psi $ . By $\psi $ -equivariance of f, the proper free factor system $\mathcal F[\mathcal T]$ is $[\psi ]$ -invariant – again, we can take this as the definition of $[\psi ]$ -invariance for proper free factor systems. Form a nonnegative integer square matrix $A[f]$ whose rows and columns are indexed by the $\mathcal F$ -orbits of edges in $\mathcal T$ ; and the entry at row- $[e]$ and column- $[e']$ is given by the number of e-translates in the interval $f(e')$ , where $e, e'$ are edges in T. The topological representative f is irreducible if the matrix $A[f]$ is irreducible; or equivalently, if, for any pair of edges $e, e'$ in $\mathcal T$ , a translate of e is contained $f^m(e')$ for some ${m = m(e,e') \ge 1}$ . It is a foundational theorem of Bestvina–Handel that automorphisms have irreducible train tracks.

Theorem 1.1 (cf. [4, Theorem 1.7]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism of a free group system and ${\mathcal {Z}}$ a $[\psi ]$ -invariant proper free factor system of $\mathcal F$ . Then there is an irreducible train track $\tau \colon \mathcal T \to \mathcal T$ for $\psi $ , where the components of ${\mathcal {Z}}$ are $\mathcal T$ -elliptic.

The proof outline of [22, Theorem I.1] explains how to deduce the theorem as currently stated from the cited theorem.

Suppose $\psi \colon \mathcal F \to \mathcal F$ is an automorphism with an irreducible topological representative $f\colon \mathcal T \to \mathcal T$ . Perron–Frobenius theory implies the matrix $A[f]$ has a unique real eigenvalue $\lambda = \lambda [f] \ge 1$ with a unique positive left eigenvector $\nu [f]$ whose entries sum to $1$ . From the eigenvector $\nu [f]$ , we get an $\mathcal F$ -invariant convex metric $d_f$ on $\mathcal T$ (well-defined up to an equivariant isometry isotopic to the identity map). The restriction of f to any edge is a $\lambda $ -homothetic embedding with respect to $d_f$ ; the metric $d_f$ is the eigenmetric (on $\mathcal T$ ) for $[f]$ . If $\lambda = 1$ , then f is a $\psi $ -equivariant simplicial automorphism of $\mathcal T$ .

1.4 Growth types and limit trees

Since the introduction of train tracks, it has been standard to construct limit forests by iterating an expanding irreducible train track (Section 2.1). Unfortunately, such a construction is not canonical as it can depend on the initial train track. The main idea of the paper: patch together a ‘descending’ sequence of limit trees to get a limit pseudoforest and inductively ‘normalize’ its hierarchy into a canonical limit pseudoforest.

Fix a free group system ${\mathcal {G}}$ of finite type (unlike $\mathcal F$ , all components of ${\mathcal {G}}$ can be cyclic), an automorphism $\psi \colon {\mathcal {G}} \to {\mathcal {G}}$ , and a metric free splitting $(\mathcal T,d)$ of ${\mathcal {G}}$ whose free factor system is $[\psi ]$ -invariant. An element $x \in {\mathcal {G}}$ [ψ]-grows exponentially rel. d with rate λ _x if it is $\mathcal T$ -loxodromic and the limit inferior of the sequence $\left (m^{-1} \log \| \psi ^m(x) \|_d\right )_{m \ge 0}$ is $\log \lambda _x> 0$ . If an element $[\psi ]$ -grows exponentially rel. d, then it $[\psi ]$ -grows exponentially rel. $d'$ with the same rate for any metric free splitting $(\mathcal T',d')$ of ${\mathcal {G}}$ with $\mathcal F[\mathcal T'] = {\mathcal {Z}}$ ; say the element [ψ]-grows exponentially rel. ${\mathcal {Z}}$ . An element $x\in {\mathcal {G}}$ [ψ]-grows polynomially rel. ${\mathcal {Z}}$ with degree < n if the sequence $\left (m^{-n} \| \psi ^m(x)\|_d\right )_{n \ge 0}$ converges to $0$ . Any element of ${\mathcal {G}} [\psi ]$ -grows either exponentially or polynomially rel. ${\mathcal {Z}}$ [22, Corollary III.4]. The growth type of an element is preserved when passing to invariant subgroup systems of finite type.

The automorphism $\psi $ is exponentially growing rel. ${\mathcal {Z}}$ if some element $[\psi ]$ -grows exponentially rel. ${\mathcal {Z}}$ ; otherwise, $\psi $ is polynomially growing rel. ${\mathcal {Z}}$ . The growth type of an outer class $[\psi ]$ is also well-defined. The ‘rel. ${\mathcal {Z}}$ ’ in our terminology may be omitted when ${\mathcal {Z}}$ is trivial. The next proposition deals with the first obstacle:

Proposition 1.2 (cf. [22, Proposition III.2]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism of a free group system and ${\mathcal {Z}}$ a $[\psi ]$ -invariant proper free factor system. Then there is a

1. 1. a minimal $\mathcal F$ -forest $(\mathcal Y, \delta )$ with trivial arc stabilizers for which ${\mathcal {Z}}$ is elliptic; and

2. 2. a unique $\psi $ -equivariant expanding homothety $h \colon (\mathcal Y, \delta ) \to (\mathcal Y, \delta )$ .

The forest $(\mathcal Y, \delta )$ is degenerate if and only if $[\psi ]$ is polynomially growing rel. ${\mathcal {Z}}$ .

The constructed $\mathcal F$ -forest $(\mathcal Y, \delta )$ is the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}'$ , for some $[\psi ]$ -invariant proper free factor system ${\mathcal {Z}}'$ that supports ${\mathcal {Z}}$ (see Sections 2.1 and 2.4). Unfortunately, these limit forests depend on the choice of ${\mathcal {Z}}'$ ; our goal is to give a canonical contruction.

Given the central tool (hierarchies) and objective (universal limit trees), we outline again how these two fit together. Gaboriau–Levitt’s index theory [11] gives a uniform bound on the complexity of the point stabilizers system ${\mathcal {G}}[\mathcal Y]$ for a minimal $\mathcal F$ -forest $(\mathcal Y,\delta )$ with trivial arc stabilizers – this is a partial generalization of Bass–Serre theory. When $\mathcal Y$ is not degenerate, the subgroup system ${\mathcal {G}}[\mathcal Y]$ has strictly lower complexity than $\mathcal F$ . This allows us to induct on complexity (see Chapter 3).

Suppose the automorphism $\psi \colon \mathcal F \to \mathcal F$ has a nondegenerate limit forest $(\mathcal Y_1,\delta _1)$ with nontrivial point stabilizers; the system of stabilizers has strictly smaller complexity than $\mathcal F$ . By $\psi $ -equivariance of $\lambda _1$ -homothety $h_1\colon (\mathcal Y_1, \delta _1) \to (\mathcal Y_1, \delta _1)$ , the $\mathcal F$ -orbits of points with nontrivial stabilizers are permuted by $[h_1]$ , the subgroup system ${\mathcal {G}}$ is $[\psi ]$ -invariant, and the restriction of $\psi $ to ${\mathcal {G}}$ determines a unique outer automorphism $[\varphi ]$ of ${\mathcal {G}}$ .

Suppose $\varphi \colon {\mathcal {G}} \to {\mathcal {G}}$ has a nondegenerate limit forest $(\mathcal Y_2, \delta _2)$ with stretch factor $\lambda _2$ . Using the blow-up construction from [22], we equivariantly blow up $\mathcal Y_1$ with respect to $h_i \colon \mathcal Y_i \to \mathcal Y_i ~ (i = 1,2)$ to get real pretrees $\mathcal T$ with a minimal rigid $\mathcal F$ -action and a $\psi $ -equivariant ‘ $\mathcal F$ -expanding’ pretree-isomorphism $f\colon \mathcal T \to \mathcal T$ induced by $h_1$ and $h_2$ . In fact, the blow-up construction implies the $\mathcal F$ -pretrees $\mathcal T$ inherit an $\mathcal F$ -invariant $2$ -level hierarchy $(\delta _1, \delta _2)$ with full support and f is an expanding homothety with respect to this hierarchy. So we have a limit pseudoforest $(\mathcal T, (\delta _1,\delta _2))$ for $[\psi ]$ (see Section 3.1). Under what conditions can we construct an $\mathcal F$ -invariant convex metric on $\mathcal T$ from $(\delta _1, \delta _2)$ ? The heart of the paper is the following observation: the two limit forests $(\mathcal Y_i, \delta _i)$ are paired with attracting laminations $\mathcal L_i^+[\psi ]$ partially ordered by inclusion; an $\mathcal F$ -invariant convex metric on $\mathcal T$ can be naturally constructed from $(\delta _1, \delta _2)$ if $\mathcal L_2^+[\psi ]$ is not in $\mathcal L_1^+[\psi ]$ (see Section 3.4) or $\lambda _1 < \lambda _2$ (see Section 3.5)!

1.5 Bounded cancellation and laminations

Suppose a minimal $\mathcal F$ -forest $(\mathcal Y, \delta )$ is very small – that is, nontrivial arc stabilizers are maximal cyclic subgroups and the fixed point subset for a nontrivial elliptic element is an arc. Let $(\mathcal T, d)$ be a metric free splitting of $\mathcal F$ and $[\cdot ,\cdot ]_T$ (resp. $[\cdot , \cdot ]_Y$ ) denote the interval function for $\mathcal T$ (resp. $\mathcal Y$ ). A map $f \colon (\mathcal T,d) \to (\mathcal Y, \delta )$ is piecewise-linear (PL) if the restriction to any closed edge is a linear map; an equivariant PL-map exists if and only if $\mathcal T$ -elliptic elements in $\mathcal F$ are $\mathcal Y$ -elliptic. Equivariant PL-maps $(\mathcal T,d) \to (\mathcal Y, \delta )$ are surjective and Lipschitz since the isometric $\mathcal F$ -action on $(\mathcal Y, \delta )$ is minimal and there are only finitely many $\mathcal F$ -orbits of edges in $\mathcal T$ ; $1$ -Lipschitz maps are also known as metric maps. Generally, if $\mathcal T, \mathcal Y$ are free splittings of $\mathcal F$ , then an equivariant function $f \colon \mathcal T \to \mathcal Y$ is a (simplicial) PL-map if its restrictions to any closed edge is isotopic to a linear map with respect to some/any $\mathcal F$ -invariant convex metrics $d, \delta $ on $\mathcal T, \mathcal Y$ , respectively.

Lemma 1.3 (bounded cancellation).

Let $f\colon (\mathcal T,d) \to (\mathcal Y, \delta )$ be an equivariant PL-map. For some constant $C[f] \ge 0$ and all points $p, q \in \mathcal T$ , the image $f([p,q]_T)$ is in the $C[f]$ -neighborhood of the interval $[f(p), f(q)]_{Y}$ .

Such a $C[f]$ is a cancellation constant for f. This proof is due to Bestvina–Feighn–Handel.

Sketch of proof [2, Lemma 3.1].

Let $\operatorname {Lip}(f)$ be the Lipschitz constant and $\operatorname {vol}(\mathcal T, d)$ the volume $\pmod {\mathcal F}$ . Then $f = g \circ h$ for some equivariant $\operatorname {Lip}(f)$ -homothety h and metric PL-map g. Suppose f is simple: its target is a metric free splitting with free factor system $\mathcal F[\mathcal T]$ . Then g factors as finitely many equivariant edge collapses and Stallings folds followed by an equivariant metric homeomorphism. The homeomorphism and each edge collapse have cancellation constants 0. A fold has a cancellation constant given by the length of folded segment. Finally, the metric PL-map g has a cancellation constant since cancellation constants are (sub)additive over compositions of metric maps. As cancellation constants are preserved by precomposition with homeomorphisms, the PL-map $f = g \circ h$ has a cancellation constant $C[f] < \operatorname {Lip}(f)\operatorname {vol}(\mathcal T, d)$ .

Otherwise, the PL-map f is not simple. For a contradiction, suppose the image $f([p,q]_T)$ is not in the $\operatorname {Lip}(f)\operatorname {vol}(\mathcal T, d)$ -neighborhood of $[f(p), f(q)]_Y$ for some $p, q \in \mathcal T$ . Let $\delta (f(r_0), [f(p), f(q)]_Y)> \operatorname {Lip}(f)\operatorname {vol}(\mathcal T, d) + \epsilon _0$ for some $\epsilon _0> 0$ and point $r_0 \in [p,q]_T$ . For any $\epsilon> 0$ , the PL-map f is approximated by an equivariant simple PL-map $f_\epsilon $ with $\operatorname {Lip}(f_\epsilon ) < \operatorname {Lip}(f) + \epsilon $ and $C[f_\epsilon ] \ge \operatorname {Lip}(f)\operatorname {vol}(\mathcal T, d) + \epsilon _0$ (see [16, Theorem 6.1]). By the previous paragraph, $C[f_\epsilon ] < \operatorname {Lip}(f_\epsilon )\operatorname {vol}(\mathcal T, d)$ for $\epsilon> 0$ . So $C[f_\epsilon ] < \operatorname {Lip}(f)\operatorname {vol}(\mathcal T, d) +\epsilon _0$ for small enough $\epsilon> 0$ – a contradiction.

Remark. The results in this section apply to $\psi $ -equivariant PL-maps $g \colon (\mathcal T, d) \to (\mathcal T, d)$ for any automorphism $\psi \colon \mathcal F \to \mathcal F$ : view g as an equivariant PL-map $(\mathcal T, d) \to (\mathcal T\psi , d)$ instead.

A line in a forest is an arc that is isometric to $(\mathbb R, d_{\mathrm {std}})$ ; a ray in a forest is an arc that is isometric to $(\mathbb R_{\ge 0}, d_{\mathrm {std}})$ , and its origin is the point corresponding to $0$ under the isometry. Two rays are end-equivalent if their intersection is a ray; an end of a forest is an end-equivalence class of rays in the forest. Note that there is a natural bijection between the set of lines in a forest and set of unordered pairs of distinct ends in the same component of the forest. For simplicial $\mathcal F$ -pretrees $\mathcal T$ , the notions of line/ray/end are well-defined for the cone of $\mathcal F$ -invariant convex metrics on $\mathcal T$ .

Corollary 1.4 (cf. [10, Lemma 3.4]).

Let $f\colon (\mathcal T,d) \to (\mathcal Y, \delta )$ be an equivariant PL-map.

1. 1. For any ray $\rho $ in $(\mathcal T, d)$ with origin $p_0$ , the image $f(\rho )$ is either bounded or in the $C[f]$ -neighborhood of a unique ray $f_*(\rho ) \subset f(\rho )$ with origin $f(p_0)$ ; moverover, if $\rho , \rho '$ represent the same end e and $f(\rho )$ is unbounded, then so is $f(\rho ')$ and $f_*(\rho ), f_*(\rho ')$ are end-equivalent – denote their end-equivalence class by $f_*(e)$ .

2. 2. For any line l in $(\mathcal T, d)$ , $f(l)$ is in a $C[f]$ -neighborhood of a unique line $f_*(l) \subset f(l)$ if both ends of l have unbounded f-images.

3. 3. For any end $\epsilon $ of $(\mathcal Y, \delta )$ , there is a unique end $f^*(\epsilon )$ of $(\mathcal T, d)$ with $\epsilon = f_*(f^*(\epsilon ))$ .

Sketch of proof

(1): Let $\rho $ be a ray in $(\mathcal T,d)$ , $p_{0} \in \rho $ its origin, $f(\rho )$ unbounded, and $s_0 = f(p_0)$ . Use Figure 1 for reference. By bounded cancellation and the Lipschitz property, $f(\rho )$ has at most one end of $(\mathcal Y, \delta )$ . For some $n \ge 0$ , assume $s_{n} \in [s_0, f(p)]_Y$ for all $p \in \rho \setminus [p_0, p_{n}]_T$ . Set . Since $f(\rho )$ is unbounded,

$$\begin{align*}\delta(s_0, f(p_{n+1}))> 2\, \delta(s_0, s_{n}) + C\end{align*}$$

for some $p_{n+1} \in \rho \setminus [p_{0},p_{n}]_T$ . Pick $s_{n+1} \in [s_0, f(p_{n+1})]_Y$ satisfying $\delta (s_0, s_{n+1})> 2\,\delta (s_0, s_{n})$ and $\delta (s_{n+1},f(p_{n+1}))> C$ ; so $s_{n} \in [s_0, s_{n+1}]_Y$ . By bounded cancellation, the interval $[s_0,s_{n+1}]_Y \text{ in } f([p_0, p_{n+1}]_T)$ is disjoint from $f(\rho \setminus [p_0,p_{n+1}]_T)$ . So the union $\bigcup _{n \ge 0} [s_0, s_n]_Y$ is a ray $f_*(\rho )$ in $f(\rho )$ with origin $s_0$ . By construction, $f(\rho )$ is in the C-neighborhood of $f_*(\rho )$ . Any bounded neighborhood of a ray contains a unique ray, up to end-equivalence.

Figure 1 The ray $f_*(\rho )$ with origin $s_0 = f(p_0)$ is built inductively in the image $f(\rho )$ .

(2): Represent both ends of l with rays $\rho ^\pm \subset l$ sharing the same origin. By Part 1 and bounded cancellation, we have rays $f_*(\rho ^\pm )$ representing unique distinct ends $\epsilon ^\pm $ of $(\mathcal Y, \delta )$ ; moreover, $f(l)=f(\rho ^-)\cup f(\rho ^+)$ is in the C-neighborhood of $f_*(\rho ^-) \cup f_*(\rho ^+) \subset f(l)$ . Let $f_*(l) \subset f_*(\rho ^-) \cup f_*(\rho ^+)$ be the line determined by the ends $\epsilon ^\pm $ . Then $f(l)$ is in the C-neighborhood of $f_*(l)$ . Any bounded neighborhood of a line contains a unique line.

(3): The argument is almost the same. Let $\rho '$ be a ray representing $\epsilon $ and $s_0 = q_0$ its origin. Pick points $q_{n+1}, s_{n+1} \in \rho '$ with $\delta (s_0, s_{n+1})> 2\,\delta (s_0, s_{n})$ , $\delta (s_0, q_{n+1})> 2\, \delta (s_0, s_n) + C$ and $\delta (s_{n+1},q_{n+1})> C$ . Since $f\colon \mathcal T \to \mathcal Y$ is surjective, we can lift $q_n$ to $p_n \in T$ . By bounded cancellation and K-Lipschitz property, the distance $d(p_0, [p_n, p_{n+m}]_T)> \frac {1}{K} \delta (s_0,s_n)$ . Thus, $(p_n)_{n\ge 0}$ determines an end e of $(\mathcal T,d)$ with unbounded f-image. Let $\rho $ be a ray representing e with origin $p_0$ . As $\rho ' \subset f(\rho )$ by construction, we get $f_*(\rho ) = \rho '$ by Part 1. By Part 2, the end e is the unique end with $f_*(e) = \epsilon $ , and we denote it by  $f^*(\epsilon )$ .

The corollary defines the equivariant lifting (resp. projecting) function $f^*$ (resp. $f_*$ ), where the domain $\operatorname {dom}(f^*)$ of $f^*$ is the set of lines in $(\mathcal Y, \delta )$ and the domain $\operatorname {dom}(f_*)$ of $f_*$ is the set of lines in $(\mathcal T, d)$ whose ends both have unbounded f-images. Note that the image $\operatorname {im}(f^*)$ is $\operatorname {dom}(f_*)$ ; moreover, $f^*$ and $f_*$ are inverses of each other. Both lifting and projecting functions are canonical: $f^* = g^*$ and $f_* = g_*$ for any equivariant maps $f,g \colon (\mathcal T, d) \to (\mathcal Y, \delta )$ since $f,g$ will be a bounded $\delta $ -distance from some equivariant PL-map; for lack of better notation, we still denote the functions by $f^*, f_*$ despite this independence.

Alternatively, we view $f^*$ and $f_*$ as functions on the sets of $\mathcal F$ -orbits of lines. We can equip these sets with a natural topology. The set $\mathbb R(\mathcal Y, \delta )$ of $\mathcal F$ -orbits of lines in $(\mathcal Y, \delta )$ has the following topology: for any $p,q \in \mathcal Y$ , let $U[p,q]$ be the $\mathcal F$ -orbit of lines that contain a translate of $[p,q]$ ; the collection $\{ U[p,q]:p,q \in \mathcal Y \}$ is a basis for the space of ( $\underline{\mathcal{F}}$ -orbits of) lines. This space is well-defined for the equivariant homothetic class of $(\mathcal Y, \delta )$ . The space of lines is also well-defined for the free splitting $\mathcal T$ and denoted $\mathbb R(\mathcal T)$ .

Claim 1.5. The canonical lifting function $f^* \colon \mathbb R(\mathcal Y, \delta ) \to \mathbb R(\mathcal T)$ is a topological embedding.

Henceforth, we identify $\mathbb R(\mathcal Y, \delta )$ with a subspace of $\mathbb R(\mathcal T)$ using the canonical embedding $f^*$ .

Sketch of proof.

We first prove the injection $f^*$ is continuous. Let $\Lambda \subset \mathbb R(\mathcal T)$ be a closed subset and . Suppose $[\gamma ]$ is in the closure of $f_*(\Lambda _f)$ in $\mathbb R(\mathcal Y,\delta )$ . For continuity, it is enough to show $f^*[\gamma ] \in \Lambda $ . Fix a long interval $I_\gamma \subset \gamma $ ; then $I_\gamma \subset [f(p), f(q)]_Y$ for some $p,q \in f^*(\gamma )$ . As $[\gamma ]$ is in the closure of $f_*(\Lambda _f)$ , the interval $I_\gamma \subset \gamma $ is in the line $f_*(l)$ for some $[l] \in \Lambda _f$ . By bounded cancellation, the f-image of the intersection contains a long interval in $I_\gamma $ . As the interval $I_\gamma $ will exhaust $\gamma $ , the interval $I_l$ exhausts $f^*(\gamma )$ ; in particular, any interval in $f^*(\gamma )$ is contained in l for some $[l] \in \Lambda $ . So $f^*[\gamma ]$ is in the closed subset $\Lambda $ .

We finally prove $f^*\colon \mathbb R(\mathcal Y, \delta ) \to \operatorname {im}(f^*)$ is an open map, where the image $\operatorname {im}(f^*) \subset \mathbb R(\mathcal T)$ has the subspace topology. Suppose $p,q\in \mathcal Y$ and $[\gamma ] \in U[p,q]$ (i.e., a line $\gamma $ in $(\mathcal Y, \delta )$ contains an interval $[p,q]_Y$ ). There is an interval $[u,v]_T \subset f^*(\gamma )$ whose f-image covers a long neighborhood of $[p,q]_Y$ . By bounded cancellation, any line $f^*(\gamma ')$ containing $[u,v]_T$ will have an $f_*$ -image $\gamma '$ containing $[p,q]_Y$ . So $f^*[\gamma ] \in U[u,v] \cap \operatorname {im}(f^*) \subset f^*(U[p,q])$ . As $[\gamma ] \in U[p,q]$ was arbitrary, the image $f^*(U[p,q])$ is open in $\operatorname {im}(f^*)$ .

Now assume $\mathcal T'$ is a free splitting of $\mathcal F$ with $\mathcal F[\mathcal T] = \mathcal F[\mathcal T']$ and let $f\colon \mathcal T \to \mathcal T'$ be an equivariant PL-map. The folds in the factorization of f never identify points in the same $\mathcal F$ -orbit. For $[l] \in \mathbb R(\mathcal T)$ , each end of l has unbounded f-image (i.e., $\operatorname {dom}(f_*) = \mathbb R(\mathcal T)$ ); so $f_* \colon \mathbb R(\mathcal T) \to \mathbb R(\mathcal T')$ is a canonical homeomorphism (with inverse $f^*$ ). Similarly, if $g \colon \mathcal T \to \mathcal T$ is a $\psi $ -equivariant PL-map for some automorphism $\psi \colon \mathcal F \to \mathcal F$ , then $g_*, g^* \colon \mathbb R(\mathcal T) \to \mathbb R(\mathcal T)$ are canonical homeomorphisms for $[\psi ]$ .

Remark. We use ambiguous terminology and say ‘line’ when we mean a line or an $\mathcal F$ -orbit of a line; our notation remains distinct: ‘l’ is always a line, while ‘ $[l]$ ’ is its $\mathcal F$ -orbit.

A lamination in $(\mathcal Y, \delta )$ (resp. $\mathcal T$ ) is a nonempty closed subset of $\mathbb R(\mathcal Y, \delta )$ (resp. $\mathbb R(\mathcal T)$ ); when the $\mathcal F$ -forest in question is clear, we say lamination with no qualifier. An element of a lamination is called a leaf; a leaf segment of a lamination $\Lambda $ is a nondegenerate closed interval in a line representing a leaf of $\Lambda $ . A lamination is minimal if each leaf is dense in the lamination; a lamination is perfect if it has no isolated leaves.

Let $[l]$ be a line and $\Lambda $ a lamination in $\mathbb R(\mathcal Y, \delta )$ (or $\mathbb R(\mathcal T)$ ). A sequence $[l_m]_{m \ge 0}$ in the space of lines weakly limits to $[l]$ if some subsequence converges to $[l]$ ; we say $[l]$ is a weak limit of the sequence. The sequence $[l_m]_{m \ge 0}$ weakly limits to $\Lambda $ if it weakly limits to every leaf of $\Lambda $ . The ‘weak’ terminology is used to highlight that the space of lines is not Hausdorff – a convergent sequence may have multiple limits!

More generally, a sequence $[p_m, q_m]_{m \ge 0}$ of intervals converges to $[l]$ if, for any closed interval $[a,b] \subset l$ , $[p_m, q_m]$ contains a translate of $[a,b]$ for $m \gg 1$ (i.e., for large enough m) – precisely, there is an $M[a,b] \ge 1$ such that $U[a, b]$ contains $U[p_m,q_m]$ for $m \ge M[a,b]$ . Again, a sequence of intervals weakly limits to $[l]$ if some subsequence converges to $[l]$ , and it weakly limits to $\Lambda $ if it weakly limits to every leaf of $\Lambda $ .

2 Limit forests

In this chapter, we sketch the proof of Proposition 1.2 (existence of limit forests) and, in the process, introduce stable laminations. The first half deals with limit forests for expanding irreducible train tracks; then, in the second half, we extend the results to all limit forests.

2.1 Constructing limit forests (1)

This is a summary of [12, Appendix]; the reader is invited to read that appendix.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ . Set and let $d_\tau $ be the eigenmetric on $\mathcal T$ for $[\tau ]$ . For $m \ge 0$ , let $d_m$ be the pullback of $\lambda ^{-m}d_\tau $ via $\tau ^m$ :

By definition, the pullback $d_m$ is an $\mathcal F$ -invariant (not necessarily convex) pseudometric on $\mathcal T$ whose quotient metric space is equivariantly isometric to $(\mathcal T\psi ^m, \lambda ^{-m}d_\tau )$ . The $\lambda $ -Lipschitz property for $\tau $ with respect to $d_\tau $ implies the sequence of pseudometrics $d_m$ is (pointwise) monotone decreasing. The limit $d_\infty $ is an $\mathcal F$ -invariant pseudometric on $\mathcal T$ , the quotient metric space $(\mathcal T_\infty , d_\infty )$ is an $\mathcal F$ -forest, and the $\psi $ -equivariant $\lambda $ -Lipschitz train track $\tau $ induces a $\psi $ -equivariant $\lambda $ -homothety $h \colon (\mathcal T_\infty , d_\infty ) \to (\mathcal T_\infty , d_\infty )$ ; in particular, the equivariant metric surjection $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal T_\infty , d_\infty )$ semiconjugates $\tau $ to h: $\pi \circ \tau = h \circ \pi $ .

As $\tau $ is a train track, the restriction of $\pi $ to any edge of $\mathcal T$ is an isometric embedding. So $\mathcal T_\infty $ is not degenerate. In fact, the $\pi $ -image of any edge of $\mathcal T$ can be extended to an axis for a $\mathcal T_\infty $ -loxodromic element in $\mathcal F$ . Thus, the $\mathcal F$ -forest $(\mathcal T_\infty , d_\infty )$ is minimal, and the uniqueness of h follows from [6, Theorem 3.7]. Finally, the minimal $\mathcal F$ -forest $(\mathcal T_\infty , d_\infty )$ has trivial arc stabilizers. This sketches the first case of Proposition 1.2. The $\mathcal F$ -forest is the (forward) limit forest for $[\tau ]$ .

2.2 Stable laminations (1)

The first part of this section is mostly adapted from Section 1 of [2]. The following definition of stable laminations is from [2, Definition 1.3].

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ . Set , let $d_\tau $ be the eigenmetric on $\mathcal T$ for $[\tau ]$ , and pick an edge $e \subset \mathcal T$ . Expanding irreducibility implies the interval $\tau ^k(e)$ contains at least three translates of e for some $k \ge 1$ . By the intermediate value theorem, $\tau ^k(p) = x \cdot p$ for some $x \in \mathcal F$ and $p \in e$ . Recall that edges are open intervals; since the restriction of $x^{-1} \cdot \tau ^k$ to the edge e is an expanding $\lambda ^k$ -homothetic embedding $e \to \mathcal T$ (with respect to $d_\tau $ ) that fixes p and has e in its image, we can extend e to a line $l_p \subset \mathcal T$ by iterating $x^{-1} \cdot \tau ^k$ . By construction, the restriction of $x^{-1} \cdot \tau ^k$ to $l_p$ is a $\lambda ^k$ -homothety $l_p \to l_p$ with respect to the eigenmetric $d_\tau $ for $[\tau ]$ ; the $\mathcal F$ -orbit $[l_p]$ is an eigenline of $[\tau ^k]$ based at $[p]$ (in $\mathcal F \backslash \mathcal T$ ). A $\underline{\mathrm{stable}\ \mathcal{T}\mathrm{-lamination}}$ $\Lambda ^+$ for $[\tau ]$ is the closure of an eigenline of $[\tau ^k]$ for some $k \ge 1$ . By $\phi $ -equivariance of $\tau $ , the restriction of $\tau $ to l representing a leaf of a stable lamination $\Lambda ^+$ is a $\lambda $ -homothetic embedding. In fact, $[\tau ]$ maps eigenlines to eigenlines, and the image is also a stable lamination for $[\tau ]$ .

As the transition matrix $A[\tau ]$ is irreducible, it is a block transitive permutation matrix, and the ‘first return’ matrix for each block is primitive (i.e., has a positive power). There is a bijective correspondence between the stable laminations for $[\tau ]$ and the blocks of $A[\tau ]$ . In particular, there are finitely many stable laminations for $[\tau ]$ , these laminations are pairwise disjoint, and $\tau _*$ transitively permutes them [2, Lemma 1.2]. By finiteness, their union $\mathcal L^+[\tau ]$ is a lamination and is called the system of stable laminations for $[\tau ]$ .

2.2.1 Quasiperiodic lines

A line $[l]$ in an $\mathcal F$ -forest is periodic if it is the axis for the conjugacy class of some loxodromic element of $\mathcal F$ . A line $[l]$ is quasiperiodic in an $\mathcal F$ -forest if any closed interval I in l has an assigned number $L(I) \ge 0$ such that any interval in l of length $L(I)$ contains a translate of I; periodic lines are quasiperiodic. If $[l]$ is a quasiperiodic line, then any leaf of its closure $\Lambda $ is quasiperiodic and hence dense in $\Lambda $ (exercise) (i.e., $\Lambda $ is minimal). If $[l]$ is quasiperiodic but not periodic, then no leaf of its closure $\Lambda $ is isolated (exercise) (i.e., $\Lambda $ is also perfect).

Remark. When the F-action on a free splitting T is free, then our definition of quasiperiodicity is equivalent to [2, Definition 1.7]; however, our definition is weaker when the action is not free.

Lemma 2.1 (cf. [2, Proposition 1.8]).

The eigenlines of $[\tau ^k]$ are quasiperiodic but not periodic for $k \ge 1$ . Thus, the stable laminations for $[\tau ]$ are minimal and perfect.

Proof. There is a length $L_0$ such that any interval in $\mathcal T$ of length $L_0$ contains an edge. Fix an $\mathcal F$ -orbit $[I]$ of intervals in an eigenline $[l]$ of $[\tau ^k]$ . By construction, I is contained in $\tau ^{km}(e)$ for some edge e in $\mathcal T$ and integer $m \ge 0$ . As the blocks in $A[\tau ^k]$ are primitive, there is an integer $m' \ge 1$ such that $\tau ^{km'}(e')$ contains a translate of e for any edge $e'$ in l. Altogether, any interval in l of length $\lambda [\tau ]^{k(m+m')} L_0$ contains a translate of I. This proves quasiperiodicity.

Now assume, for a contradiction, that the eigenline $[l]$ was periodic (i.e., l is an axis for a $\mathcal T$ -loxodromic element $x \in \mathcal F$ ). By construction, the $\mathcal F$ -orbit $[l]$ is $\tau ^k$ -invariant, and hence, the cyclic subgroup $\langle x \rangle $ is $[\psi ^k]$ -invariant. So x is $[\psi ]$ -periodic as $\psi $ is an automorphism; yet x must $[\psi ]$ -grow exponentially since its axis is an eigenline and $\tau $ is expanding.

Fix an equivariant PL-map $f \colon (\mathcal T, d) \to (\mathcal Y, \delta )$ and canonically embed $\mathbb R(\mathcal Y, \delta )$ into $\mathbb R(\mathcal T)$ via $f^*$ (Claim 1.5). If a quasiperiodic line $[l] \in \mathbb R(\mathcal T)$ is in the subspace $\mathbb R(\mathcal Y, \delta ) = \operatorname {im}(f^*)$ , then so its closure $\Lambda $ in $\mathbb R(\mathcal T)$ (exercise). Returning to limit forests, the equivariant metric PL-map $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ restricts to an isometric embedding on the leaves of $\mathcal L^+[\tau ]$ ; therefore, the stable lamination $\mathcal L^+[\tau ]$ is in $\mathbb R(\mathcal Y_\tau , d_\infty ) \subset \mathbb R(\mathcal T)$ .

2.2.2 Characterizing loxodromics

Let $(\mathcal Y_\tau , d_\infty )$ be the limit forest for $[\tau ]$ , $h: (\mathcal Y_\tau , d_\infty ) \to (\mathcal Y_\tau , d_\infty )$ the unique $\psi $ -equivariant $\lambda $ -homothety, and $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ the constructed equivariant metric PL-map. By Lemma 1.3, the map $\tau \colon (\mathcal T, d_\tau ) \to (\mathcal T, d_\tau )$ has a cancellation constant . Set and denote the interval functions for $\mathcal T$ by $[\cdot , \cdot ]$ . The equivariant metric maps $\tau ^m\colon (\mathcal T, d_\tau ) \to (\mathcal T\psi ^m, \lambda ^{-m}d_\tau )$ have cancellation constants $\sum _{i=1}^m \lambda ^{-i}C \le C'$ ; so their limit $\pi $ has a cancellation constant .

Let $P \subset \mathcal Y_\tau $ be $\mathcal F$ -orbit representatives of points with nontrivial stabilizers. Define the subgroup system , where is the stabilizer in $\mathcal F$ of $p \in P$ . As the action on $\mathcal Y_\tau $ has trivial arc stabilizers, the system ${\mathcal {G}}[\mathcal Y_\tau ]$ is malnormal: each component is malnormal (as a subgroup of the appropriate component of $\mathcal F$ ) and conjugates of distinct components (in the same component of $\mathcal F$ ) have trivial intersections. The $\psi $ -equivariance of homothety h implies ${\mathcal {G}}[\mathcal Y_\tau ]$ is $[\psi ]$ -invariant. By Gaboriau–Levitt’s index theory, the complexity of ${\mathcal {G}}[\mathcal Y_\tau ]$ is strictly less than that of $\mathcal F$ [11, Theorem III.2]. In particular, ${\mathcal {G}}[\mathcal Y_\tau ]$ has finite type: P is finite, and each component $G_p$ is finitely generated. The restriction of $\psi $ to ${\mathcal {G}}[\mathcal Y_\tau ]$ determines a unique outer automorphism of the system.

We now characterize the elliptic/loxodromic elements in $\mathcal F$ in the limit forest $(\mathcal Y_\tau , d_\infty )$ :

Proposition 2.2 (cf. [2, Proposition 1.6]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ , and $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ .

If $x \in \mathcal F$ is a $\mathcal T$ -loxodromic element, then the following statements are equivalent:

1. 1. the element x is $\mathcal Y_\tau $ -loxodromic;

2. 2. the element $x [\psi ]$ -grows exponentially rel. $\mathcal T$ : $\underset {m \to \infty } \lim \, \frac {1}{m} \log \| \psi ^m(x)\|_{\mathcal T} = \log \lambda [\tau ]$ ; and

3. 3. the $\mathcal T$ -axis for $\psi ^m(x)$ has an arbitrarily long leaf segment of $\mathcal L^+[\tau ]$ for $m \gg 1$ .

The restriction of $\psi $ to the $[\psi ]$ -invariant subgroup system ${\mathcal {G}}[\mathcal Y_\tau ]$ of $\mathcal Y_\tau $ -point stabilizers has constant growth rel. $\mathcal T$ : $\{ \| \psi ^m(x)\|_{\mathcal T} : m \ge 0 \}$ is bounded for all $x \in {\mathcal {G}}[\mathcal Y_\tau ]$ .

Proof. Let , be a cancellation constant for $\tau \colon (\mathcal T,d_\tau ) \to (\mathcal T,d_\tau )$ , and a cancellation constant for $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ . Fix a line l in $\mathcal T$ , and let $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ be the constructed equivariant metric PL-map.

Case 1: $d_\infty (\pi (p), \pi (q))> 2C' + L$ for some $k \ge 0$ , $p,q \in \tau _*^k(l)$ , and $L> 0$ . By definition of $d_\infty $ (construction of $\pi $ ), $d_\tau (\tau ^m(p),\tau ^m(q))> \lambda ^m(2C' + L)$ for $m \gg 1$ . Pick $m \gg 1$ and $r_m, s_m \text{ in } [\tau ^m(p), \tau ^m(q)]$ so that $d_\tau (\tau ^m(p), r_m), d_\tau (s_m, \tau ^m(q))> \lambda ^m C'$ and $d_\tau (r_m, s_m)> \lambda ^m L$ . By bounded cancellation (for $\tau ^m$ ), the interval is disjoint from $\tau ^m(\tau _*^k(l) \setminus [p,q])$ in $(T, d_\tau )$ . So $I_m$ is an interval in $\tau _*^{m+k}(l)$ .

Let be the number of vertices in the interval $(p,q)$ . Then $I_m$ is covered by $N+1$ leaf segments (of $\mathcal L^+[\tau ]$ ) as $\tau $ is a train track. By the pigeonhole principle, $I_m$ (and hence $\tau _*^{m+k}(l)$ ) contains a leaf segment with $d_\tau $ -length $> \frac {\lambda ^m L}{N+1}$ ; therefore, the line $\tau _*^n(l)$ in $\mathcal T$ contains arbitrarily long leaf segments for $m \gg 1$ .

Case 2: $\pi (\tau _*^m(l))$ has diameter $\le 2C'$ for all $m \ge 0$ . We claim that any leaf segment in the line $\tau _*^m(l)~(m\ge 0)$ has $d_\tau $ -length $ \le 2C'$ . For the contrapositive, suppose some $\tau _*^m(l)$ has a leaf segment with $d_\tau $ -length $L> 2C'$ . By the train track property and bounded cancellation, $\tau _*^{m+1}(l)$ has a leaf segment with $d_\tau $ -length $\ge \lambda L - 2C> L$ . By induction, $\pi (\tau _*^{m+m'}(l))$ has diameter $\ge \lambda ^{m'} (L - 2C')$ for $m' \ge 0$ and $\lambda ^{m'} (L - 2C')> 2C'$ for $m' \gg 1$ .

We finally return to the proof of the proposition. Fix a $\mathcal T$ -loxodromic element $x \in \mathcal F$ and let $l \subset \mathcal T$ be its axis; in particular, $\pi (l)$ is x-invariant by equivariance of $\pi $ . As $\tau $ is $\lambda $ -Lipschitz with respect to $d_\tau $ , $\underset {m \to \infty } \limsup \, \frac {1}{m} \log \| \psi ^m(x)\|_{d_\tau } \le \log \lambda $ .

Case–i: $d_\infty (\pi (p), \pi (q))> 2C'$ for some $k \ge 0$ and $p,q \in \tau _*^k(l)$ . The line $\tau _*^m(l)$ , the axis for $\phi ^m(x)$ in $\mathcal T$ , contains an arbitrarily long leaf segment for $m \gg 1$ by the Case 1 analysis. By bounded cancellation (for $\pi $ ), some nondegenerate interval I in $[\pi (p), \pi (q)]_{\infty }$ is disjoint from $\pi (\tau _*^k(l) \setminus [p,q])$ . Since $\tau _*^k(l)$ is the axis for $\psi ^k(x)$ , it contains disjoint translates $[p,q]$ , $\psi ^k(x^{-n}) \cdot [p,q]$ , $\psi ^k(x^{n}) \cdot [p,q]$ for some $n \gg 1$ . Then $\psi ^k(x^{-n}) \cdot I $ and $\psi ^k(x^{n}) \cdot I$ are in distinct components of $\mathcal Y_\tau \setminus I$ and $\psi ^k(x)$ is $\mathcal Y_\tau $ -loxodromic. Since $\|\cdot \|_{d_\infty } \le \| \cdot \|_{d_\tau }$ and $\|\psi (\cdot )\|_{d_\infty } = \lambda \|\cdot \|_{d_\infty }$ , we get $ \log \lambda \le \underset {m \to \infty } \liminf \, \frac {1}{m} \log \| \psi ^m(x)\|_{d_\tau }$ and x is $\mathcal Y_\tau $ -loxodromic. Finally, $\log \lambda = \underset {m \to \infty } \lim \, \frac {1}{m} \log \| \psi ^m(x)\|_{\mathcal T}$ since $d_\tau $ and the combinatorial metric on $\mathcal T$ are bilipschitz.

Case–ii: $\pi (\tau _*^m(l))$ has diameter $\le 2C'$ for all $m \ge 0$ . Any leaf segment in $\tau _*^m(l)~(m \ge 0)$ has $d_\tau $ -length $\le 2C'$ by Case 2 analysis. Let N be the number of vertices in a fundamental domain of x acting on l. By the train track property, the fundamental domain of $\tau _*^m(l)$ is covered by $N+1$ leaf segments and $\|\psi ^m(x)\|_{\mathcal T} \le K \|\psi ^m(x)\|_{d_\tau } \le 2C'K(N+1)$ for some $K \ge 1$ and all $m \ge 0$ . But x acts on $\mathcal Y_\tau $ by an isometry, and $\pi (l) \subset \mathcal Y_\tau $ is x-invariant; so x must be $\mathcal Y_\tau $ -elliptic.

We now introduce the notion of factored forests. Suppose the stable laminations $\mathcal L^+[\tau ]$ have components $\Lambda _i^+~(1 \le i \le k)$ . The $\mathcal F$ -orbits of edges in $\mathcal T$ can be partitioned into blocks $\mathcal B_i$ indexed by the components $\Lambda _i^+ \subset \mathcal L^+[\tau ]$ . For $p,q \in \mathcal T$ , let $d_\tau ^{(i)}$ be the $d_\tau $ -length of the intersection of the interval $[p,q]$ and the subforest spanned by $\mathcal B_i$ ; this defines an $\mathcal F$ -invariant convex pseudometric $d_\tau ^{(i)}$ on $\mathcal T$ . The metric $d_\tau $ is a sum of the pseudometrics $d_\tau ^{(i)}$ , denoted $\Sigma _{i=1}^k d_\tau ^{(i)}$ ; we call $\Sigma _{i=1}^k d_\tau ^{(i)}$ a factored $\mathcal F$ -invariant convex metric and $(\mathcal T, \Sigma _{i=1}^k d_\tau ^{(i)})$ a factored $\mathcal F$ -forest. This factored metric is special: the factors $d_\tau ^{(i)}~(1 \le i \le k)$ are pairwise mutually singular: for $i \neq j$ , there are intervals (e.g., the leaf segments) with positive $d_\tau ^{(i)}$ -length and 0 $d_\tau ^{(j)}$ -length – such factored pseudometrics will be denoted by $\oplus _{i=1}^k d_\tau ^{(i)}$ to invoke the idea of independence in direct sums. The limit pseudometrics $d_\infty ^{(i)}$ are pairwise mutually singular since $\pi $ is surjective and isometric on leaf segments; thus, $d_\infty = \oplus _{i=1}^k d_\infty ^{(i)}$ . The next lemma is the cornerstone of our universality result:

Lemma 4.3 (cf. [2, Lemma 3.4]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ with eigenmetric $d_\tau $ , $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ , and .

If $(\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ is an equivariant PL-map and the k-component lamination $\mathcal L^+[\tau ]$ is in $\mathbb R(\mathcal Y, \delta ) \subset \mathbb R(\mathcal T)$ , then the sequence $(\mathcal Y \psi ^{mk}, \lambda ^{-mk} \delta )_{m \ge 0}$ converges to $(\mathcal Y_\tau , \oplus _{i=1}^k c_i \, d_\infty ^{(i)})$ , where $d_\infty = \oplus _{i=1}^k \, d_\infty ^{(i)}$ and $c_i> 0$ .

Remark. Factored $\mathcal F$ -forests are needed for this lemma when $k \ge 2$ ; moreover, the sequence $(\mathcal Y \psi ^m, \lambda ^{-m}\delta )_{m \ge 0}$ will not converge in general (but is asymptotically periodic) when $k \ge 2$ . Convergence is in the subspace of translation distance functions in $\mathbb R_{\ge 0}^{\mathcal F}$ with the product topology.

We give the proof in Section 4.1. In particular, if $\tau '\colon \mathcal T' \to \mathcal T'$ is another expanding irreducible train track for $\psi $ and $\mathcal F[\mathcal T'] = \mathcal F[\mathcal T]$ , then the limit forest for $[\tau ']$ is equivariantly homothetic to $(\mathcal Y_\tau , d_\infty )$ – set , apply the lemma, then observe that the sequence $(c_i)_{i=1}^k$ must be constant in this case. A minimal very small $\mathcal F$ -forest $(\mathcal Y, \delta )$ is an expanding forest for $[\psi ]$ like $\mathcal Y_\tau $ if it is nondegenerate and there is

1. 1. a $\psi $ -equivariant expanding homothety $(\mathcal Y, \delta ) \to (\mathcal Y, \delta )$ ; and

2. 2. an equivariant PL-map $(\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ .

Corollary 2.3. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ , and $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ . Any expanding forest for $[\psi ]$ like $\mathcal Y_\tau $ is uniquely equivariantly homothetic to $(\mathcal Y_\tau , d_\infty )$ .

Proof. Let $(\mathcal Y, \delta )$ be an expanding forest for $[\psi ]$ like $\mathcal Y_\tau $ , $f \colon (\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ an equivariant PL-map with cancellation constant , $g \colon (\mathcal Y, \delta ) \to (\mathcal Y, \delta )$ the $\psi $ -equivariant expanding s-homothety, $x \in \mathcal F$ a $\mathcal Y$ -loxodromic element. By equivariance of f, the element x is $\mathcal T$ -loxodromic with axis $l_x \subset \mathcal T$ . Let $[p_0, x \cdot p_0] \subset l_x$ be (the closure of) a fundamental domain of x acting on $l_x$ . The interval $[p_0,x \cdot p_0]$ is a concatenation of $N \ge 1$ leaf segments (of $\mathcal L^+[\tau ]$ ). Choose $m \gg 1$ so that $\| \psi ^m(x) \|_{\delta } = s^m \| x \|_\delta> 2C N$ . Note that the action of $\psi ^m(x)$ on its axis has a fundamental domain $[p_m, \psi ^m(x) \cdot p_m]$ covered by N leaf segments as $\tau $ is a train track. So $\delta (f(p_m), f(\psi ^m(x) \cdot p_m))> 2C N$ and, by the pigeonhole principle, $[p_m, \psi ^m(x) \cdot p_m]$ contains a leaf segment $[q, r]$ with $\delta (f(q), f(r))> 2C$ .

Let $l \supset [q,r]$ represent some leaf $[l] \in \mathcal L^+[\tau ]$ . Bounded cancellation implies the components of $l \setminus [q,r]$ have f-images with disjoint closures. By quasiperiodicity of $[l]$ and equivariance of f, both ends of l have unbounded f-image ( i.e., $[l] \in \operatorname {dom}(f_*) = \mathbb R(\mathcal Y, \delta )$ ) (Corollary 1.4, Claim 1.5). Finally, the closure of $[l]$ in $\mathbb R(\mathcal T)$ , a component $\Lambda _i^+ \subset \mathcal L^+[\tau ]$ , is a subset of $\mathbb R(\mathcal Y, \delta )$ by quasiperiodicity of $[l]$ . Note that the $\psi $ -equivariant homothety g induces a homeomorphism $g_* \colon \mathbb R(\mathcal Y, \delta ) \to \mathbb R(\mathcal Y, \delta )$ that is the restriction of the homeomorphism $\tau _* \colon \mathbb R(\mathcal T) \to \mathbb R(\mathcal T)$ . So $\mathcal L^+[\tau ] \subset \mathbb R(\mathcal Y, \delta )$ since $\tau _*$ acts transitively on the k components of $\mathcal L^+[\tau ]$ . Set ; by Lemma 4.3, the sequence $(\mathcal Y\psi ^{mk}, \lambda ^{-mk}\delta )_{m \ge 0}$ converges to the factored $\mathcal F$ -forest $(\mathcal Y_\tau , \oplus _{i=1}^k c_i \, d_\infty ^{(i)})$ for some $c_i> 0$ . Yet $(\mathcal Y, \delta )$ is equivariantly isometric to $(\mathcal Y\psi , s^{-1} \delta )$ ; thus, $s = \lambda $ , $c_i = c_{i+1}~(i<k)$ , $(\mathcal Y, \delta )$ is equivariantly isometric to $(\mathcal Y_\tau , c_1 \, d_\infty )$ , and the equivariant isometry is unique [6, Theorem 3.7].

2.2.3 Iterated turns

We have already shown how iterating an edge in $\mathcal T$ by the train track $\tau $ produces the system of stable laminations $\mathcal L^+[\tau ]$ . Later, we will consider how $\mathcal L^+[\tau ]$ determines laminations in (a free splitting of) the subgroup system ${\mathcal {G}}[\mathcal Y_\tau ]$ .

Let $\mathcal T'$ be a free splitting of $\mathcal F$ whose free factor system $\mathcal F[\mathcal T']$ is trivial. Then there is an equivariant PL-map $f \colon (\mathcal T', d') \to (\mathcal T, d_\tau )$ . Let $\gamma $ be a line in $(\mathcal Y_\tau , d_\infty )$ , $\pi ^*(\gamma )$ its lift to $(\mathcal T, d_\tau )$ , and $f^*(\pi ^*(\gamma ))$ its lift to $(\mathcal T', d')$ . Denote the ends of $\gamma $ by $\varepsilon _i~(i = 1,2)$ . Let $T \subset \mathcal T$ be the component containing $\pi ^*(\gamma )$ , and $T' \subset \mathcal T'$ , $Y_\tau \subset \mathcal Y_\tau $ , and $F \subset \mathcal F$ be the matching components. Denote the first return maps of $\tau $ , h and $\psi $ on T, $Y_\tau $ and F by $\tilde \tau, \tilde h$ and $\varphi $ respectively. For the rest of the section, redefine $\lambda $ to be the stretch factor of the expanding homothety $\tilde h$ .

Suppose $\circ $ is a point on the line $\gamma $ with a nontrivial stabilizers . Let $d_i~(i = 1,2)$ be the direction at $\circ $ containing $\varepsilon _i$ . By Gaboriau–Levitt index theory, $\tilde h^j(\circ ) = y \cdot \circ $ and $\tilde h^j(d_i) = ys_i \cdot d_i$ for some $y \in F$ , $s_i \in G_\circ $ , and minimal $j \ge 1$ . Since F acts on $Y_\tau $ with trivial arc stabilizers, the elements $ys_1, ys_2$ are unique and $s_1^{-1}s_2 \in G_\circ $ is independent of the chosen $y \in F$ .

Set to be the trivial element and for $m \ge 0$ . Let $T'(G)$ be the characteristic convex subset of $T'$ for a nontrivial subgroup $G \le F$ . Since $T'$ is simplicial, the characteristic convex subset $T'(G)$ is closed, and we have the closest point retraction $T' \to T'(G)$ ; it extends uniquely to the ends-completions. Let $q_{i,m}'$ be the closest point projection of $f^*(\pi ^*(\tilde h_*^{mj}(\varepsilon _i)))$ to $T'(\varphi ^{mj}(G_\circ ))$ . Set and to be $\psi _\circ $ -equivariant maps for an automorphism $\psi _\circ \colon F \to F$ in the outer class $[\varphi ^j]$ . As $h_\circ $ fixes $\circ $ , we get $\psi _\circ (G_\circ ) = G_\circ $ and $y_{m}^{-1} \cdot T'(\varphi ^{mj}(G_\circ ))$ is the characteristic convex subset for $\psi _\circ ^{m}(G_\circ ) = G_\circ $ . Thus, is in $T'(G_\circ )$ for $i = 1,2$ and $m \ge 0$ . The interval $[q_{1,m}, q_{2,m}]$ in $T'(G_\circ )$ (i.e., the closest point projection of $f^*(\pi ^*(h_\circ ^m(\gamma )))$ ) is the turn in $f^*(\pi ^*(h_\circ ^m(\gamma )))$ about $T'(G_\circ )$ .

Since $h_\circ (d_1) = d_1$ , the ends $h_{\circ *}^m(\varepsilon _1)~(m \ge 0)$ are in fact ends of $d_1$ . If $h_{\circ *}(\varepsilon _1) = \varepsilon _1$ , then the sequence $(q_{1,m})_{m \ge 0}$ is constant. Otherwise, the ends $h_{\circ *}^m(\varepsilon _1)$ are distinct for $m \ge 0$ . Let $\gamma _{1,m}$ be the line in $d_1$ determined by $h_{\circ *}^{m+1}(\varepsilon _1)$ and $h_{\circ *}^m(\varepsilon _1)$ . As $h_\circ $ is an expanding homothety, the distance $d_\infty (\circ , \gamma _{1,m})> 0$ from $\circ $ to $\gamma _{1,m}$ grows exponentially in m. So $d_\infty (\circ , \gamma _{1,M_1})> 2C[\pi \circ f]$ for some minimal $M_1 \ge 0$ , and the line $f^*(\pi ^*(\gamma _{1,m}))$ is disjoint from $T'(G_\circ )$ for $m \ge M_1$ by bounded cancellation (see Figure 2). In particular, the ends $f^*(\pi ^*(h_{\circ *}^{m+1}(\varepsilon _1)))$ and $f^*(\pi ^*(h_{\circ *}^m(\varepsilon _1)))$ have the same closest point projection to $T'(G_\circ )$ , and the sequence $(q_{1,m})_{m \ge M_1}$ is constant.

Figure 2 For $m \ge M_1$ , the line $f^*(\pi ^*(\gamma _{1,m}))$ cannot intersect $T'(G_\circ )$ .

Since $h_\circ (d_2) = s_1^{-1}s_2 \cdot d_2$ , the ends $f^*(\pi ^*(h_{\circ *}^{m+1}(\varepsilon _2)))$ and $\psi _\circ ^{m}(s_1^{-1}s_2) \cdot f^*(\pi ^*(h_{\circ *}^{m}(\varepsilon _2)))$ have the same closest point projection to $T'(G_\circ )$ for $m \gg 1$ by similar bounded cancellation reasoning (i.e., $q_{2,m+1} = \psi _\circ ^m(s_1^{-1}s_2)\cdot q_{2,m}$ for some minimal $M_2 \ge 0$ and all $m \ge M_2$ ).

Set $M = \max (M_1, M_2)$ . The sequence $[q_{1,M+m}, q_{2,M+m}]_{m \ge 0}$ of intervals is well-defined for the line $\gamma $ and point $\circ \in \gamma $ as $M_1$ and $M_2$ were chosen minimally. An iterated turn over $T'(G_\circ )$ rel. $\left .\psi _\circ \right |{}_{G_\circ }$ is any such sequence of intervals. More generally, we define an iterated turn over T rel. $\varphi $ : pick arbitrary points $p_i \in T~(i=1,2)$ and elements $x_i \in F$ ; set and for $m \ge 0$ ; the sequence $[p_{1,m}, p_{2,m}]_{m \ge 0}$ is the iterated turn denoted by $(p_1, p_2: x_1, x_2; \varphi )_{T}$ . Any iterated turn $(p_1, p_2: x_1, x_2; \varphi )_{T}$ translates to a unique normal form $(p_1, p_2: \epsilon , x_1^{-1}x_2; \tilde \varphi )_{T}$ with $\tilde \varphi \colon y \mapsto x_1^{-1}\varphi (y)x_1$ .

We now characterize the growth of an iterated turn over T rel. $\varphi $ :

Proposition 2.4. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ with eigenmetric $d_\tau $ , and $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ . Choose a nondegenerate component $T \subset \mathcal T$ , corresponding components $F \subset \mathcal F$ , $Y_\tau \subset \mathcal Y_\tau $ , and a positive iterate $\psi ^k$ that preserves F. Let $\tilde h \colon (Y_\tau , d_\infty ) \to (Y_\tau , d_\infty )$ be the $\varphi $ -equivariant $\lambda $ -homothety, where $\varphi $ is in the outer automorphism $[\left .\psi ^k\right |{}_F]$ and . Finally, for $i = 1,2$ , pick $p_i \in T$ and $x_i \in F$ .

The point in $(T\varphi ^m, \lambda ^{-m}d_\tau )$ converges to $\star _i$ in $(\overline {Y}_\tau , d_\infty )$ as $m \to \infty $ , where $\star _i$ is the unique fixed point of $x_i^{-1} \cdot \tilde h$ in the metric completion $(\overline Y_\tau , d_\infty )$ ; concretely:

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} d_\tau(p_{1,m}, p_{2,m}) = d_\infty(\star_1, \star_2).\end{align*}$$

If $x_1^{-1}x_2$ fixes $\star _1$ , then $\star _1 = \star _2$ and the $m^{th}$ term $[p_{1,m}, p_{2,m}]$ of the iterated turn $(p_1, p_2: x_1, x_2; \varphi )_T$ has $d_\tau $ -length $ \le (m+1)A$ for some constant $A \ge 1$ . Otherwise, $\star _1 \neq \star _2$ , and the iterated turn has arbitrarily long leaf segments of $\mathcal L^+[\tau ]$ .

The limit $[\star _1, \star _2] \subset \overline {Y}_\tau $ of an iterated turn is independent of the points $p_1, p_2 \in T$ . Thus, we introduce the notion of an algebraic iterated turn over F rel. $\varphi $ , denoted $(x_1, x_2; \varphi )_F$ .

Proof. Let $p_1, p_2 \in T$ , $x_1, x_2 \in F$ and $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ be the constructed equivariant metric PL-map. For $i = 1,2$ , set and for $m \ge 0$ . Recall that T and $T\varphi ^m$ are the same pretrees but with different actions; thus, in $T\varphi ^m$ , we have $p_{i,m} = \varphi ^{-1}(x_i)\cdots \varphi ^{-m}(x_i) \cdot p_i$ for $m \ge 0$ . As $\pi \colon (T, d_\tau ) \to (Y_\tau , d_\infty )$ is an equivariant metric PL-map, so is the composition

$$\begin{align*}\pi_m \colon (T\varphi^m, \lambda^{-m}d_\tau) \overset{\pi}\longrightarrow (Y_\tau\varphi^m, \lambda^{-m}d_\infty) \overset{\tilde h^{-m}}\longrightarrow (Y_\tau, d_\infty). \end{align*}$$

The point $p_i$ in $(T, d_\tau )$ projects (via $\pi $ ) to $\pi (p_i)$ in $(Y_\tau , d_\infty )$ ; the point $p_{i,m}$ in $(T \varphi ^m, \lambda ^{-m}d_\tau )$ projects (via $\pi _m$ ) to

in $(Y_\tau , d_\infty )$ for $m \ge 1$ – in the last line, $x_i^{-1} \cdot \tilde h$ is a $\lambda $ -homothety $(Y_\tau , d_\infty ) \to (Y_\tau , d_\infty )$ . Since $(x_i^{-1} \cdot \tilde h)^{-1}$ is contracting, the point $\pi _m(p_{i,m})$ converges (as $m \to \infty $ ) to the unique fixed point $\star _i$ of $(x_i^{-1}\cdot \tilde h)^{-1}$ (and $x_i^{-1} \cdot \tilde h$ ) in the metric completion $(\overline {Y}_\tau , d_\infty )$ by the contraction mapping theorem; note that $x_1^{-1}x_2 \cdot \star _1 = \star _1$ if and only if $\star _1 = \star _2$ . Thus, the $\pi _m$ -projection of the point $p_{i,m}$ in $(T\varphi ^{m}, \lambda ^{-m} d_\tau )$ converges (as $m \to \infty $ ) to $\star _i$ in $(\overline {Y}_\tau , d_\infty )$ ; in particular,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} d_\infty(\pi(p_{1,m}), \pi(p_{2,m})) = \lim_{m \to \infty} d_\infty(\pi_m(p_{1,m}), \pi_m(p_{2,m})) = d_\infty(\star_1, \star_2). \end{align*}$$

Let $\tilde \tau \colon T \to T$ be the $\varphi $ -equivariant translate of a component of $\tau ^k$ . The interval $[p_{1,m}, p_{2,m}] \subset T$ , the $m^{th}$ term in $(p_1, p_2:x_1,x_2;\varphi )_T$ , is covered by these $2m+1$ intervals:

$$\begin{align*}\begin{aligned} &\varphi^{m-1}(x_1) \cdots \varphi(x_1) \cdot [x_1 \cdot p_1, \tilde \tau(p_1)], \ldots,~\varphi^{m-1}(x_1) \cdot [\tilde \tau^{m-2}(x_1 \cdot p_1), \tilde \tau^{m-1}(p_1)],\\ &[\tilde \tau^{m-1}(x_1\cdot p_1), \tilde \tau^{m}(p_1)],~[\tilde \tau^{m}(p_1), \tilde \tau^{m}(p_2)],~[\tilde \tau^{m}(p_2), \tilde \tau^{m-1}(x_2 \cdot p_2)], \\ &\varphi^{m-1}(x_2) \cdot [\tilde \tau^{m-1}(p_2), \tilde \tau^{m-2}(x_2 \cdot p_2)],\ldots,~\varphi^{m-1}(x_2) \cdots \varphi(x_2) \cdot [\tilde \tau(p_2), x_2\cdot p_2]. \end{aligned} \end{align*}$$

Set and .

Recall that $\underset {m' \to \infty }\lim \lambda ^{-m'} d_\tau (\tilde \tau ^{m'}(p_{1,m}), \tilde \tau ^{m'}(p_{2,m})) = d_\infty (\pi (p_{1,m}), \pi (p_{2,m}))$ . For $m' \ge 0$ , we get a similar covering of $[p_{1,m+m'}, p_{2,m+m'}]$ by $2m'+1$ intervals with the ‘middle’ $[\tilde \tau ^{m'}(p_{1,m}), \tilde \tau ^{m'}(p_{2,m})]$ . Since $\tilde \tau $ is $\lambda $ -Lipschitz with respect to $d_\tau $ , the sum of the $d_\tau $ -lengths of all intervals but the middle in this covering is $\le \lambda ^{m'} 2D'$ . By the triangle inequality,

$$\begin{align*}\lambda^{-(m+m')} \left| d_\tau(p_{1,m+m'}, p_{2,m+m'}) - d_\tau(\tilde \tau^{m'}(p_{1,m}), \tilde \tau^{m'}(p_{2,m})) \right| \le \lambda^{-m} 2D'. \end{align*}$$

Fix $\epsilon> 0$ ; then $\lambda ^{-m} 2D' < \epsilon $ and $\left | \lambda ^{-m} d_\infty (\pi (p_{1,m}), \pi (p_{2,m})) - d_\infty (\star _1, \star _2) \right | < \epsilon $ for some $m \gg 1$ .

Similarly,

$$\begin{align*}\begin{aligned} \lambda^{-m}\left| \lambda^{-m'} d_\tau(\tilde \tau^{m'}(p_{1,m}), \tilde \tau^{m'}(p_{2,m})) - d_\infty(\pi(p_{1,m}), \pi(p_{2,m})) \right| &< \epsilon \\ \text{and } \left|\lambda^{-(m+m')} d_\tau(p_{1,m+m'}, p_{2,m+m'}) - d_\infty(\star_1, \star_2)\right| &< 3\epsilon \text{ for } m' \gg 1; \\ \text{that is,}~ \lim_{m \to \infty} \lambda^{-m} d_\tau(p_{1,m}, p_{2,m}) = d_\infty(\star_1, \star_2).& \end{aligned} \end{align*}$$

Let $N(u,v)$ be the number of vertices in an interval $(u,v) \subset T$ ; set N to be the maximum of $N(p_1,p_2)$ , $N(x_1 \cdot p_1, \tilde \tau (p_1))$ , and $N(\tilde \tau (p_2), x_2 \cdot p_2)$ . As $\tilde \tau $ is a train track, the interval $[p_{1,m}, p_{2,m}]$ is covered by $(2m+1)(N+1)$ leaf segments.

Suppose $\star _1 = \star _2$ . We claim that any leaf segment (of $\mathcal L^+[\tau ]$ ) in $[p_{1,m}, p_{2,m}]$ has uniformly (in $m\ge 0$ ) bounded $d_\tau $ -length – this implies $[p_{1,m}, p_{2,m}]$ has $d_\tau $ -length $\le (2m+1)(N+1)B$ for some bounding constant $B \ge 1$ . We mimic Case 2 from the proof of Proposition 2.2. For the contrapositive, suppose some term $[p_{1,m}, p_{2,m}]$ has a leaf segment with $d_\tau $ -length $L> 2(C[\pi ] + D')$ . By the train track property, bounded cancellation and interval covering, $[p_{1, m+m'}, p_{2, m+m'}]$ has a leaf segment with $d_\tau $ -length $\ge \lambda ^{m'} (L - 2C[\pi ] - 2D')$ for $m' \ge 0$ ; in $(T\varphi ^{m+m'}, \lambda ^{-(m+m')}d_\tau )$ , $[p_{1, m+m'}, p_{2, m+m'}]$ has length $ \ge \lambda ^{-m}(L-2C[\pi ] - 2D')$ . In the limit (as $m' \to \infty $ ), $d_\infty (\star _1, \star _2) \ge \lambda ^{-m}(L-2C[\pi ] - 2D')> 0$ .

Suppose $\star _1 \neq \star _2$ . Set ; then $\lambda ^{-m}d_\tau (p_{1,m}, p_{2,m})> L$ for some $m \gg 1$ . By the pigeonhole principle, the interval $[p_{1,m}, p_{2,m}]$ has a leaf segment with $d_\tau $ -length $\frac {\lambda ^{m}L}{(2m+1)(N+1)}$ , which can be arbitrarily large (in m).

2.2.4 Nested iterated turns

The first part of the previous subsection explains how a line in $(\mathcal Y_\tau , d_\infty )$ determines algebraic iterated turns over ${\mathcal {G}}[\mathcal Y_\tau ]$ . We now give a similar discussion for an iterated turn over $\mathcal T'$ .

Recall how $f, T, T', Y_\tau , F, \tilde \tau $ , $\tilde h$ and $\varphi $ were chosen, and $\lambda $ was redefined in the previous subsection. Pick points $p_1', p_2' \in T'$ and elements $x_1, x_2 \in F$ . Set , , , in $T_m'$ and $p_{i,m} = f(p_{i,m}')$ for $m \ge 1$ and $i = 1,2$ . By Proposition 2.4, the point $p_{i,m}$ in $(T_m, \lambda ^{-m} d_\tau )$ converges (as $m\to \infty $ ) to $\star _i$ , the unique fixed point of $x_i^{-1}\cdot \tilde h$ in the metric completion $(\overline {Y}_\tau , d_\infty )$ . The $\lambda $ -homothety is $\varphi _i$ -equivariant for some automorphism $\varphi _i \colon F \to F$ in the outer class $[\varphi ]$ . Set .

Case–a: . Suppose $G_1$ is not trivial, and let $a_{i,m}$ be the closest point projection of $p_{i,m}'$ to $T'(\varphi ^m(G_1))$ for $m \ge 0$ . As $\tilde h(\star _1) = x_1 \cdot \star _1$ and $\tilde h$ is $\varphi $ -equivariant, we get $T'(\varphi ^{m+1}(G_1)) = \varphi ^m(x_1) \cdot T'(\varphi ^m(G_1))$ , $a_{1,m+1} = \varphi ^m(x_1) \cdot a_{1,m}$ , and

$$\begin{align*}\begin{aligned} a_{2,m+1} &= \varphi^m(x_1) \varphi^m(s) \cdot a_{2,m} \\ &= \varphi^m(x_1)\cdots \varphi(x_1) x_1 \varphi_1^m(s)\cdots \varphi_1^m(s)s \cdot a_{2,0} \quad \text{ for } m \ge 0. \end{aligned}\end{align*}$$

Thus, the closest point projection to $T'(\varphi ^m(G_1))$ of the $m^{th}$ term of the given iterated turn $(p_1',p_2':x_1,x_2;\varphi )_{T'}$ is a translate of the $m^{th}$ term in $(a_{1,0}, a_{2,0} : \epsilon , s;\left .\varphi _1\right |{}_{G_1})_{T'(G_1)}$ , where $m \ge 0$ and $\epsilon $ is the trivial element. Hence, we have an algebraic iterated turn $(\epsilon , s;\left .\varphi _1\right |{}_{G_1})_{G_1}$ that is well-defined for the algebraic iterated turn $(x_1, x_2;\varphi )_F$ .

Case–b: $\star _1 \neq \star _2$ . Suppose $G_1$ is not trivial – the argument is symmetric if $\operatorname {Stab}_F(\star _2)$ is not trivial – and let d be the direction at $\star _1$ containing $\star _2$ . By Gaboriau–Levitt index theory, $h_1^j(d) = t \cdot d$ for some $t \in G_1$ and minimal $j \ge 1$ . For $m \gg 1$ , $\pi _m(p_{2,m}) = h_2^{-m}(\pi (p_2))$ is in the direction d since $h_2^{-m}(\pi (p_2)) \to \star _2$ in $(\overline {Y}_\tau , d_\infty )$ . For $m \gg 1$ and $m{' } \ge 0$ , the interval $[p_{2, m+m{' }j}, \tilde \tau ^{m{' }j}(p_{2,m})]$ in $(T_{m+m{' }j}, \lambda ^{-m-m{' }j}d_\tau )$ is disjoint from $T_{m+m{' }j}(G_1)$ by bounded cancellation (see Figure 3, top); or equivalently, the interval $[p_{2,m+m{' }j}, \tilde \tau ^{m{' }j}(p_{2,m})]$ in $T_m$ is disjoint from $T_m(\varphi ^{m{' }j}(G_1))$ . In fact, the $\lambda ^{-m}d_\tau $ -distance in $T_m$ from $[p_{2,m+m{' }j}, \tilde \tau ^{m{' }j}(p_{2,m})]$ to $T_m(\varphi ^{m{' }j}(G_1))$ can be arbitrarily large for $m{' } \gg 1$ .

Figure 3 The two figures illustrating certain closest point projections are the same.

Set and . Let $b_{i,m{' }}{' }~(i=1,2)$ be the closest point projection of $p_{i,m+m{' }j}{' }$ to $T_m{' }(\varphi ^{m{' }j}(G_1)) = z_{m{' }j} \cdot T_m{' }(G_1)$ and set in $T_m{' }(G_1)$ . Following the definitions, $z_{m{' }j}^{-1} \cdot p_{1,m+m{' }j}{' } = p_{1,m}{' }$ in $T_m{' }$ and $z_{m{' }j}^{-1} \cdot \tilde \tau ^{m{' }j} = \tau _1^{m{' }j}$ in $T_m$ , where ; in particular, $b_{1,m{' }} = b_{1,0}$ for $m{' } \ge 0$ . Since $h_1^j(d) = t \cdot d$ , bounded cancellation implies the $\lambda ^{-m}d_\tau $ -distance in $T_m$ from $[\tau _1^{(m{' }+1)j}(p_{2,m}), \varphi _1^{m{' }j}(t) \cdot \tau _1^{m{' }j}(p_{2,m})]$ to $T_m(G_1)$ is arbitrarily large for $m{' } \gg 1$ (see Figure 3, bottom).

So $[z_{(m{' }+1)j}^{-1} \cdot p_{2,m+(m{' }+1)j}, \varphi _1^{m{' }j}(t)z_{m{' }j}^{-1} \cdot p_{2,m+m{' }j}]$ is arbitrarily far from $T_m(G_1)$ by transitivity. By bounded cancellation, $[z_{(m{' }+1)j}^{-1} \cdot p_{2,m+(m{' }+1)j}{' }, \varphi _1^{m{' }j}(t)z_{m{' }j}^{-1}\cdot p_{2, m+m{' }j}{' }]$ is disjoint from $T_m{' }(G_1)$ for $m{' } \gg 1$ (i.e., $b_{2,m{' }+1} = \varphi _1^{m{' }j}(t) \cdot b_{2,m{' }}$ for $m{' } \gg 1$ ). Thus, for some $M{' } \gg 1$ , the sequence $[b_{1,M{' }+m{' }}, b_{2,M{' }+m{' }}]_{m{' }\ge 0}$ is an iterated turn over $T_m{' }(G_1)$ rel. $\varphi _1^j|_{G_1}$ , denoted $(b_{1,M{' }}, b_{2,M{' }}: \epsilon , t;\varphi _1^j|_{G_1})_{T_m{' }(G_1)}$ . The corresponding algebraic iterated turn $(\epsilon , t; \varphi _1^j|_{G_1})_{G_1}$ is well-defined for $(x_1, x_2;\varphi )_F$ .

Now suppose $\circ \in (\star _1, \star _2)$ has a nontrivial stabilizer . Let $d_i~(i = 1,2)$ be the direction at $\circ $ containing $\star _i$ . By index theory again, $h_1^{\,l}(\circ ) = x \cdot \circ $ and $h_1^{\,l}(d_i) = xs_i \cdot d_i$ for some $x \in F$ , $s_i \in G_\circ $ , and minimal $l \ge 1$ . Since F acts on $Y_\tau $ with trivial arc stabilizers, the elements $xs_1, xs_2$ are unique and $s_1^{-1}s_2 \in G_\circ $ is independent of the chosen $x \in F$ . For $m \gg 1$ , $\pi _m(p_{i,m})$ is in the direction $d_i$ since $\pi _m(p_{i,m}) \to \star _i$ . A variation of the bounded cancellation argument used in the preceding paragraphs proves the following. For $m, m{' } \gg 1$ , the interval $[p_{i,m+m{' }l}, \tilde \tau ^{m{' }l}(p_{i,m})]$ in $(T_m, \lambda ^{-m}d_\tau )$ is far from $T_m(\varphi ^{m{' }l}(G_\circ ))$ .

Set , , , and to be $\varphi _{\circ }$ -equivariant maps for an automorphism $\varphi _{\circ } \colon F \to F$ in the outer class $[\varphi _1^j]$ . Let $c_{i,m{' }}{' } \in T_m{' }(\varphi ^{m{' }l}(G_\circ ))$ be the closest point projection of $p_{i,m+m{' }l}{' }$ and set . Then is the closest point projection of $y_{m{' }}^{-1}z_{m{' }l}^{-1}\cdot p_{i,m+m{' }l}{' }$ . Since $h_\circ (d_i) = s_i \cdot d_i$ , the interval $[\tau _\circ ^{m{' }+1}(p_{i,m}), \varphi _\circ ^{m{' }}(s_i)\cdot \tau _\circ ^{m{' }}(p_{i,m})]$ is arbitrarily far from $T_m(G_\circ )$ for $m{' } \gg 1$ . By transitivity, $[y_{m{' }+1}^{-1}z_{(m{' }+1)l}^{-1} \cdot p_{i,m+(m{' }+1)l}, \varphi _\circ ^{m{' }}(s_i)y_{m{' }}^{-1}z_{m{' }}^{-1} \cdot p_{i,m+m{' }l}]$ is arbitrarily far from $T_m(G_\circ )$ . As before, $[y_{m{' }+1}^{-1}z_{(m{' }+1)l}^{-1}\cdot p_{i,m+(m{' }+1)l}{' }, \varphi _\circ ^{m{' }}(s_i)y_{m{' }}^{-1}z_{m{' }}^{-1}\cdot p_{i, m+m{' }l}{' }]$ is disjoint from $T_m{' }(G_\circ )$ for $m{' } \gg 1$ (i.e., $c_{i,m{' }+1} = \varphi _\circ ^{m{' }}(s_i)\cdot c_{i,m{' }}$ for $m{' } \gg 1$ ). Thus, for some $M" \gg 1$ , the sequence $[c_{1,M{' }{' }+m{' }}, c_{2,M{' }{' }+m{' }}]_{m{' }\ge 0}$ is an iterated turn over $T_m{' }(G_\circ )$ rel. $\left .\varphi _\circ \right |{}_{G_\circ }$ : $(c_{1,M{' }{' }}, c_{2,M{' }{' }}: s_1, s_2; \left .\varphi _\circ \right |{}_{G_\circ } )_{T_m{' }(G_\circ )}$ . It is a ‘translate’ of the normalized iterated turn: $(c_{1,M{' }{' }}, c_{2,M{' }{' }}:\epsilon , s_1^{-1}s_2; \left .\varphi _\circ \right |{}_{G_\circ })_{T_m{' }(G_\circ )}$ . The corresponding algebraic iterated turn $(\epsilon , s_1^{-1}s_2; \left .\varphi _\circ \right |{}_{G_\circ })_{G_\circ }$ is well-defined for $(x_1, x_2;\varphi )_F$ and $\circ \in (\star _1, \star _2)$ .

2.3 Coordinate-free laminations

We have only defined the stable laminations for an expanding irreducible train track $[\tau ]$ representing $[\psi ]$ . The free splitting $\mathcal T$ of $\mathcal F$ can be seen as a coordinate system, and we need a coordinate-free definition of stable laminations that applies to all outer automorphisms.

Fix a proper free factor system ${\mathcal {Z}}$ of $\mathcal F$ and consider the set $scv(\mathcal F, {\mathcal {Z}})$ of all free splittings $\mathcal T'$ of $\mathcal F$ with $\mathcal F[\mathcal T'] = {\mathcal {Z}}$ (i.e., an element of $\mathcal F$ is $\mathcal T'$ -elliptic if and only if it is conjugate to an element of ${\mathcal {Z}}$ ); this set with some natural partial order is the spine of relative outer space [7]. For any pair of free splittings $\mathcal T_1, \mathcal T_2 \in scv(\mathcal F, {\mathcal {Z}})$ , there are changes of coordinates, equivariant PL-maps $\mathcal T_1 \rightleftarrows \mathcal T_2$ . We saw in the discussion following Claim 1.5 that a change of coordinates $f\colon \mathcal T_1 \to \mathcal T_2$ induces a canonical homeomorphism $f_* \colon \mathbb R(\mathcal T_1) \to \mathbb R(\mathcal T_2)$ on the space of lines. Denote the canonical homeomorphism class of $\mathbb R(\mathcal T_1) \cong \mathbb R(\mathcal T_2)$ by $\mathbb R(\mathcal F, {\mathcal {Z}})$ . If ${\mathcal {Z}}$ is the trivial free factor system of $\mathcal F$ , then we denoted the canonical homeomorphism class by $\mathbb R(\mathcal F)$ instead.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ and a $[\psi ]$ -invariant proper free factor system ${\mathcal {Z}}$ . Let $\psi _* \colon \mathbb R(\mathcal F, {\mathcal {Z}}) \to \mathbb R(\mathcal F, {\mathcal {Z}})$ be the canonical induced homeomorphism on the space of lines: $f_* \circ g_{1*} = g_{2*} \circ f_*$ for any $\mathcal T_1, \mathcal T_2 \in scv(\mathcal F, {\mathcal {Z}})$ , equivariant PL-map $f \colon \mathcal T_1 \to \mathcal T_2$ , and $\psi $ -equivariant PL-maps $g_i \colon \mathcal T_i \to \mathcal T_i~(i=1,2)$ . A line $[l] \in \mathbb R(\mathcal F, {\mathcal {Z}})$ weakly ψ _*-limits to a lamination $\Lambda \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ if the sequence $(\psi _*^n[l])_{n \ge 0}$ weakly limits to $\Lambda $ .

A coordinate-free definition of stable laminations boils down to characterizing the lines of a stable $\mathcal T$ -lamination for $[\tau ]$ in a way that is independent of coordinates. For the rest of the section, assume there is an equivariant PL-map $(\mathcal T, d) \to (\mathcal Y, \delta )$ and consider the canonical embedding $\mathbb R(\mathcal Y, \delta ) \subset \mathbb R(\mathcal T)$ . Note that a lamination $\Lambda \subset \mathbb R(\mathcal Y, \delta )$ is contained in a canonical lamination $\mathcal L \subset \mathbb R(\mathcal T)$ : set $\mathcal L$ to be the closure of $\Lambda $ in $\mathbb R(\mathcal T)$ .

Claim (cf. [2, Lemma 1.9(2)]).

A line is quasiperiodic in $\mathbb R(\mathcal Y, \delta )$ if it is quasiperiodic in $\mathbb R(\mathcal T)$ . (exercise)

So quasiperiodicity is a well-defined property for a line in $\mathbb R(\mathcal F, {\mathcal {Z}})$ ; moreover, the induced homeomorphism $\psi _* \colon \mathbb R(\mathcal F, {\mathcal {Z}}) \to \mathbb R(\mathcal F, {\mathcal {Z}})$ preserves quasiperiodicity for any automorphism $\psi \colon \mathcal F \to \mathcal F$ that preserves ${\mathcal {Z}}$ (up to conjugacy).

Suppose there is an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ for $\psi $ with $\mathcal F[\mathcal T] = {\mathcal {Z}}$ . Recall that the eigenlines of $[\tau ^k]$ (for some $k \ge 1$ ) are constructed by iterating an expanding edge; more precisely, an eigenline $[l]$ of $[\tau ^k]$ is the union $\bigcup _{n \ge 1} \tau ^{kn}(\mathcal F \cdot e)$ for some edge $e \subset l$ . The leaf segments $\tau ^{kn}(e)$ determine a neighborhood basis for $[l]$ in the space of lines.

For a line $[l] \in \mathbb R(\mathcal F, {\mathcal {Z}})$ , a subset $U \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ is a $\psi _*^k$ -attracting neighborhood of $[l]$ if $\psi _*^k(U) \subset U$ and $\{\psi _{*}^{kn}(U):n \ge 1\}$ is a neighborhood basis for $[l]$ in the space of lines. A stable lamination for $[\psi ]$ rel. ${\mathcal {Z}}$ is the closure of a quasiperiodic line in $\mathbb R(\mathcal F, {\mathcal {Z}})$ with a $\psi _*^k$ -attracting neighborhood for some $k \ge 1$ . Note that the homeomorphism $\psi _* \colon \mathbb R(\mathcal F, {\mathcal {Z}}) \to \mathbb R(\mathcal F, {\mathcal {Z}})$ permutes the stable laminations for $[\psi ]$ rel. ${\mathcal {Z}}$ and, by Lemma 2.1, each stable $\mathcal T$ -lamination for $[\tau ]$ is identified with some stable lamination for $[\psi ]$ rel. ${\mathcal {Z}}$ . Let $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ be the union of stable laminations for $[\psi ]$ rel. ${\mathcal {Z}}$ .

Lemma 2.5 (cf. [2, Lemma 1.12]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ , and . The stable laminations $\mathcal L^+[\tau ]$ for $[\tau ]$ are identified with the stable laminations $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ for $[\psi ]$ rel. ${\mathcal {Z}}$ .

So $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ is a lamination system whose finitely many components are the stable laminations for $[\psi ]$ rel. ${\mathcal {Z}}$ , and these are transitively permuted by $\psi _* \colon \mathbb R(\mathcal F, {\mathcal {Z}}) \to \mathbb R(\mathcal F, {\mathcal {Z}})$ .

Sketch of proof.

Suppose a quasiperiodic line $[l]$ in $\mathcal T$ has a $\tau _*^k$ -attracting neighborhood U for some $k \ge 1$ . This forces any $\mathcal T$ -loxodromic conjugacy class $[x]$ with axis in U to have a translation distance that (eventually) grows under forward $[\psi ^k]$ -iteration. In particular, the conjugacy class $[x]$ is $\mathcal Y_\tau $ -loxodromic, and the line $[l]$ , a weak $\psi _*^k$ -limit of the $\mathcal T$ -axis for $[x]$ , is a leaf in $\mathcal L^+[\tau ]$ by Proposition 2.2.

The stable laminations $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ are in the subspace $\mathbb R(\mathcal Y_\tau , d_\infty ) \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ .

2.4 Constructing limit forests (2)

This chapter has thus far focused on automorphims with expanding irreducible train tracks. For the rest of the chapter, we extend our focus to all automorphisms.

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ , set , , and let ${\mathcal {Z}}$ be a $[\psi _1]$ -invariant proper free factor system. By Theorem 1.1, there is an irreducible train track $\tau _1 \colon \mathcal T_1 \to \mathcal T_1$ for $\psi _1$ . By $\psi _1$ -equivariance of $\tau _1$ , the nontrivial vertex stabilizers of $\mathcal T_1$ determine a $[\psi _1]$ -invariant proper free factor system . The restriction of $\psi _1$ to $\mathcal F_2$ determines a unique outer class of automorphisms $\psi _2 \colon \mathcal F_2 \to \mathcal F_2$ . We can repeatedly apply Theorem 1.1 to $\psi _{i+1}~(i\ge 1)$ as long as $\lambda [\tau _i] = 1$ and contains a noncyclic component. Bass-Serre theory implies this process must stop; we end up with a maximal sequence $(\tau _i)_{i=1}^n$ of irreducible train tracks with $\lambda [\tau _i] = 1$ for $1 \le i < n$ – such a maximal sequence is called a descending sequence of irreducible train tracks for $[\psi ]$ rel. ${\mathcal {Z}}$ .

This leads to our working definition of growth type: $[\psi ]$ is polynomially growing rel. ${\mathcal {Z}}$ if and only if $\lambda [\tau _n] = 1$ [22, Proposition III.1]. For automorphisms that are polynomially growing rel. ${\mathcal {Z}}$ , define the limit forest to be degenerate.

Suppose $[\psi ]$ is exponentially growing rel. ${\mathcal {Z}}$ and $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ is a descending sequence of irreducible train tracks for $[\psi ]$ rel. ${\mathcal {Z}}$ . Sections 2.1–2.2 already cover the case $n = 1$ , so we may assume $n> 1$ for the rest of the chapter. Set , , , and $d_n^\circ $ the eigenmetric on $\mathcal T_n^\circ $ for $\tau _n^\circ $ . For $1 \le i < n$ , we inductively form an equivariant simplicial blow-up $\mathcal T_{i}^\circ $ of $\mathcal T_{i}$ rel. $\mathcal T_{i+1}^\circ $ : the vertices with nontrivial stabilizers are equivariantly replaced by copies of corresponding components of $\mathcal T_{i+1}^\circ $ , and arbitrary vertices in $\mathcal T_{i+1}^\circ $ are chosen as attaching points for the edges of $\mathcal T_i$ . Let $\tau _i^\circ \colon \mathcal T_i^\circ \to \mathcal T_i^\circ $ be the topological representative for $\psi _i$ induced by $\tau _i$ and $\tau _{i+1}^\circ $ . As $\tau _i$ is a simplicial automorphism, we can make $\tau _i^\circ $ a $\lambda $ -Lipschitz map by assigning the same large enough length to the edges of $\mathcal T_i$ in the blow-up $\mathcal T_i^\circ $ when extending the metric $d_{i+1}^\circ $ on $\mathcal T_{i+1}^\circ $ to a metric $d_i^\circ $ on $\mathcal T_i^\circ $ . The topological representative on is an equivariant blow-up of the descending sequence $(\tau _i)_{i=1}^n$ . Set and identify $(\mathcal T_i^\circ , d_i^\circ )$ with the characteristic subforest of $(\mathcal T^\circ , d^\circ )$ for $\mathcal F_{i}$ . We will abuse terminology and refer to $d^\circ $ as the eigenmetric as well. Translates of edges in $\mathcal T^\circ $ coming from $\mathcal T_i$ form the $\underline{i^{\textit{th}}\ \mathrm{stratum}\ \mathrm{of}\ \mathrm{T}^{\circ}}$ : the $n^{th}$ stratum is exponential, while the rest are (relatively) polynomial.

As in Section 2.1, the maps $\tau ^{\circ m}\colon (\mathcal T^\circ , d^\circ ) \to (\mathcal T^\circ \psi ^m, \lambda ^{-m}d^\circ )$ converge (as $m\to \infty $ ) to an equivariant metric surjection $\pi ^\circ \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal Y, \delta )$ . The map $\tau ^\circ $ induces a $\psi $ -equivariant $\lambda $ -homothety $h \colon (\mathcal Y, \delta ) \to (\mathcal Y, \delta )$ and $\pi ^\circ $ semiconjugates $\tau ^\circ $ to h. By restricting to $\mathcal T_i^\circ $ , we have also constructed an equivariant metric surjection $\pi _i^\circ \colon (\mathcal T_i^\circ , d^\circ ) \to (\mathcal Y_i, \delta )$ and $\psi _i$ -equivariant $\lambda $ -homothety $h_i$ on $(\mathcal Y_i, \delta )$ for $2 \le i \le n$ .

The $\mathcal F_n$ -forest $(\mathcal Y_n, \delta )$ is the limit forest for $[\tau _n^\circ ]$ ; so it is a nondegenerate minimal $\mathcal F_n$ -forest with trivial arc stabilizers. For induction, assume $(\mathcal Y_i, \delta )$ is a nondegenerate minimal $\mathcal F_i$ -forest with trivial arc stabilizers for $2 \le i \le n$ . Equivariantly collapse $\mathcal T_2^\circ $ in $(\mathcal T^\circ , d^\circ )$ to get the $\mathcal F$ -forest $(\mathcal T_1, d_1)$ . For $m \ge 0$ , the metric free splitting $(\mathcal T^\circ \psi ^m, \lambda ^{-m} d^\circ )$ is an equivariant metric blow-up of $(\mathcal T_1 \psi ^m, \lambda ^{-m} d_1)$ rel. $(\mathcal T_2^\circ \psi _2^{m}, \lambda ^{-m} d^\circ )$ . Since $\tau _1 \colon (\mathcal T_1, d_1) \to (\mathcal T_1 \psi , d_1)$ is an equivariant isometry, the limit $(\mathcal Y, \delta )$ is equivariantly isometric to an equivariant metric blow-up of $(\mathcal T_1, d_1)$ rel. $(\mathcal Y_2, \delta )$ whose top stratum (edges coming from $\mathcal T_1$ ) have then been equivariantly collapsed, also known as a graph of actions (with degenerate skeleton) – more details are given in the next subsection. Thus, $(\mathcal Y, \delta )$ is a nondegenerate minimal $\mathcal F$ -forest with trivial arc stabilizers. See [22, Theorem IV.1] for a direct construction of $(\mathcal Y, \delta )$ as a graph of actions. This sketches the general case of Proposition 1.2. The $\mathcal F$ -forest $(\mathcal Y,\delta )$ is the limit forest for $[\tau _i]_{i=1}^n$ .

2.4.1 Decomposing limit forests

We now give a hierarchical decomposition of the limit forest $(\mathcal Y, \delta )$ and its space of lines.

Choose an iterate $[\tau _1^{k'}]$ that fixes all $\mathcal F$ -orbits of branches in $\mathcal T_1$ . Pick an edge e in $\mathcal T_1$ and one of its endpoints p. Replace $\psi ^{k'}$ with an automorphism in its outer class $[\psi ^{k'}]$ if necessary and assume $\tau _1^{k'}$ fixes p and e. Identify $(\mathcal T^\circ \psi ^{mk'}, \lambda ^{-mk'} d^\circ )$ with an equivariant metric blow-up of $(\mathcal T_1, \lambda ^{-mk'} d_1)$ rel. $(\mathcal T_2^\circ \psi _2^{mk'}, \lambda ^{-mk'} d^\circ )$ for $m \ge 0$ , then let $p_m \in \mathcal T_2^\circ \psi _2^{mk'}$ be the attaching point of e to $\mathcal T_2^\circ \psi _2^{mk'}$ corresponding to the endpoint p. Since $\tau _1^{k'}$ fixes e and p, we get $p_m = p_0$ for $m \ge 1$ . As in the first part of the proof for Proposition 2.4, the sequence $(p_m)_{m \ge 0}$ converges to the unique fixed point $\star $ of $h_2^{k'}$ in the metric completion $(\overline {\mathcal Y}_2, \delta )$ . So, in the description of $(\mathcal Y, \delta )$ as a graph of actions, the edge e is collapsed and identified with $\star $ . Thus, the closure $\widehat {\mathcal Y}_2$ of $\mathcal Y_2$ in $(\mathcal Y, \delta )$ is the union of $\mathcal Y_2$ with the $\mathcal F_2$ -orbits of attaching points $\star $ as the pair $(e, p)$ ranges over the $\mathcal F$ -orbit representatives e of edges and their endpoints p. For the same reasons, we inductively get a similar description of the closure $\widehat {\mathcal Y}_{i+1}$ of $\mathcal Y_{i+1}$ in $(\mathcal Y_i, \delta )$ for $2 \le i < n$ .

Remark. Constructing $(\mathcal Y, \delta )$ directly by iterating $\tau ^\circ $ allows us to lift metric properties of $(\mathcal Y,\delta )$ to dynamical properties of $\tau ^\circ $ through the semiconjugacy $\pi ^\circ \circ \tau ^\circ = h \circ \pi ^\circ $ ; this viewpoint is used in the Section 2.5. On the other hand, constructing $(\mathcal Y, \delta )$ directly as we did in [22, Theorem IV.1] (and sketched in this subsection) gives us a nice structural description of intervals in the limit forest. This is explained in the next subsection and will be a key component of Chapter 3!

For $1 < i \le n$ , any two translates of $\mathcal T_i^\circ \subset \mathcal T^\circ $ by elements of $\mathcal F$ either coincide or are disjoint by construction. This induces a canonical closed embedding of $\mathbb R(\mathcal F_{i}, {\mathcal {Z}}) $ into $\mathbb R(\mathcal F, {\mathcal {Z}})$ (exercise). Similarly, any two intersecting translates of $\mathcal Y_i \subset \mathcal Y$ by elements of $\mathcal F$ either coincide or have degenerate intersection. This also induces a canonical closed embedding $\mathbb R(\mathcal Y_i, \delta ) \subset \mathbb R(\mathcal Y, \delta )$ . Finally, the constructed equivariant metric map $\pi ^\circ $ induces a canonical embedding of the topological pair $(\mathbb R(\mathcal Y, \delta ), \mathbb R(\mathcal Y_i, \delta ))$ into $(\mathbb R(\mathcal F, {\mathcal {Z}}), \mathbb R(\mathcal F_i, {\mathcal {Z}}))$ .

2.4.2 Intervals in limit forests

Here is an inductive description of intervals in the limit forest $(\mathcal Y, \delta )$ in terms of the limit forest for $[\tau _n^\circ ]$ . For $1 \le i \le n$ , the characteristic subforest $(\mathcal Y_i, \delta )$ of $(\mathcal Y, \delta )$ for $\mathcal F_{i}$ is the limit forest for $[\tau _j]_{j=i}^n$ . For $1 < i \le n$ , let $\widehat {\mathcal Y}_i$ be the closure of $\mathcal Y_i$ in $(\mathcal Y_{i-1}, \delta )$ .

It follows from the blow-up (and collapse) description of $\mathcal Y_{i-1}$ that its closed intervals are finite concatenations of closed intervals in translates of $\widehat {\mathcal Y}_i$ . As shown in the previous subsection, the $\mathcal F_{i}$ -orbits $[p]$ of points in $\widehat {\mathcal Y}_i \setminus \mathcal Y_i$ are fixed by the extension of $h_i^{k'}$ to $\widehat {\mathcal Y}_i$ for some $k' \ge 1$ . As $p \notin \mathcal Y_i$ , it has exactly one direction $d_p$ in $\widehat {\mathcal Y}_i$ . This direction’s $\mathcal F_{i}$ -orbit $[d_p]$ is also fixed (setwise) by the expanding homothety $h_i^{k'}$ , and $d_p$ determines a singular eigenray $\rho _p \subset \widehat {\mathcal Y}_i$ of $[h_i^{k'}]$ based at p. For any point $q \in {\mathcal Y}_i$ , the closed interval $[p,q] \subset \widehat {\mathcal Y}_i$ is a concatenation of an initial segment of the singular eigenray $\rho _p$ and a closed interval in ${\mathcal Y}_i$ ; therefore, closed intervals in ${\mathcal Y}_{i-1}$ are finite concatenations of translates of closed intervals in ${\mathcal Y}_i$ and initial segments of singular eigenrays of $[h_i^{k'}]$ for some $k' \ge 1$ .

Let $\mathcal L_{{\mathcal {Z}}}^+[\psi _n] = \mathcal L^+[\tau _n]$ be the k-component stable laminations for $[\tau _n^\circ ] = [\tau _n]$ and $\oplus _{j=1}^k \delta _j$ the factored $\mathcal F_n$ -invariant convex metric on $\mathcal Y_n$ indexed by components $\Lambda _j^+ \subset \mathcal L_{{\mathcal {Z}}}^+[\psi _n]$ . By the inductive description of intervals in $\mathcal Y$ and the fact $h_n^k$ is a $\lambda ^k$ -homothety with respect to each factor $\delta _j$ , we get $\delta _j$ equivariantly extends to $\mathcal Y$ ; $\delta = \oplus _{j=1}^k \delta _j$ is a factored $\mathcal F$ -invariant convex metric on $\mathcal Y$ ; and $h^k$ is a $\lambda ^k$ -homothety with respect to each factor $\delta _j$ .

The lamination $\mathcal L_{{\mathcal {Z}}}^+[\psi _n] \subset \mathbb R(\mathcal Y_n, \delta )$ can be seen as a $(\mathcal Y, \delta )$ -lamination since $\mathbb R(\mathcal Y_n, \delta )$ is a closed subspace of $\mathbb R(\mathcal Y, \delta )$ . Note that closed edges of $\mathcal T_n = \mathcal T_n^\circ $ are leaf segments (of $\mathcal L_{{\mathcal {Z}}}^+[\psi _n]$ ); thus, any closed interval in $\mathcal T_n^\circ $ is a finite concatenation of leaf segments. As the equivariant PL-map $\pi _n^\circ \colon (\mathcal T_n^\circ , d^\circ ) \to (\mathcal Y_n, \delta )$ is surjective and isometric on leaf segments, we get the following:

Lemma 2.6. Let $\tau _n \colon \mathcal T_n \to \mathcal T_n$ be an expanding irreducible train track and $(\mathcal Y_n, \delta )$ its limit forest. Any closed interval in $\mathcal Y_n$ is a finite concatenation of leaf segments of $\mathcal L^+[\tau _{n}]$ .

This lemma no longer holds when $n \ge 2$ and we consider closed intervals in $\widehat {\mathcal Y}_n$ . To account for this, let n ^th level leaf blocks in $\mathcal Y$ be leaf segments. By the lemma, any interval of $\mathcal Y_n$ is a finite concatenation of $n^{th}$ level leaf blocks.

Inductively define the (i−1) ^st level leaf blocks in $\mathcal Y~(1 < i \le n)$ to be the $i^{th}$ level leaf blocks or (translates of) closed intervals in singular eigenrays $\rho \subset \widehat {\mathcal Y}_i$ of $[h_{i}]$ -iterates. By the earlier description of intervals and induction hypothesis, any interval of $\mathcal Y_{i-1}$ is a finite concatenation of $(i-1)^{st}$ level leaf blocks. The $1^{st}$ level leaf blocks are simply leaf blocks of $\mathcal L_{{\mathcal {Z}}}^+[\psi _{n}]$ . Altogether, we have a generalization of Lemma 2.6 in terms of leaf blocks:

Lemma 2.7. Let $(\tau _i \colon \mathcal T_i \to \mathcal T_i)_{i=1}^n$ be a descending sequence of irreducible train tracks for an automorphism $\psi \colon \mathcal F \to \mathcal F$ and $(\mathcal Y, \delta )$ the corresponding limit forest. Any closed interval in $\mathcal Y$ is a finite concatenation of leaf blocks of $\mathcal L_{{\mathcal {Z}}}^+[\psi _{n}]$ , where .

2.5 Stable laminations (2)

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an invariant proper free factor system ${\mathcal {Z}}'$ . Let $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ be a descending sequence of irreducible train tracks rel. ${\mathcal {Z}}'$ with $\lambda [\tau _n]> 1$ , $(\mathcal Y,\delta )$ the limit forest for $[\tau _i]_{i=1}^n$ , $\mathcal T^\circ $ an equivariant blow-up of free splittings $(\mathcal T_i)_{i=1}^n$ with eigenmetric $d^\circ $ , and . The characteristic convex subsets of $\mathcal T^\circ $ for are identified with the free splitting $\mathcal T_n$ .

Claim 2.8. The stable laminations $\mathcal L^+_{{\mathcal {Z}}}[\psi _{n}]$ for $[\psi _n]$ in $\mathbb R(\mathcal F_{n}, {\mathcal {Z}})$ are identified with the stable laminations $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ for $[\psi ]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ .

Note that $\mathcal L_{{\mathcal {Z}}}^+[\psi ] = \mathcal L_{{\mathcal {Z}}}^+[\psi _n]$ is in the subspace $\mathbb R(\mathcal Y_n, \delta ) \subset \mathbb R(\mathcal Y, \delta ) \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ .

Sketch of proof.

Since $\lambda [\tau _i]=1$ for $i < n$ , no quasiperiodic line in $\mathbb R(\mathcal F, \mathcal F_{n})$ has a $\psi _*^k$ -attracting neighborhood for any $k \ge 1$ . Thus, any stable lamination for $[\psi ]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ is contained in $\mathbb R(\mathcal F_{n}, {\mathcal {Z}})$ and corresponds to a stable lamination for $[\psi _{n}]$ .

We generalize Proposition 2.4 by characterizing limits of iterated turns over $\mathcal T^\circ $ :

Theorem 2.9. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism with an invariant proper free factor system ${\mathcal {Z}}'$ , $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks rel. ${\mathcal {Z}}'$ with $\lambda [\tau _n]>1$ , $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ , and $\mathcal T^\circ $ an equivariant blow-up of free splittings $(\mathcal T_i)_{i=1}^n$ with eigenmetric $d^\circ $ . Choose a nondegenerate component $T^\circ \subset \mathcal T^\circ $ , corresponding components $F \subset \mathcal F$ , $Y \subset \mathcal Y$ , and a positive iterate $\psi ^k$ that preserves F. Let $\tilde h \colon (Y, \delta ) \to (Y, \delta )$ be the $\varphi $ -equivariant $\lambda $ -homothety, where $\varphi $ is in the outer class $[\left .\psi ^k\right |{}_F]$ and . Finally, for $\iota = 1,2$ , pick $p_\iota \in T^\circ $ and $x_\iota \in F$ .

The point in $(T^\circ \varphi ^m, \lambda ^{-m}d^\circ )$ converges to $\star _\iota $ in $(\overline {Y}, \delta )$ as $m \to \infty $ , where $\star _\iota $ is the unique fixed point of $x_\iota ^{-1} \cdot \tilde h$ in the metric completion $(\overline Y, \delta )$ .

If $x_1^{-1}x_2$ fixes $\star _1$ , then $\star _1 = \star _2$ and the term $[p_{1,m}, p_{2,m}]~(m \ge 0)$ of the iterated turn $(p_1, p_2: x_1, x_2; \varphi )_{T^\circ }$ has $d^\circ $ -length $ \le \alpha (m)$ for some (degree n) polynomial $\alpha $ . Otherwise, $\star _1 \neq \star _2$ , and the iterated turn weakly limits to a component of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ , where .

An iterated turn $[p_{1,m}, p_{2,m}]_{m \ge 0}$ weakly limits to a component $\Lambda ^+ \subset \mathcal L_{{\mathcal {Z}}}^+[\psi ]$ if the term $[p_{1,m}, p_{2,m}]$ contains a leaf segment of $\Lambda ^+$ with arbitrarily large $d^\circ $ -length as $m \to \infty $ .

Sketch of proof.

Let $\tilde \tau ^{\circ } \colon (T^\circ , d^\circ ) \to (T^\circ , d^\circ )$ be the $\varphi $ -equivariant $\lambda $ -Lipschitz topological representative induced by the irreducible train tracks $(\tau _i)_{i=1}^n$ and $\pi ^\circ \colon (T^\circ , d^\circ ) \to (Y, \delta )$ the equivariant metric map constructed using $\tilde \tau ^\circ $ -iteration. Even though $\pi ^\circ $ may fail to be a PL-map, it still has a cancellation constant $C[\pi ^\circ ] \ge 0$ as a limit of equivariant metric maps with uniformly bounded cancellation constants. The proof of the first part is the same as in Proposition 2.4 using $\pi ^\circ $ , $\tilde \tau ^{\circ }$ and the $\varphi $ -equivariant $\lambda $ -homothety $\tilde h$ .

The interval $[p_{1,m}, p_{2,m}] \subset T^\circ $ , a term in the sequence $(p_1, p_2:x_1,x_2;\varphi )_{T^\circ }$ , is covered by certain $2m+1$ intervals as in the proof of Proposition 2.4. Since $\tilde \tau ^{\circ }$ is induced by a descending sequence $(\tau _i)_{i=1}^n$ of irreducible train tracks, the intervals $[\tilde \tau ^{\circ (l-1)}(x_1 \cdot p_1), \tilde \tau ^{\circ l}(p_1)]$ , $[\tau ^{\circ l}(p_1), \tilde \tau ^{\circ l}(p_2)]$ , and $[\tilde \tau ^{\circ l}(p_2), \tilde \tau ^{\circ (l-1)}(x_2 \cdot p_2)]$ are covered by $\alpha (l)$ polynomial strata edges and leaf segments (of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ ) for some degree $(n-1)$ polynomial $\alpha $ . So the interval $[p_{1,m}, p_{2,m}]$ is covered by $\alpha (m) + \sum _{l=1}^{m} 2\alpha (l)$ polynomial strata edges and leaf segments (of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ ). Note that $\alpha (m) + \sum _{l=1}^{m} 2\alpha (l) \le \beta (m)$ for some degree n polynomial $\beta $ .

Assume $\star _1 = \star _2$ , where $\star _\iota $ is the unique fixed point of in metric completion $(\overline Y, \delta )$ for $\iota = 1,2$ . The proof given in Proposition 2.4 implies there is a uniform bound on the $d^\circ $ -length of leaf segments in $[p_{1,m}, p_{2,m}]$ . Consequently, the $d^\circ $ -length of $[p_{1,m}, p_{2,m}]$ is $\le \beta (m) B$ for some constant $B \ge 1$ .

Assume $\star _1 \neq \star _2$ . Set ; then $\delta (\tilde h_1^{-m}(\pi ^\circ (p_1)), \tilde h_2^{-m}(\pi ^\circ (p_2)))> L$ and $d^\circ (p_{1,m}, p_{2,m})> \lambda ^{m} L$ for $m \gg 1$ . The contribution of polynomial strata to the $d^\circ $ -length of $[p_{1,m}, p_{2,m}]$ is at most $\beta (m) B'$ for some constant $B' \ge 1$ ; the exponential stratum edges intersecting the interval are covered by $\beta (m)$ leaf segments. By the pigeonhole principle, the interval $[p_{1,m}, p_{2,m}]$ , a term in the iterated turn $(p_1, p_2: x_1, x_2; \varphi )_{T^\circ }$ , has a leaf segment of $d^\circ $ -length $\ge \frac {\lambda ^m L - \beta (m)B'}{\beta (m)} \gg 1$ . Quasiperiodicity implies the iterated turn weakly limits to a component of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ .

Remark. The argument given in Subsection 2.2.4 applies in this general context involving a descending sequence of irreducible train tracks; it describes how an iterated turn over $\mathcal F$ determines (nested) iterated turns over ${\mathcal {G}}[\mathcal Y]$ .

As in Proposition 2.2, we can characterize the elements in $\mathcal F$ that are $\mathcal Y$ -loxodromic:

Theorem 2.10. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism with an invariant proper free factor system ${\mathcal {Z}}'$ , $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks rel. ${\mathcal {Z}}'$ with , $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ , $\mathcal T^\circ $ an equivariant blow-up of free splittings $(\mathcal T_i)_{i=1}^n$ , and .

If $x \in \mathcal F$ is a $\mathcal T^\circ $ -loxodromic element, then the following statements are equivalent:

1. 1. the element x is $\mathcal Y$ -loxodromic;

2. 2. the element $x [\psi ]$ -grows exponentially rel. ${\mathcal {Z}}$ with rate $\lambda $ ; and

3. 3. the axis for the conjugacy class $[x]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ weakly $\psi _*$ -limits to $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ .

The restriction of $\psi $ to the $[\psi ]$ -invariant subgroup system ${\mathcal {G}}[\mathcal Y]$ of $\mathcal Y$ -point stabilizers is polynomially growing rel. ${\mathcal {Z}}$ with degree $< n$ .

Sketch of proof.

Set , , and for $1 \le i < n$ . Under the canonical embedding $\mathbb R(\mathcal F_i, {\mathcal {Z}}) \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ , we identify the stable laminations $\mathcal L^+_{{\mathcal {Z}}}[\psi ]$ and $\mathcal L^+_{{\mathcal {Z}}}[\psi _i]$ . Let $\mathcal T^\circ $ be an equivariant blow-up of free splittings $(\mathcal T_i)_{i=1}^n$ and $\mathcal T_i^\circ \subset \mathcal T^\circ $ the characteristic convex subsets for $\mathcal F_i$ . Suppose $x \in \mathcal F_1$ is a $\mathcal T^\circ $ -loxodromic element. The equivalence between Conditions 1–3 is given by Proposition 2.2 if x is conjugate to an element of $\mathcal F_{n}$ . Assume $n \ge 2$ and, up to conjugacy, $x \in \mathcal F_i$ is $\mathcal T_i$ -loxodromic for some $i < n$ .

Recall that $\tau ^\circ \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal T^\circ , d^\circ )$ is a $\psi $ -equivariant $\lambda $ -Lipschitz topological representative induced by the irreducible train tracks $(\tau _i)_{i=1}^n$ , and $\pi ^\circ \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal Y, \delta )$ is the constructed equivariant metric map. In particular, $\underset {m \to \infty }\limsup \, \frac {1}{m} \log \| \psi ^m(x)\|_{d^\circ } \le \log \lambda $ .

Suppose $[\tau _i^{k'}]$ (for some $k' \ge 1$ ) fixes all $\mathcal F_i$ -orbits of vertices and edges in $\mathcal T_i$ . Let $l^\circ \subset \mathcal T_i^\circ $ be the axis for $x \in \mathcal F_i$ . The axis $l^\circ $ projects to the axis l of x in $\mathcal T_i$ ; write l as a biinfinite concatenation of edges $\cdots e_{-1} \cdot e_0 \cdot e_1 \cdots $ and identify $e_j \subset \mathcal T_i$ with its lift to $\mathcal T_i^\circ $ . For $m \ge 0$ and any integer j, let $w_{j,m}$ be the closed interval in $\mathcal T_i^\circ $ between (lifts of) $\tau _i^{m}(e_j)$ and $\tau _i^{m}(e_{j+1})$ ; in fact, $w_{j,m}$ is in a component of $\mathcal F_i \cdot \mathcal T_{i+1}^\circ \subset \mathcal T_i^\circ $ . Since $[\tau _i^{k'}]$ fixes the $\mathcal F_i$ -orbits $[e], [e']$ and the vertex of $\mathcal T_i$ between them, the sequence $(w_{j,mk'+r})_{m \ge 0}$ , up to translation, is an iterated turn over $\mathcal T_{i+1}^\circ $ rel. $\psi _{i+1}^{k'}$ for $0 \le r < k'$ ; by Theorem 2.9, the iterated turn limits to an interval $w_{j,r}^*$ in a translate of a component of $\widehat {\mathcal Y}_{i+1} \subset \mathcal Y_i$ .

The intervals $w_{j,m}, w_{j+1,m}$ are always in distinct components of $\mathcal F_i \cdot \mathcal T_{i+1}^\circ $ ; therefore, the limit intervals $w_{j,r}^*, w_{j+1,r}^*$ have degenerate intersection. By the equivariance of the limits, the union is an x-invariant arc. If some limit interval $w_{j,0}^*$ is not degenerate, then x is $\mathcal Y_i$ -loxodromic and $l_*$ is its $\mathcal Y_i$ -axis; otherwise, $l_*$ is degenerate and x is $\mathcal Y_i$ -elliptic.

Case 1: x is $\mathcal Y_i$ -loxodromic (i.e., some limit interval $w_{j,0}^*$ is not degenerate). By Theorem 2.9, the iterated turn $(w_{j,mk'})_{m \ge 0}$ over $\mathcal T_{i+1}^\circ $ rel. $\psi _{i+1}^{k'}$ weakly limits to a component of $\mathcal L_{{\mathcal {Z}}}^+[\psi _{i+1}]$ . So $[l^\circ ] \in \mathbb R(\mathcal F, {\mathcal {Z}})$ weakly $\psi _*^{k'}$ -limits to a component of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ . Finally, $[l^\circ ]$ weakly $\psi _*$ -limits to $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ since $\psi _*$ acts transitively on the components of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ . As $\pi ^\circ $ is an equivariant metric map, $\|\cdot \|_{\delta } \le \| \cdot \|_{d^\circ }$ and $\log \lambda \le \underset {m \to \infty } \liminf \, \frac {1}{m} \log \| \psi ^m(x)\|_{d^\circ }$ .

Case 2: x is $\mathcal Y_i$ -elliptic (i.e., each limit interval $w_{j,0}^*$ is degenerate). By Theorem 2.9, the interval $w_{j,mk}$ has $d^\circ $ -length is bounded above by some degree $(n-i)$ polynomial (in m). Thus, $\|\psi ^{mk}(x)\|_{d^\circ }$ is bounded above by a degree $(n-i)$ polynomial. By $\psi $ -equivariance of the homothety $h_i$ , the elements $\psi (x), \ldots , \psi ^{k-1}(x)$ are $\mathcal Y_i$ -elliptic as well. The same argument implies $\|\psi ^{m}(x)\|_{d^\circ }$ is bounded above by a degree $(n-i)$ polynomial.

We conclude the chapter by stating the extension of Lemma 4.3 to all limit forests:

Lemma 4.5. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, ${\mathcal {Z}}'$ a $[\psi ]$ -invariant proper free factor system, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ rel. ${\mathcal {Z}}'$ with , $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ , $(\mathcal Y', \delta ')$ a minimal $\mathcal F$ -forest with trivial arc stabilizers, and .

If ${\mathcal {Z}}$ is $\mathcal Y'$ -elliptic and the k-component lamination $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is in $\mathbb R(\mathcal Y', \delta ') \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ , then the limit of $(\mathcal Y' \psi ^{mk}, \lambda ^{-mk} \delta ')_{m \ge 0}$ is $(\mathcal Y, \oplus _{j=1}^k c_j \, \delta _j)$ , where $\delta = \oplus _{j=1}^k \, \delta _j$ and $c_j> 0$ .

Again, we postpone the proof to Section 4.2. If $(\tau _i')_{i=1}^{n'}$ is another descending sequence for $[\psi ]$ with $\mathcal F[\mathcal T_{n' }'] = {\mathcal {Z}}$ , then its limit forest $(\mathcal Y', \delta ')$ is equivariantly homothetic to $(\mathcal Y, \delta )$ ; therefore, $(\mathcal Y, \delta )$ is the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ (up to rescaling of $\delta $ ). A nondegenerate minimal very small $\mathcal F$ -forest $(\mathcal Y', \delta ')$ is an expanding forest for $[\psi ]$ rel. ${\mathcal {Z}}$ if

1. 1. the $\mathcal F$ -forest $(\mathcal Y'\psi , \delta ')$ is equivariantly isometric to $(\mathcal Y', s \delta ')$ for some $s> 1$ ; and

2. 2. the free factor system ${\mathcal {Z}}$ is $\mathcal Y'$ -elliptic.

Corollary 2.11. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ with $\lambda [\tau _n]> 1$ . Any expanding forests for $[\psi ]$ rel. $\mathcal F[\mathcal T_n]$ is the limit forest for $[\psi ]$ rel. $\mathcal F[\mathcal T_n]$ .

We will end the paper with a complete generalization of this corollary (Theorem 5.3).

Sketch of proof.

Let $(\mathcal Y', \delta ')$ be an expanding forest for $[\psi ]$ rel. and $x \in \mathcal F$ a $\mathcal Y'$ -loxodromic element. The proof is essentially the proof of Corollary 2.3 with two main changes. First, choose $m \gg 1$ so that $\| \psi ^m(x) \|_{\delta '}> \alpha (m)(2C[f]+B')$ for some polynomial $\alpha $ and constant $B' \ge 1$ determined by x; therefore, a fundamental domain of $\psi ^m(x)$ acting on its axis has a leaf segment $[q, r]$ with $\delta '(f(q), f(r))> 2C[f]$ by the pigeonhole principle. For the second change, we need $(\mathcal Y', \delta ')$ to have trivial arc stabilizers in order to conclude the proof by invoking Lemma 4.5 instead of Lemma 4.3.

The minimal very small $\mathcal F$ -forest $(\mathcal Y', \delta ')$ has finitely many orbits of branch points [11]; it decomposes as some graph of actions whose skeleton is not degenerate in the forest if and only if the forest does not have dense orbits [18]. Any $\psi $ -equivariant homothety must be an isometry if the skeleton were not degenerate. Since $(\mathcal Y', \delta ')$ admits a $\psi $ -equivariant expanding s-homothety, the skeleton must be degenerate and the forest has dense orbit. Very small $\mathcal F$ -forests with dense orbits have trivial arc stabilizers [19, Lemma 4.2].

For a nondegenerate minimal $\mathcal F$ -forest $(\mathcal Y', \delta ')$ , the projective stabilizer $\operatorname {Stab}[\mathcal Y', \delta ']$ is the subgroup of automorphisms $\varphi \colon \mathcal F \to \mathcal F$ for which $\|\varphi (\cdot )\|_{\delta '} = s_\varphi \| \cdot \|_{\delta '}$ for some $s_\varphi> 0$ . The function $\operatorname {SF} \colon \operatorname {Stab}[\mathcal Y', \delta '] \to \mathbb R_{>0}$ that maps $\varphi \mapsto s_\varphi $ is a homomorphism called the stretch factor homomorphism – $\mathbb R_{>0}$ is considered multiplicatively.

Corollary 2.12. Let $\operatorname {SF} \colon \operatorname {Stab}[\mathcal Y', \delta '] \to \mathbb R_{>0}$ be the stretch factor homomorphism for some nondegenerate minimal very small $\mathcal F$ -forest $(\mathcal Y', \delta ')$ . The image of $\operatorname {SF}$ is cyclic.

Proof. Suppose $\operatorname {SF}(\psi )> 1$ for some $\psi \in \operatorname {Stab}[\mathcal Y', \delta ']$ . Then $\psi $ is exponentially growing since any $\mathcal Y'$ -loxodromic element $[\psi ]$ -grows exponentialy with rate at least $\operatorname {SF}(\psi )$ . Set , , and let $(\mathcal Y_1, \delta _1)$ be the limit forest for $[\psi _1]$ rel. some $[\psi _1]$ -invariant proper free factor system $\mathcal F_2$ . If $\mathcal F_2$ is not $\mathcal Y'$ -elliptic, then the restrictions $\psi _2$ of $\psi _1$ to $\mathcal F_2$ are in the projective stabilizer of the nondegenerate characteristic subforest of $(\mathcal Y', \delta ')$ for $\mathcal F_2$ and have the same stretch factor $\operatorname {SF}(\psi )$ .

By repeatedly considering limit forests and taking restrictions, we may assume some free factor system $\mathcal F_n$ is not $\mathcal Y'$ -elliptic while a nested proper free factor system $\mathcal F_{n+1}$ is for some $n \ge 1$ . Then the characteristic subforest of $(\mathcal Y', \delta ')$ for the free factor system $\mathcal F_n$ is an expanding forest for $[\psi _n]$ rel. $\mathcal F_{n+1}$ . By Corollary 2.11, this subforest is equivariantly homothetic to the limit forest $(\mathcal Y_n, \delta _n)$ for $[\psi _n]$ rel. $\mathcal F_{n+1}$ . In particular, $\operatorname {SF}(\psi )$ is the exponential growth rate for $[\psi _n]$ rel. $\mathcal F_{n+1}$ and is bounded away from $1$ by a uniform constant that depends only on $\mathcal F$ . Thus, the image of $\operatorname {SF}$ is discrete, and discrete subgroups of $\mathbb R_{> 0}$ are cyclic.

3 Main constructions

The limit forest produced by our proof of Proposition 1.2 is universal for an outer automorphism and some choice of an invariant proper free factor system (Corollary 2.11). Our goal is to remove the latter dependence on an invariant proper free factor system.

3.1 Assembling limit hierarchies

This section first summarizes the main result of the paper’s prequel [22]. The general strategy follows closely the construction of limit forests sketched in Section 2.4.

Fix an exponentially growing automorphism $\psi \colon \mathcal F \to \mathcal F$ and set , . By our proof of Proposition 1.2, there is a nondegenerate limit forest $(\mathcal Y_1, \delta _1)$ for $[\psi _1]$ rel. ${\mathcal {Z}}_1$ (some proper free factor system of ${\mathcal {G}}_1$ ) and a unique $\psi _1$ -equivariant expanding $\lambda _1$ -homothety $h_1 \colon (\mathcal Y_1, \delta _1) \to (\mathcal Y_1, \delta _1)$ . Thus, $\mathcal Y_1$ -loxodromic elements in $\mathcal F [\psi ]$ -grow exponentially rel. ${\mathcal {Z}}_1$ with rate $\lambda _1$ . By Gaboriau–Levitt index theory and $\psi _1$ -equivariance of $\tau _1$ , the nontrivial point stabilizers of $\mathcal Y_1$ determine a $[\psi _1]$ -invariant malnormal subgroup system with strictly lower complexity than ${\mathcal {G}}_1$ . The restriction of $\psi _1$ to ${\mathcal {G}}_2$ determines a unique outer class of automorphisms $\psi _2 \colon {\mathcal {G}}_2 \to {\mathcal {G}}_2$ .

We can repeatedly apply Proposition 1.2 to $\psi _{i+1}~(i \ge 1)$ as long as $\psi _{i+1}$ is exponentially growing. This inductive invocation of Proposition 1.2 eventually stops since the complexity of ${\mathcal {G}}_i$ is a strictly decreasing (in i) positive integer. In the end, we have a maximal sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of nondegenerate limit forests for $[\psi _i]$ rel. ${\mathcal {Z}}_i$ each with a unique $\psi _i$ -equivariant expanding $\lambda _i$ -homothety $h_i$ on $(\mathcal Y_i, d_i)$ – such a maximal sequence of limit forests is a descending sequence of limit forests for $[\psi ]$ . By construction, an element $x \in \mathcal F$ has a conjugate in ${\mathcal {G}}_{n+1}$ if and only if $x [\psi ]$ -grows polynomially!

In Section 2.4, the blow-ups of free splittings $(\mathcal T_i)_{i=1}^n$ were arbitary and done inductively upwards (i.e., started with $i = n$ ). We then used a limiting argument to produce the final limit forest $(\mathcal Y, \delta )$ . For this section, the blow-ups of limit forests $(\mathcal Y_i, \delta _i)_{i=1}^n$ will not be arbitrary but will make use of the expanding homotheties $(h_i)_{i=1}^n$ ; moreover, it will be done inductively downwards (i.e., starts with $i = 1$ ) to produce an $\mathcal F$ -pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ .

Set and . For $1 < i \le n$ , we inductively construct the equivariant pseudoforest blow-up $(\mathcal X^{(i)}, (\delta _j)_{j=1}^i)$ of the $\mathcal F$ -pseudoforest $(\mathcal X^{(i-1)}, (\delta _j)_{j=1}^{i-1})$ rel. the ${\mathcal {G}}_i$ -forest $(\mathcal Y_i, \delta _i)$ and expanding homotheties $g^{(i-1)}$ and $h_i$ . Here is a sketch:

Let $(\overline {\mathcal Y}_i, \delta _i)$ be the metric completion and $\bar h_i$ the extension to the metric completion. For $1 \le j < i$ , assume that $(\mathcal Y_j, \delta _j)$ is equivariantly isometric to the associated ${\mathcal {G}}_j$ -forest for the ${\mathcal {G}}_j$ -invariant convex pseudometric $\delta _j$ restricted to $\mathcal X^{(i-1)}({\mathcal {G}}_j)$ , the characteristic convex subsets of $\mathcal X^{(i-1)}$ for ${\mathcal {G}}_j$ . Since the hierarchy $(\delta _j)_{j=1}^{i-1}$ has full support, $(\mathcal X^{(i-1)}({\mathcal {G}}_{i-1}), \delta _{i-1})$ is equivariantly isometric to $(\mathcal Y_{i-1}, \delta _{i-1})$ , and the nontrivial point stabilizers of $\mathcal X^{(i-1)}$ are conjugates in $\mathcal F$ of ${\mathcal {G}}_i$ -components. The points of $\mathcal X^{(i-1)}$ with nontrivial stabilizers are replaced by corresponding copies of $\overline {\mathcal Y}_i$ -components; this produces a unique set system $\widehat {\mathcal X}^{(i)}$ with an $\mathcal F$ -action that is the equivariant set blow-up of $\mathcal X^{(i-1)}$ rel. $\overline {\mathcal Y}_i$ : it comes with an equivariant injection $\iota _i \colon \overline {\mathcal Y}_i \to \widehat {\mathcal X}^{(i)}$ and an equivariant surjection $\kappa _i \colon \widehat {\mathcal X}^{(i)} \to \mathcal X^{(i-1)}$ that is a bijection on the complement $\widehat {\mathcal X}^{(i)} \setminus \mathcal F \cdot \iota _i(\overline {\mathcal Y}_i)$ . Consequently, there is a unique $\psi $ -equivariant induced permutation $g^{(i)} \colon \widehat {\mathcal X}^{(i)} \to \widehat {\mathcal X}^{(i)}$ induced by $g^{(i-1)}$ and $\bar h_i$ – $\kappa _i$ semiconjugates $\hat g^{(i)}$ to $g^{(i-1)}$ , while $\iota _i$ conjugates $\bar h_i$ to the restriction $\left .g^{(i)}\right |{}_{\iota _i(\overline {\mathcal Y}_i)}$ .

There are plenty of equivariant interval functions $[\cdot, \cdot ]^{(i)}$ on $\widehat {\mathcal X}^{(i)}$ compatible with $\mathcal X^{(i-1)}$ and $\mathcal Y_i$ – compatibility means the injection $\iota _i$ and surjection $\kappa _i$ map intervals to intervals. Some compatible $\mathcal F$ -pretrees $(\widehat {\mathcal X}^{(i)}, [\cdot , \cdot ]^{(i)})$ are real [22, Proposition IV.3], and they naturally inherit an $\mathcal F$ -invariant hierarchy $(\hat \delta _j)_{j=1}^i$ with full support: $(\hat \delta _j)_{j=1}^{i-1}$ is the pullback $\kappa _i^*(\delta _j)_{j=1}^{i-1}$ and $\hat \delta _i$ is the pushforward $\iota _{i *} \delta _i$ extended equivariantly to the orbit $\mathcal F \cdot \iota _i(\overline {\mathcal Y}_i)$ ; moreover, for $1 \le j \le i$ , $(\mathcal Y_j, \delta _j)$ is equivariantly isometric to the associated ${\mathcal {G}}_j$ -forest for the ${\mathcal {G}}_j$ -invariant convex pseudometric $\hat \delta _j$ restricted to $\widehat {\mathcal X}^{(i)}({\mathcal {G}}_j)$ .

Claim [22, Theorem IV.4].

Since $\bar h_i$ is expanding, the permutation $g^{(i)}$ is a pretree-automorphism for a unique real compatible $\mathcal F$ -pretree $(\widehat {\mathcal X}^{(i)}, [\cdot , \cdot ]_g^{(i)})$ .

Remark. This is the main technical result of [22]. Its proof uses Gaboriau–Levitt’s index inequality and the contraction mapping theorem.

We now fix the interval function $[\cdot , \cdot ]_g^{(i)}$ but omit it for brevity. By construction, the $\mathcal F$ -pseudoforest $(\widehat {\mathcal X}^{(i)}, (\hat \delta _j)_{j=1}^i)$ has trivial arc stabilizers, and $g^{(i)}$ is an expanding homothety with respect to $(\hat \delta _j)_{j=1}^i$ . Finally, let $\mathcal X^{(i)} \subset \widehat {\mathcal X}^{(i)}$ be the characteristic convex subsets for $\mathcal F$ and $(\delta _j)_{j=1}^i$ the restriction of the hierarchy $(\hat \delta _j)_{j=1}^i$ to $\mathcal X^{(i)}$ . Then replace the maps $\iota _i, \kappa _i$ and $g^{(i)}$ with their restrictions to $\mathcal X^{(i)}$ ; so $({\mathcal X}^{(i)}, (\delta _j)_{j=1}^i)$ is a minimal $\mathcal F$ -pseudoforest.

At the $n^{th}$ iteration, we have a minimal $\mathcal F$ -pseudoforest with trivial arc stabilizers, unique for the descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ . The $\psi $ -equivariant pretree-automorphism on $(\mathcal T, (\delta _i)_{i=1}^n)$ is a $(\lambda _i)_{i=1}^n$ -homothety, where $\lambda _i>1$ is the scaling factor for the homothety $h_i$ . Lastly, an element $x \in \mathcal F$ is $\mathcal T$ -elliptic if and only if x has a conjugate in ${\mathcal {G}}_{n+1}$ . The real $\mathcal F$ -pretrees $\mathcal T$ are the limit pretrees for $(\mathcal Y_i)_{i=1}^n$ , and the $\mathcal F$ -pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ is the limit pseudoforest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ . To summarize,

Theorem 3.1 (cf. [22, Theorem III.3]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism. Then there is

1. 1. a minimal $\mathcal F$ -pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ with trivial arc stabilizers;

2. 2. a $\psi $ -equivariant expanding homothety $h \colon (\mathcal T, (\delta _i)_{i=1}^n) \to (\mathcal T, (\delta _i)_{i=1}^n)$ ; and

3. 3. an element $x \in \mathcal F$ is $\mathcal T$ -loxodromic if and only if $x [\psi ]$ -grows exponentially.

The real pretrees $\mathcal T$ are degenerate if and only if $[\psi ]$ is exponentially growing.

Without metrics, there is not much one can do to compare limit pretrees. On the other hand, we do not expect limit pseudoforests to be well-defined (even up to homothety) for a given outer automorphism – this would be equivalent to the existence of a canonical descending sequence of limit forests. The new idea is to pick a limit pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ and normalize its hierarchy $(\delta _i)_{i=1}^n$ using the attracting laminations for $[\psi ]$ . For the normalized hierarchy, the associated top level forest will be universal; in particular, it is independent of any choices made in its construction.

3.2 Attracting laminations

Fix an exponentially growing automorphism $\psi \colon \mathcal F \to \mathcal F$ with a descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of limit forests. Let ${\mathcal {G}}_1 = \mathcal F$ , ${\mathcal {G}}_{i+1} = {\mathcal {G}}[\mathcal Y_i]$ , and $[\psi _i]$ be the restriction of $[\psi ]$ to ${\mathcal {G}}_i$ for $i \ge 1$ . Each limit forest $(\mathcal Y_i, \delta _i)$ has matching stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ for $[\psi _i]$ rel. ${\mathcal {Z}}_i$ , where ${\mathcal {Z}}_i$ is a $[\psi _i]$ -invariant proper free factor system of ${\mathcal {G}}_i$ . By Claim 1.5, $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ is canonically identified with a subspace of $\mathbb R({\mathcal {G}}_i)$ via a lifting map. As ${\mathcal {G}}_{i+1}$ is a malnormal subgroup system of ${\mathcal {G}}_i$ , the space of lines $\mathbb R({\mathcal {G}}_{i+1})$ is canonically identified with a closed subspace of $\mathbb R({\mathcal {G}}_i)$ (exercise). By transitivity, $\mathbb R({\mathcal {G}}_n) \subset \mathbb R({\mathcal {G}}_{n-1}) \subset \cdots \subset \mathbb R({\mathcal {G}}_0) = \mathbb R(\mathcal F)$ .

Consider this chain of canonical embeddings: $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i) \subset \mathbb R({\mathcal {G}}_i) \subset \mathbb R(\mathcal F)$ . Quasiperiodicity is not preserved by the first embedding, but a weaker form of it is. A line $[l]$ is birecurrent in an $\mathcal F$ -forest if any closed interval $I \subset l$ has infinitely many translates contained in both ends of l; quasiperiodic lines are birecurrent.

An attracting lamination for $[\psi ]$ in $\mathbb R(\mathcal F)$ is the closure of a birecurrent line in $\mathbb R(\mathcal F)$ with a $\psi _*^k$ -attracting neighborhood for some $k\ge 1$ . The set of all attracting laminations for $[\psi ]$ is canonical as it is defined using canonical constructs: $\mathbb R(\mathcal F)$ and the homeomorphism $\psi _* \colon \mathbb R(\mathcal F) \to \mathbb R(\mathcal F)$ . Note that $\psi _*$ permutes the attracting laminations for $[\psi ]$ .

Remark. This definition is from [3, Definition 3.1.5]. Shortly, we will define topmost attracting laminations as done in [3, Section 6].

Lemma 3.2 (cf. [3, Lemma 3.1.4]).

Let $f \colon (\mathcal T,d) \to (\mathcal Y, \delta )$ be an equivariant PL-map. A line is birecurrent in $\mathbb R(\mathcal Y,\delta )$ if and only if it is birecurrent in $\mathbb R(\mathcal T)$ . (exercise)

So leaves of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ are birecurrent in $\mathbb R({\mathcal {G}}_i)$ , and hence, $\mathbb R(\mathcal F)$ ; moreover, a $\psi _{i*}^k$ -attracting neighborhood of a line in $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ will lift to a $\psi _*^k$ -attracting neighborhood of the same line in $\mathbb R(\mathcal F)$ . (exercise) Thus, the closure in $\mathbb R(\mathcal F)$ of a stable lamination for $[\psi _i]$ rel. ${\mathcal {Z}}_i$ (i.e., a component of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ ) is an attracting lamination for $[\psi ]$ .

Lemma 3.3 (cf. [3, Lemma 3.1.10]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an exponentially growing automorphism with a descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of limit forests. The components of stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]~(1 \le i \le n)$ determine all the attracting laminations for $[\psi ]$ .

Sketch of proof.

Suppose that $[l] \in \mathbb R(\mathcal F)$ is a birecurrent line with a $\psi _*^k$ -attracting neighborhood for some $k \ge 1$ . If ${\mathcal {G}}_{n+1} \neq \emptyset $ , then either it consists of only cyclic components or the restriction of $\psi _{n}$ to ${\mathcal {G}}_{n+1}$ is polynomially growing. Either way, ${\mathcal {G}}_{n+1}$ cannot support an attracting lamination of $\psi _n$ . Let $i \le n$ be the maximal index for which $\mathbb R({\mathcal {G}}_{i}) \subset \mathbb R(\mathcal F)$ contains $[l]$ . Birecurrence in $\mathbb R(\mathcal F)$ and Lemma 3.2 imply $[l]$ is birecurrent in $\mathbb R({\mathcal {G}}_{i}, {\mathcal {Z}}_i)$ with a $\psi _{i*}^k$ -attracting neighborhood for some $k \ge 1$ . Following the proof of Claim 2.8, assume some descending chain $(\mathcal F_{i,j})_{j=2}^{n_i}$ of proper free factor systems of was used to construct $(\mathcal Y_i, \delta _i)$ ; then any birecurrent line in $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ with a $\psi _{i*}^k$ -attracting neighborhood is in $\mathbb R(\mathcal F_{i, n_i}, {\mathcal {Z}}_i)$ . The proof of Lemma 2.5 (with ‘birecurrence’ in place of ‘quasiperiodicity’) implies $[l] \in \mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ .

The finite set of all attracting laminations for $[\psi ]$ is canonical (by definition) and partially ordered by inclusion; an attracting lamination for $[\psi ]$ is topmost if it is maximal in this partial order. By Lemma 2.5, $\psi _{i*}$ transitively permutes the components of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ ; so the closure in $\mathbb R(\mathcal F)$ of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i] \subset \mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ is a $\psi _*$ -orbit $\mathcal L_i^+[\psi ]$ of attracting laminations for $[\psi ]$ . The goal is to normalize any limit pseudoforest $(\mathcal T, (d_i)_{i=1}^n)$ so that the levels are related to the partial order of the attracting laminations.

The next proposition is a repackaging of Theorem 2.10 in the language of this chapter:

Proposition 3.4. Let $\psi \colon \mathcal F \to \mathcal F$ be an exponentially growing automorphism with a limit pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ .

For a nontrivial element $x \in \mathcal F$ , the following statements are equivalent:

1. 1. the element x is $\mathcal T$ -loxodromic;

2. 2. the element $x [\psi ]$ -grows exponentially; and

3. 3. the axis for x in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to an attracting lamination.

Proof. The equivalence between Conditions 1–2 is part of Theorem 3.1. Suppose $x \in \mathcal F$ is $\mathcal T$ -loxodromic and the limit pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ is constructed from the descending sequence of limit forests $(\mathcal Y_i, \delta _i)$ for $1 \le i \le n$ . By construction, the element x is conjugate to a $\mathcal Y_i$ -loxodromic element $y \in {\mathcal {G}}_i$ for some $i \le n$ ; in particular, x and y have the same axis in $\mathbb R({\mathcal {G}}_i) \subset \mathbb R(\mathcal F)$ . The axis for y in $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i) \subset \mathbb R({\mathcal {G}}_i)$ weakly $\psi _{i *}$ -limits to the stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i] \subset \mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ by Theorem 2.10; therefore, the shared axis for y and x in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to the attracting laminations for $[\psi ]$ determined by $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ (i.e., the closure of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ in $\mathbb R(\mathcal F)$ ).

Conversely, suppose $x \in \mathcal F$ is $\mathcal T$ -elliptic. Then x is must be conjugate to a $\mathcal Y_n$ -elliptic element $y \in {\mathcal {G}}_n$ . If y is conjugate to an element of ${\mathcal {Z}}_i$ , then the shared axis for y and x in the closed subspace $\mathbb R({\mathcal {Z}}_i) \subset \mathbb R(\mathcal F)$ cannot weakly $\psi _*$ -limit to the attracting lamination for $[\psi ]$ determined by a component of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ – such an attracting lamination contains lines not in $\mathbb R({\mathcal {Z}}_i)$ . If y is not conjugate to an element of ${\mathcal {Z}}_i$ , then the axis for y in $\mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ does not weakly $\psi _{i *}$ -limit to $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ by Theorem 2.10; therefore, the shared axis for y and x in $\mathbb R(\mathcal F)$ cannot weakly $\psi _*$ -limit to the attracting lamination for $[\psi ]$ determined by a component of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ . By Lemma 3.3, we have exhausted all possibilities when $1 \le i \le n$ , and the axis for x in $\mathbb R(\mathcal F)$ cannot weakly $\psi _*$ -limit to an attracting lamination for $[\psi ]$ .

3.3 Pseudolaminations

Fix an exponentially growing automorphism $\psi \colon \mathcal F \to \mathcal F$ with a descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of limit forests, and let $(\mathcal T, (\delta _i)_{i=1}^n)$ be the limit pseudoforest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ . Recall that ${\mathcal {G}}_1 = \mathcal F$ , ${\mathcal {G}}_{i+1} = {\mathcal {G}}[\mathcal Y_i]$ , and $[\psi _i]$ is the restriction of $[\psi ]$ to ${\mathcal {G}}_i$ for $i \ge 1$ . For $1 \le i \le n$ , the stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ are contained in $\mathbb R(\mathcal Y_i, \delta _i) \subset \mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ , where ${\mathcal {Z}}_i$ is some $[\psi _i]$ -invariant proper free factor system of ${\mathcal {G}}_i$ .

Let $\mathcal T_i \subset \mathcal T$ be the characteristic convex subsets for ${\mathcal {G}}_i$ . By construction of $(\mathcal T, (\delta _i)_{i=1}^n)$ , $\delta _i$ restricts to a ${\mathcal {G}}_i$ -invariant convex pseudometric on $\mathcal T_i$ whose associated ${\mathcal {G}}_i$ -forest can be equivariantly identified with $(\mathcal Y_i, \delta _i)$ . Fix such an identification, and let $\kappa _i \colon \mathcal T_i \to \mathcal Y_i$ denote the natural equivariant collapse map. The stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ are in $\mathbb R(\mathcal Y_i, \delta _i)$ ; their leaves have unique lifts (via $\kappa _i$ ) to $\mathcal T_i \subset \mathcal T$ ; we call these pseudoleaves of $\mathcal L_{\mathcal T}^+[\psi _i]$ . A pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _i]$ is a closed interval in a (representative of a) pseudoleaf with nondegenerate $\kappa _i$ -image in $\mathcal Y_i$ .

Remarkably, the pseudoleaf segments detect weak $\psi _*$ -limits of elements in attracting laminations. Let $\mathcal L_i^+[\psi ]$ be the attracting laminations for $[\psi ]$ determined by $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ (i.e., the closure in $\mathbb R(\mathcal F)$ of the stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ ).

Proposition 3.5. Let $\psi \colon \mathcal F \to \mathcal F$ be an exponentially growing automorphism with a limit pseudoforest $(\mathcal T, (\delta _i)_{i=1}^n)$ . For $1 \le j \le n$ and $\mathcal T$ -loxodromic $x \in \mathcal F$ , the axis for x in $\mathcal T$ contains a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _j]$ if and only if the axis for x in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to the attracting laminations $\mathcal L_j^+[\psi ]$ .

Proof. Let $(\mathcal T, (\delta _i)_{i=1}^n)$ be the limit pseudoforest for the descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of limit forests for $[\psi ]$ . For $i \le n$ , pick a descending sequence $(\tau _{i,j})_{j=1}^{n_i}$ of irreducible train tracks for $[\psi _i]$ rel. ${\mathcal {Z}}_i$ ; we can assume $\tau _{i+1,j}$ is defined on a free splitting of ${\mathcal {Z}}_i$ for some $j<n_{i+1}$ since $[\psi _{i+1}]$ is polynomially growing rel. ${\mathcal {Z}}_i$ (Theorem 2.10). The train tracks $(\tau _{i,j})_{j=1}^{n_i}$ induce a $\psi _i$ -equivariant $\lambda _i$ -Lipschitz PL-map $\tau _i^\circ \colon (\mathcal T_i^\circ , d_i^\circ ) \to (\mathcal T_i^\circ , d_i^\circ )$ . Fix a metric free splitting $(\mathcal T^\star , d^\star )$ of $\mathcal F$ that is the metric blow-up of $(\mathcal T_1^\circ , d_1^\circ )$ , $(\mathcal T_{i+1}^\circ ({\mathcal {Z}}_{i}), d_{i+1}^\circ )$ for $i < n$ , and some metric free splitting $(\mathcal T_{n+1}^\circ , d_{n+1}^\circ )$ of ${\mathcal {Z}}_n$ whose free factor system $\mathcal F[\mathcal T_{n+1}^\circ ]$ is trivial. As the ${\mathcal {G}}_i$ -orbit of $\mathcal T_i^\circ ({\mathcal {Z}}_{i-1})$ is $\tau _i^\circ $ -invariant, the maps $(\tau _i^\circ )_{i=1}^n$ induce a $\psi $ -equivariant PL-map $\tau ^\star $ on $(\mathcal T^\star , d^\star )$ .

Let $x \in \mathcal F$ be a $\mathcal T$ -loxodromic element. By construction, the element x is conjugate to a $\mathcal Y_i$ -loxodromic $y_i \in {\mathcal {G}}_i$ for some $i \le n$ ; let $l_i^\circ $ be the $\mathcal T_i^\circ $ -axis for $y_i$ . If $j = i$ , then the equivalence in the proposition’s statement follows from Theorem 2.10. For the rest of the proof, we prove the equivalence when $j> i$ . As we are going to invoke the same argument in the next proof, we mostly forget that $l_i^\circ $ is a $\mathcal T_i^\circ $ -axis for a $\mathcal Y_i$ -loxodromic element and only use the fact $[l_i^\circ ] \in \mathbb R(\mathcal Y_i, \delta _i)$ (i.e., $l_i^\circ $ projects to a line $\gamma _i$ in $(\mathcal Y_i, \delta _i)$ ).

Suppose the $\mathcal T$ -axis for x contains a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _j]$ for some $j> i$ . Then the $\mathcal T$ -axis for $y_i$ contains a pseudoleaf segment $\sigma _j$ of $\mathcal L_{\mathcal T}^+[\psi _j]$ , and $\kappa _i(\sigma _{j})$ is a point $\circ _i \in \gamma _i$ with nontrivial point stabilizer . In Subsection 2.2.3, we describe how the line $\gamma _i$ in $(\mathcal Y_i, \delta _i)$ and point $\circ _i \in \gamma _i$ determine an algebraic iterated turn $(\epsilon , s_{i+1,1}^{-1}s_{i+1,2}; \varphi _{i+1})_{{\mathcal {G}}_{i+1}}$ . Any iterated turn $(\beta _{i+1,m})_{m \ge 0}$ over $\mathcal T_{i+1}^\circ $ realizing this algebraic iterated turn limits to an interval $[\star _{i+1,1}, \star _{i+1,2}]$ in the metric completion $(\overline {\mathcal Y}_{i+1}, \delta _{i+1})$ by Theorem 2.9, and $[\star _{i+1,1}, \star _{i+1,2}]$ contains $\kappa _{i+1}(\sigma _j)$ .

If $j = i+1$ , then $[\star _{i+1,1}, \star _{i+1,2}] \supset \kappa _{i+1}(\sigma _{i+1})$ is not degenerate and $(\beta _{i+1,m})_{m \ge 0}$ weakly limits to a component of $\mathcal L_{{\mathcal {Z}}_{i+1}}^+[\psi _{i+1}]$ by Theorem 2.9. Otherwise, for $k \ge i+1$ , assume $\kappa _{k}(\sigma _j)$ is a point $\circ _{k}$ in the interval $[\star _{k,1}, \star _{k,2}] \subset \overline {\mathcal Y}_{k}$ corresponding to the algebraic iterated turn $(\epsilon , s_{k,1}^{-1}s_{k,2}; \varphi _{k})_{{\mathcal {G}}_{k}}$ , where $\circ _{k}$ has nontrivial stabilizer $G_{\circ _{k}}$ . By the discussion in Subsection 2.2.4 (and remark after Theorem 2.9), the algebraic iterated turn over ${\mathcal {G}}_{k}$ and point $\circ _{k}$ in $[\star _{k,1}, \star _{k,2}]$ determine an algebraic iterated turn $(\epsilon , s_{k+1,1}^{-1}s_{k+1,2}; \varphi _{k+1})_{{\mathcal {G}}_{k+1}}$ that limits to $[\star _{k+1,1}, \star _{k+1,2}] \subset \overline {\mathcal Y}_j$ ; morevoer, $[\star _{k+1,1}, \star _{k+1,2}]$ contains $\kappa _{k+1}(\sigma _j)$ . By induction, $[\star _{j,1}, \star _{j,2}]$ contains $\kappa _{j}(\sigma _{j})$ . Since $\kappa _{j}(\sigma _{j})$ is not degenerate, any realization $(\beta _{j,m})_{m \ge 0}$ over $\mathcal T_j^\circ $ of the algebraic iterated turn $(\epsilon , s_{j,1}^{-1}s_{j,2}; \varphi _{j})_{{\mathcal {G}}_{j}}$ weakly limits to a component of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ by Theorem 2.9.

In either case ( $j \ge i+1$ ), any realization over $\mathcal T^\star $ of $(\epsilon , s_{j,1}^{-1}s_{j,2}; \varphi _{j})_{{\mathcal {G}}_j}$ weakly limits to (the closure in $\mathbb R(\mathcal F)$ of) a component of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ (bounded cancellation). If $j> i+1$ , any realization over $\mathcal T^\star $ of $(\epsilon , s_{i+1,1}^{-1}s_{i+1,2}; \varphi _{i+1})_{{\mathcal {G}}_{i+1}}$ weakly limits to a component of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ by transitivity. Hence, the shared axis for $y_i$ and x in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to a component of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ . As $\psi _{j*} \colon \mathbb R({\mathcal {G}}_j, {\mathcal {Z}}_j) \to \mathbb R({\mathcal {G}}_j, {\mathcal {Z}}_j)$ acts transitively on the components of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ , the axis for x in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal L_{j}^+[\psi ]$ , the closure in $\mathbb R(\mathcal F)$ of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ .

Conversely, suppose the axis $[l^\star ]$ for $y_i$ (and x) in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal L_j^+[\psi ]$ for some $j> i$ . Using $(\mathcal T^\star , d^\star )$ -coordinates, the axis $\tau _*^{\star m}(l^\star )$ contains arbitrarily $d_j^\circ $ -long leaf segments of $\mathcal L_j^+[\psi ]$ for $m \gg 1$ . So $\tau _*^{\star M}(l^\star )$ has a $\mathcal L_{j}^+[\psi ]$ -leaf segment $I^\star \subset \mathcal T^\star ({\mathcal {Z}}_{j-1})$ with $d_{j}^\circ $ -length for $M \gg 1$ . As $\tau _{j}^\circ $ is a train track on leaves of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ , $\tau _*^{\star m}(l^\star )$ has a $\mathcal L_{j}^+[\psi ]$ -leaf segment surviving from $I^\star $ with $d_j^\circ $ -length $>\lambda _{j}^{M-m}(L-C')$ for $m \ge M$ .

Let $\rho _{i} \colon (\mathcal T^\star ({\mathcal {G}}_{i}), d^\star ) \to (\mathcal T_{i}^\circ , d_{i}^\circ )$ be an arbitrary equivariant PL-map. The $\rho _i$ -image of ${I^\star \subset \tau _*^{\star M}(l^\star )}$ is a vertex $v \in \tau _{i *}^{\circ M}(l_i^\circ )$ with nontrivial stabilizer. Since a nondegenerate part of $I^\star $ survives in $\psi _*^{m}(l^\star )$ for all $m \ge M$ , we have $\tau _i^{\circ (m-M)}(v) \in \tau _{i *}^{\circ m}(l_i^\circ )$ for all $m \ge M$ and $h_i^{-M}(\pi _i^\circ (v)) \in \gamma _i$ has a nontrivial stabilizer , where $h_i$ is the $\psi _i$ -equivariant $\lambda _i$ -homothety on $(\mathcal Y_i, \delta _i)$ . As before, the line $\gamma _i$ , point $h_i^{-M}(\pi _i^\circ (v)) \in \gamma _i$ , and equivariant PL-maps $\rho _{i+1}$ , …, $\rho _{j}$ determine nested iterated turns over $\mathcal T_{i+1}^\circ $ , …, $\mathcal T_{j}^\circ $ limiting to intervals in $\overline {\mathcal Y}_{i+1}$ , …, $\overline {\mathcal Y}_{j}$ . By the computation in the previous paragraph and quasiperiodicity of stable laminations, the last iterated turn over $\mathcal T_j^\circ $ weakly limits to a component of $\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}]$ . So the corresponding interval $[\star _{j,1}, \star _{j,2}] \subset \overline {\mathcal Y}_{j}$ is not degenerate (Theorem 2.9), and the $\mathcal T$ -axis for $y_i$ has an intersection with $\mathcal T_{j}$ whose $\kappa _{j}$ -image is $[\star _{j,1}, \star _{j,2}]$ . By the description of intervals in $\mathcal Y_{j}$ , $[\star _{j,1}, \star _{j,2}]$ contains a leaf segment of $\pi _{j *}^\circ (\mathcal L_{{\mathcal {Z}}_{j}}^+[\psi _{j}])$ ; therefore, the $\mathcal T$ -axes of $y_i$ and x contain pseudoleaf segments of $\mathcal L_{\mathcal T}^+[\psi _j]$ .

In fact, the containment relation on pseudoleaf segments detects the partial order on the set of attracting laminations:

Claim 3.6. For $1\le i, j \le n$ , a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _i]$ contains a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _j]$ if and only if $\mathcal L_i^+[\psi ]$ contains $\mathcal L_j^+[\psi ]$ .

We only sketch the proof as it is almost identical to the proof of Proposition 3.5.

Sketch of proof.

There is nothing to show if $i = j$ . Without loss of generality, assume $i < j$ ; certainly, $\mathcal L_j^+[\psi ]$ does not contain $\mathcal L_i^+[\psi ]$ , and no pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _j]$ can contain a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _i]$ . Let $[l_i^\circ ]$ be an eigenline in $(\mathcal T_i^\circ , d_i^\circ )$ of $[\tau _i^{\circ k}]$ for some $k \ge 1$ , and $l^\star $ be the lift of $l_i^\circ $ to $(\mathcal T^\star , d^\star )$ . The projection $\gamma _i$ (of $l_i^\circ $ ) is a line in $(\mathcal Y_i, \delta _i)$ , and we denote by $l_i \subset \mathcal T_i$ its lift via $\kappa _i$ to a pseudoleaf of $\mathcal L_{\mathcal T}^+[\psi _i]$ .

Suppose the pseudoleaf $l_i$ of $\mathcal L_{\mathcal T}^+[\psi _i]$ contains a pseudoleaf segment $\sigma _j$ of $\mathcal L_{\mathcal T}^+[\psi _j]$ . Then $\kappa _i(\sigma _{j})$ is a point $\circ _i \in \gamma _i$ with nontrivial point stabilizer $G_{\circ _i}$ . By the same argument as in the previous proof, the line $l^\star $ in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal L_j^+[\psi ]$ . Note that $\psi _*^k[l^\star ] = [l^\star ]$ in $\mathbb R(\mathcal F)$ as $[l_i^\circ ]$ is an eigenline for $[\tau _i^{\circ k}]$ ; moreover, $\mathcal L_i^+[\psi ]$ consists of the closures in $\mathbb R(\mathcal F)$ of $[l^\star ]$ , …, $\psi _*^{k-1}[l^\star ]$ since $\mathcal L_i^+[\psi ]$ is a $\psi _*$ -orbit of attracting laminations. So $\mathcal L_i^+[\psi ] \supset \mathcal L_{j}^+[\psi ]$ .

Conversely, suppose $\mathcal L_i^+[\psi ] \supset \mathcal L_{j}^+[\psi ]$ . As $\mathcal L_i^+[\psi ]$ and $\mathcal L_{j}^+[\psi ]$ are $\psi _*$ -orbits of attracting laminations, the line $l^\star $ contains arbitrarily $d_{j}^\circ $ -long leaf segments of $\mathcal L_{j}^+[\psi ]$ . By the same argument as in the previous proof, the pseudoleaf $l_i$ , and hence some pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _i]$ , contains a pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _j]$ .

3.4 Topmost forests

Fix an exponentially growing automorphism $\psi \colon \mathcal F \to \mathcal F$ with a descending sequence $(\mathcal Y_i, \delta _i)_{i=1}^n$ of limit forests, and let $(\mathcal T, (\delta _i)_{i=1}^n)$ be the limit pseudoforest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ . Each limit forest $(\mathcal Y_i, \delta _i)$ has stable laminations $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ for $[\psi _i]$ rel. ${\mathcal {Z}}_i$ . Let $\mathcal L_{\mathcal T}^+[\psi _i]$ be the lifts to $\mathcal T$ of leaves in $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ , $\mathcal L_i^+[\psi ]$ the closure in $\mathbb R(\mathcal F)$ of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ , and $\{\mathcal A_{j}^{top}[\psi _i]\}_{j=1}^{k_i}$ the subset of $\{\mathcal L_j^+[\psi ]\}_{j=i}^n$ consisting of all topmost attracting laminations for $[\psi _i]$ . So $\mathcal A_j^{top}[\psi _i] = \mathcal L_{\iota (i,j)}^+[\psi ]$ for some subsequence $(\iota (i,j))_{j=1}^{k_i}$ of $(j)_{j=i}^n$ with $\iota (i,1) = i$ , and $(\iota (i,j))_{j=2}^{k_i}$ is a subsequence of $(\iota (i+1,j))_{j=1}^{k_{i+1}}$ if $k_i \ge 2$ .

For $i \ge 1$ , we say the ${\mathcal {G}}_i$ -invariant hierarchy $(\delta _j)_{j=i}^n$ on the characteristic convex subsets $\mathcal T_i \subset \mathcal T$ for ${\mathcal {G}}_i$ normalizes to a factored ${\mathcal {G}}_i$ -invariant convex pseudometric $\Sigma _{j=1}^{k_i} \delta _{\iota (i,j)}$ if the ${\mathcal {G}}_{\iota (i,j)}$ -invariant convex pseudometric $\delta _{\iota (i,j)}$ can be extended to a ${\mathcal {G}}_i$ -invariant convex pseudometric, also denoted $\delta _{\iota (i,j)}$ , on $\mathcal T_i$ . The $\mathcal F$ -invariant hierarchy $(\delta _i)_{i=1}^n$ normalizes to $\delta _1$ if (and only if) $k_1=1$ .

We may assume $k_1 \ge 2$ and the ${\mathcal {G}}_2$ -invariant hierarchy $(\delta _i)_{i=2}^n$ normalizes to $\oplus _{j=1}^{k_2} \delta _{\iota (2,j)}$ . Let $\widehat {\mathcal T}_2$ be the $\kappa _1$ -preimage of the characteristic convex subsets $\mathcal Y_1({\mathcal {G}}_2)$ . Suppose , …, $\mathcal F_{1, m}$ are the proper free factor systems of $\mathcal F$ used to construct $(\mathcal Y_1, \delta _1)$ and let $\mathcal T_{1, 1}$ , …, $\mathcal T_{1, m}$ be their corresponding characteristic convex subsets in $\mathcal T$ . By Lemma 2.6, every closed interval in the characteristic convex subsets $\mathcal Y_1(\mathcal F_{1,m})$ is a finite concatenation of leaf segments of $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ . Thus, every closed interval in $\mathcal T_{1, m}$ is a finite concatenation of pseudoleaf segments of $\mathcal L_{\mathcal T}^+[\psi _1]$ and closed intervals in $\mathcal F_{1,m} \cdot \widehat {\mathcal T}_2$ .

Fix $j \in \{2, \ldots , k_1\}$ . Since $\mathcal L_1^+[\psi ]$ does not contain $\mathcal L_{\iota (1,j)}^+[\psi ]$ , Claim 3.6 implies the intersection of any pseudoleaf segment of $\mathcal L_{\mathcal T}^+[\psi _1]$ with $\widehat {\mathcal T}_2$ has 0 diameter with respect to the convex pseudometric $\delta _{\iota (1,j)}$ ; we say that $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ and $\delta _{\iota (1,j)}$ are independent. So the intersection of any closed interval in $\mathcal T_{1,m}$ with $\mathcal F_{1,m} \cdot \widehat {\mathcal T}_2$ has finitely many components that are translates of closed intervals in $\widehat {\mathcal T}_2$ with positive $\delta _{\iota (1,j)}$ -diameter. Thus $\delta _{\iota (1,j)}$ can be extended to an $\mathcal F_{1,m}$ -invariant convex pseudometric on $\mathcal T_{1,m}$ that is mutually singular with $\delta _1$ . By our inductive description of intervals in $\mathcal Y_1$ (Lemma 2.7), the convex pseudometric $\delta _{\iota (1,j)}$ extends equivariantly to $\mathcal T$ as $\lambda _{\iota (1,j)}> 1$ .

As j was arbitrary, the $\mathcal F$ -invariant hierarchy $(\delta _i)_{i=1}^n$ normalizes to the factored convex pseudometric $\oplus _{j=1}^k \delta _{\iota (j)}$ , where and . Let $(\mathcal Y, \oplus _{j=1}^{k} \delta _{\iota (j)})$ be the associated factored $\mathcal F$ -forest. The real $\mathcal F$ -pretrees $\mathcal Y$ are minimal and have trivial arc stabilizers since the pseudometric $\oplus _{j=1}^{k} \delta _{\iota (j)}$ on $\mathcal T$ is convex. The $\psi $ -equivariant $(\lambda _i)_{i=1}^n$ -homothety h induces a $\psi $ -equivariant ⊕ _j=1 ^k λ _ι(j)-dilation on $(\mathcal Y, \oplus _{j=1}^{k} \delta _{\iota (j)})$ : a $\lambda _{\iota (j)}$ -homothety with respect to each factor $\delta _{\iota (j)}$ . By Proposition 3.5, a nontrivial element of $\mathcal F$ is $\delta _{\iota (j)}$ -loxodromic if and only if its axis in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal A_{j}^{top}[\psi _1]$ – here, $\delta _{\iota (j)}$ -loxodromic means the element acts loxodromically on the associated $\mathcal F$ -forest for $\delta _{\iota (j)}$ . The factored $\mathcal F$ -forest $(\mathcal Y, \oplus _{j=1}^{k} \delta _{\iota (j)})$ is the complete topmost limit forest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ . Given any subset of the $\psi _*$ -orbits of topmost attracting laminations for $[\psi ]$ , then one may consider the associated factored $\mathcal F$ -forest for the sum of corresponding pseudometrics:

Theorem 3.7. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\{ \mathcal A_j^{top}[\psi ]\}_{j=1}^k$ a (possibly empty) subset of $\psi _*$ -orbits of topmost attracting laminations for $[\psi ]$ .

Then there is

1. 1. a minimal factored $\mathcal F$ -forest $(\mathcal Y, \oplus _{j=1}^k \delta _j)$ with trivial arc stabilizers;

2. 2. a unique $\psi $ -equivariant expanding dilation $f \colon (\mathcal Y, \oplus _{j=1}^k \delta _j) \to (\mathcal Y, \oplus _{j=1}^k \delta _j)$ ; and

3. 3. for $1 \le j \le k$ , a nontrivial element $x \in \mathcal F$ is $\delta _j$ -loxodromic if and only if its axis in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal A_j^{top}[\psi ]$ .

Fix some index $\iota (j) \neq 1$ , and let $\mathcal X_{1,m}$ be the associated $\mathcal F_{1,m}$ -forest for $\delta _1 \oplus \delta _{\iota (j)}$ on $\mathcal T_{1,m}$ . Two lines in $(\mathcal X_{1,m}, \delta _1 \oplus \delta _{\iota (j)})$ representing leaves in $\mathcal L_{{\mathcal {Z}}_1}^+[\psi ]$ overlap if they have a nondegenerate intersection; overlapping generates an equivalence relation, and each overlapping class is identified with its union in $\mathcal X_{1,m}$ . Let $\operatorname {\underline {supp}}[\psi _1; {{\mathcal {Z}}_1}]$ denote the subgroup system corresponding to the (setwise) stabilizers of overlapping classes $L_{{\mathcal {Z}}_1}^+$ – this subgroup system, called the lower-support of $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ , is $[\psi ]$ -invariant. The system $\operatorname {\underline {supp}}[\psi _1; {{\mathcal {Z}}_1}]$ is not empty as there are $\mathcal Y_1$ -loxodromic elements whose axis in $\mathcal Y_1^*$ is contained in $L_{{\mathcal {Z}}_1}^+$ . Note the number of components in $\operatorname {\underline {supp}}[\psi _1; {{\mathcal {Z}}_1}]$ is at most the number of components in $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ . Let $(\widehat {\mathcal X}_{1,m}({\mathcal {G}}_2), \delta _{\iota (j)})$ be the closure in $(\mathcal X_{1,m}, \delta _1 \oplus \delta _{\iota (j)})$ of the characteristic subforest for ${\mathcal {G}}_{2}$ . By Lemma 2.6 again, intervals in $\mathcal X_1(\mathcal F_{1,n_1})$ are finite concatenations of leaf segments of $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ and closed intervals in $\widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ .

The overlapping classes $L_{{\mathcal {Z}}_1}^+$ and the $\mathcal F_{1,m}$ -orbits of components of $\widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ form an $\mathcal F_{1,m}$ -invariant transverse covering of $\mathcal X_{1,m}$ (see [14, Definition 4.6]). Let $\mathcal S'$ be a simplicial $\mathcal F_{1,m}$ -pretree: vertices (‘component-vertices’) in equivariant bijective correspondence with the components of the transverse covering (overlapping classes $L_{{\mathcal {Z}}_1}^+$ and translates of components of $\widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ ); for each point in $\mathcal X_{1,m}$ contained in exactly two components of the transverse covering, there is an edge between the corresponding component-vertices; for each point contained in more than two components, there is a new vertex (‘intersection-vertex’) and an edge connecting it to each relevant component-vertex. By the blow-up construction, translates of components of $\widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ either coincide or are disjoint. In particular, each intersection-vertex $v \in \mathcal S'$ with a nontrivial stabilizer is adjacent to a unique vertex $w \in \mathcal S'$ corresponding to a component of $\mathcal F_{1,m} \cdot \widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ , and the stabilizer of v fixes w; therefore, we can collapse all such edges $[v,w]$ to form a simplicial $\mathcal F_{1,m}$ -pretree $\mathcal S$ whose intersection-vertices have trivial stabilizers.

The $\mathcal F_{1,m}$ -forest $(\mathcal X_{1,m}, \delta _1 \oplus \delta _{\iota (j)})$ is a graph of actions with skeleton $\mathcal S$ , and the nondegenerate ‘vertex trees’ are the components of the transverse covering [14, Lemma 4.7]. As the $\psi _{1,m}$ -equivariant expanding dilation on $(\mathcal X_{1,m}, \delta _1 \oplus \delta _{\iota (j)})$ permutes the overlapping classes (and components of $\mathcal F_{1,m} \cdot \widehat {\mathcal X}_{1,m}({\mathcal {G}}_{2})$ ), it induces a $\psi _{1,m}$ -equivariant simplicial automorphism $\sigma \colon \mathcal S \to \mathcal S$ that preserves the ‘type’ of a vertex.

Let $\mathcal T_1^\diamond $ be an equivariant blow-up of $(\mathcal T_{1, j})_{j=1}^{m-1}$ , $\mathcal S$ , and $\mathcal X_{1,m}({\mathcal {G}}_2)$ . When the metric $\delta _{\iota (j)}$ is extended appropriately to $\mathcal T_1^\diamond $ , the simplicial automorphisms $(\tau _{1,j})_{j=1}^{m-1}$ , $\sigma $ , and the homothety $f_2$ on $\mathcal X_{1,m}({\mathcal {G}}_2)$ induce a $\psi $ -equivariant $\lambda _{\iota (j)}$ -Lipschitz map $\tau ^\diamond \colon (\mathcal T_1^\diamond , \delta _{\iota (j)}^\diamond ) \to (\mathcal T_1^\diamond , \delta _{\iota (j)}^\diamond )$ that linearly extends $f_2$ . Using $\tau ^\diamond $ -iteration, we define the limit forest $(\mathcal X_1, \delta _{\iota (j)})$ for $[\tau _i]_{i=1}^{n-1}$ , $\sigma $ , and $f_2$ . Like the previous convergence criteria, the proof of the following lemma is postponed to Section 4.3.

Lemma 4.7. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ , , ${\mathcal {G}}$ the nontrivial point stabilizer system for the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ , $[\psi _{{\mathcal {G}}}]$ the $[\psi ]$ -restriction to ${\mathcal {G}}$ , $(\mathcal Y_{{\mathcal {G}}}, \delta )$ a minimal ${\mathcal {G}}$ -forest with trivial arc stabilizers such that $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are independent, $h_{{\mathcal {G}}} \colon (\mathcal Y_{{\mathcal {G}}}, \delta ) \to (\mathcal Y_{{\mathcal {G}}}, \delta )$ a $\psi _{{\mathcal {G}}}$ -equivariant $\lambda $ -homothety, $\mathcal S$ a minimal simplicial $\mathcal F[\mathcal T_{n-1}]$ -forest that is the skeleton for the graph of actions for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ , $\sigma \colon \mathcal S \to \mathcal S$ the corresponding simplicial automorphism, and $(\mathcal X, \delta )$ the limit forest for $[\tau _i]_{i=1}^{n-1}$ , $\sigma $ , and $h_{{\mathcal {G}}}$ .

If $(\mathcal Y', \delta ')$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers, the characteristic subforest of $(\mathcal Y', \delta ')$ for ${\mathcal {G}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {G}}}, \delta )$ and the lower-support $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is $\mathcal Y'$ -elliptic, then the limit of $(\mathcal Y' \psi ^{m}, \lambda ^{-m} \delta ')_{m \ge 0}$ is $(\mathcal X, \delta )$ .

Fix a subset $\{\mathcal A_j^{top}[\psi ]\}_{j=1}^k$ of $\psi _*$ -orbits of topmost attracting laminations for $[\psi ]$ ; a topmost forest for $[\psi ]$ is a factored $\mathcal F$ -forest satisfying the conclusion of Theorem 3.7 with respect to this subset. Lemma 4.7 is enough to prove the universality of topmost forests:

Theorem 3.8. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\{\mathcal A_j^{top}[\psi ]\}_{j=1}^k$ a (possibly empty) subset of $\psi _*$ -orbits of topmost attracting laminations for $[\psi ]$ . Any topmost forest for $[\psi ]$ with respect to the given subset has a unique equivariant dilation to any corresponding topmost limit forest for $[\psi ]$ .

Thus, the factored $\mathcal F$ -forest $(\mathcal Y, \oplus _{j=1}^{k} \delta _{\iota (j)})$ is the complete topmost forest for $[\psi ]$ (up to rescaling of the factors). We omit the proof since we are about to prove something stronger in the next section (see Theorem 3.11).

Suppose $(\mathcal T, (\delta _i)_{i=1}^n)$ and $(\mathcal T', (\delta _i)_{i=1}^{n'})$ are two limit pseudoforests for $[\psi ]$ . Then $n = n'$ as they are exactly the number of $\psi _*$ -orbits of attracting laminations for $[\psi ]$ . Using Theorem 3.7, the hierarchies can be inductively normalized to $(\oplus _{j=1}^{k_i} \delta _{i,j})_{i=1}^d$ and $(\oplus _{j=1}^{k_i} \delta _{i,j}')_{i=1}^d$ , respectively, where d is the length of the longest chain in the partial order of attracting laminations for $[\psi ]$ and $\delta _{i,j}, \delta _{i,j}'$ are indexed by the same $\psi _*$ -orbit $\mathcal A_{i,j}[\psi ]$ of attracting laminations. By inductively invoking Theorem 3.8 and uniqueness of the blow-up construction, the normalized pseudoforests $(\mathcal T, (\oplus _{j=1}^{k_i} \delta _{i,j})_{i=1}^d)$ and ( $\mathcal T', (\oplus _{j=1}^{k_i} \delta _{i,j}')_{i=1}^d)$ are in the same equivariant dilation class, and invariants of this class are invariants of $[\psi ]$ ! In particular, $\mathcal T$ and $\mathcal T'$ are equivariantly pretree-isomorphic.

Corollary 3.9. Any two limit pretrees for an automorphism $\psi \colon \mathcal F \to \mathcal F$ are equivariantly pretree-isomorphic.

We can now define more invariants of an attracting lamination: let A be an attracting lamination for $[\psi ]$ , $\mathcal A[\psi ]$ its $\psi _*$ -orbit and $(\delta _{i,j}, \lambda _{i,j})$ the corresponding pair of pseudometric, and stretch factor in the normalized pseudoforest $(\mathcal T, (\oplus _{j=1}^{k_i} \delta _{i,j})_{i=1}^d)$ ; then is a well-defined stretch factor for A. Now let A be topmost, $\{ \mathcal B_{i{' },j{' }} \}$ be the whole subset of $\psi _*$ -orbits of attracting laminations not contained in $\mathcal A[\psi ]$ , and $(\mathcal T_A, (\oplus _{j{' }=1}^{k_i{' }} \delta _{i{' },j{' }}{' })_{i{' }=1}^{d{' }})$ the associated normalized pseudoforest. Then the upper-support of $\mathcal A[\psi ]$ is the subgroup system of point stabilizers . Unlike the lower-support, the upper-support is always a malnormal subgroup system of finite type. Note that components of the lower-support are conjugate into components of the upper-support.

3.5 Dominating forests

Fix an exponentially growing automorphism $\psi \colon \mathcal F \to \mathcal F$ . Let $A \subset \mathbb R(\mathcal F)$ be an attracting lamination for $[\psi ]$ and $\lambda (A)$ its stretch factor. We say A is dominating if $\lambda (A)> \lambda (A{' })$ whenever $A{' }$ is an attracting lamination for $[\psi ]$ containing A and $A{' } \neq A$ ; topmost attracting laminations are vacuously dominating. We will extend Theorem 3.7 to dominating attracting laminations by mimicking the reasoning in the previous section, focusing only on the changes needed for dominating attracting laminations.

Let $(\mathcal Y_i, \delta _i)_{i = 1}^n$ be a descending sequence of limit forests for $[\psi ]$ , $(\mathcal L_i^+[\psi ])_{i=1}^n$ the corresponding sequence of $\psi _*$ -orbits of attracting laminations for $[\psi ]$ , $(\mathcal T, (\delta _i)_{i=1}^n)$ the limit pseudoforest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ , and $\{\mathcal A_j^{dom}[\psi _i]\}_{j=1}^{k_i}$ the subset of $\{\mathcal L_j^+[\psi ]\}_{j=i}^n$ consisting of all dominating attracting laminations for $[\psi _i]$  – recall that $\mathcal Y_i$ are ${\mathcal {G}}_i$ -pretrees and $[\psi _i]$ is the restriction of $[\psi ]$ to ${\mathcal {G}}_i$ . As before, $\mathcal A_j^{dom}[\psi _i] = \mathcal L_{\iota (i,j)}^+[\psi ]$ for some subsequence $(\iota (i,j))_{j=1}^{k_i}$ of $(j)_{j=i}^n$ with $\iota (i,1) = i$ .

Suppose $k_1 \ge 2$ and the ${\mathcal {G}}_2$ -invariant hierarchy $(\delta _i)_{i=2}^n$ normalizes to the factored ${\mathcal {G}}_2$ -invariant convex pseudometric $\Sigma _{j=1}^{k_2} \delta _{\iota (2,j)}$ on the characteristic convex subsets $\mathcal T_2 \subset \mathcal T$ for ${\mathcal {G}}_2$ . Fix some $j \in \{2, \ldots , k_1\}$ . The previous section discusses how to equivariantly extend $\delta _{\iota (1,j)}$ to $\mathcal T$ when $\mathcal L_1^+[\psi ]$ does not contain $\mathcal L_{\iota (1,j)}^+[\psi ]$ . Assume for the rest of this section that $\mathcal L_{\iota (1,j)}^+[\psi ] \subset \mathcal L_1^+[\psi ]$ ; thus, $\lambda _1 < \lambda _{\iota (1,j)}$ as $\mathcal L_{\iota (1,j)}^+[\psi ]$ is dominating. Let $(\mathcal Y^*, (\delta _1, \delta _{\iota (1,j)}))$ be the associated $\mathcal F$ -pseudoforest for the $\mathcal F$ -invariant 2-level hierarchy $(\delta _1, \delta _{\iota (1,j)})$ and $h^*$ the $\psi $ -equivariant pretree-automorphism on $\mathcal Y^*$ induced by $h \colon \mathcal T \to \mathcal T$ .

Let $\tau _1 \colon (\Gamma _1, d_1) \to (\Gamma _1, d_1)$ be the $\lambda _1$ -Lipschitz topological representative for $\psi $ used to construct $(\mathcal Y_1, \delta _1)$ through iteration. Pick an equivariant blow-up $\Gamma ^\circ $ of $\Gamma _1$ rel. $\mathcal Y^*({\mathcal {Z}}) \subset \mathcal Y^*$ , the characteristic convex subsets for the proper free factor system . Since ${\mathcal {Z}}$ is $\delta _1$ -elliptic, $\delta _{\iota (1,j)}$ is a metric on $\mathcal Y^*({\mathcal {Z}})$ . The blow-up inherits an $\mathcal F$ -invariant 2-level hierarchy $(d_1, \delta _{\iota (1,j)})$ with full support. As $\Gamma _1$ is simplicial, this hierarchy extends to a factored $\mathcal F$ -invariant convex metric $d_1 \oplus \delta _{\iota (1,j)}$ on $\Gamma ^\circ $ .

Let $[\psi _{{\mathcal {Z}}}]$ be the restriction of $[\psi ]$ to ${\mathcal {Z}}$ and $h_{{\mathcal {Z}}}^*$ the $\psi _{{\mathcal {Z}}}$ -equivariant ‘restriction’ of $h^*$ to $(\mathcal Y^*({\mathcal {Z}}), \delta _{\iota (1,j)})$ . For a parameter $c> 0$ , the topological representative $\tau _1$ induces a $\psi $ -equivariant map $\tau _c^\circ $ on $\Gamma ^\circ $ that extends $h_{{\mathcal {Z}}}^*$ and is linear with respect to $(c \, d_1) \oplus \delta _{\iota (1,j)}$ on edges from $\Gamma _1$ . If $c \gg 1$ , then $\tau _c^\circ $ is $\lambda _{\iota (1,j)}$ -Lipschitz with respect to $(c \, d_1 ) \oplus \delta _{\iota (1,j)}$ since $\lambda _1 < \lambda _{\iota (1,j)}$ . Through $\tau _c^\circ $ -iteration, we define a limit forest $(\mathcal Y, \delta _{\iota (1,j)})$ for $\tau _1$ and $h_{{\mathcal {Z}}}^*$ whose characteristic subforest for ${\mathcal {Z}}$ is identified with $(\mathcal Y^*({\mathcal {Z}}), \delta _{\iota (1,j)})$ – up to equivariant isometry, this limit forest is independent of the parameter c; moreover, there is an induced $\psi $ -equivariant $\lambda $ -homothety h on $(\mathcal Y, \delta _{\iota (1,j)})$ that restricts to $h_{{\mathcal {Z}}}^*$ on $\mathcal Y^*({\mathcal {Z}})$ .

We now refine this construction of a limit forest. For $n \ge 1$ , set and . The map $\tau ^\circ \colon (\Gamma ^\circ , d_0^\circ ) \to (\Gamma ^\circ \psi , d_1^\circ )$ is equivariant, and $(1+D)$ -Lipschitz for some $D \ge 0$ . In fact, $\tau ^\circ \colon (\Gamma ^\circ , d_n^\circ ) \to (\Gamma ^\circ \psi , d_{n+1}^\circ )$ is $(1+D r^n)$ -Lipschitz, where . Set ; then $\tau ^{\circ n} \colon (\Gamma ^\circ , d_0^\circ ) \to (\Gamma ^\circ \psi ^n, p_n^{-1} d_n^\circ )$ is equivariant and metric; moreover, the pullback of $p_n^{-1} d_n^\circ $ to $\Gamma ^\circ $ along $\tau ^{\circ n}$ converges to an $\mathcal F$ -invariant pseudometric on $\Gamma ^\circ $ as $n \to \infty $ . Since $|r| < 1$ , the sequence $(p_n)_{n=1}^\infty $ converges, and the pullback of $d_n^\circ $ to $\Gamma ^\circ $ along $\tau ^{\circ n}$ converges to a factored $\mathcal F$ -invariant pseudometric $\delta _1^\circ + \delta _{\iota (1,j)}^\circ $ on $\Gamma ^\circ $ . Let $(\Gamma ^*, \delta _1^\circ + \delta _{\iota (1,j)}^\circ )$ be the associated factored $\mathcal F$ -forest for this factored pseudometric on $\Gamma ^\circ $ – as $\mathcal L_{\iota (1,j)}^+[\psi ] \subset \mathcal L_1^+[\psi ]$ , one can show that $\delta _1^\circ $ and $\delta _{\iota (1,j)}^\circ $ are not mutually singular and $\delta _{\iota (1,j)}^\circ $ is actually a metric on $\Gamma ^*$ . By construction, the characteristic subforest for ${\mathcal {Z}}$ in $(\Gamma ^*, \delta _1^\circ + \delta _{\iota (1,j)}^\circ )$ is equivariantly isometric to $(\mathcal Y^*({\mathcal {Z}}), \delta _{\iota (1,j)})$ . Similarly, the $\mathcal F$ -forest $(\mathcal Y_1, \delta _1)$ is equivariantly isometric to the associated metric space for the pseudometric $\delta _1^\circ $ on $\Gamma ^*$ or $\Gamma ^\circ $ (Corollary 2.11).

Lemma 4.9. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, ${\mathcal {Z}}$ a $[\psi ]$ -invariant proper free factor system, $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ a minimal ${\mathcal {Z}}$ -forest with trivial arc stabilizers, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ with $\mathcal F[\mathcal T_n] = {\mathcal {Z}}$ , $h_{{\mathcal {Z}}} \colon (\mathcal Y_{{\mathcal {Z}}}, \delta ) \to (\mathcal Y_{{\mathcal {Z}}}, \delta )$ a $\psi _{{\mathcal {Z}}}$ -equivariant $\lambda $ -homothety, and $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ and $h_{{\mathcal {Z}}}$ , where $\lambda> \lambda [\tau _n]$ and $[\psi _{{\mathcal {Z}}}]$ is the $[\psi ]$ -restriction to ${\mathcal {Z}}$ .

If $(\mathcal Y', \delta ')$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers and the characteristic subforest of $(\mathcal Y', \delta ')$ for ${\mathcal {Z}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ , then the limit of $(\mathcal Y' \psi ^{m}, \lambda ^{-m} \delta ')_{m \ge 0}$ is $(\mathcal Y, \delta )$ .

Again, the proof is postponed to Section 4.4. Since the restriction of $[\psi ]$ to ${\mathcal {G}}_1$ is polynomially growing rel. ${\mathcal {Z}}$ , Lemma 4.9 implies the characteristic subforests $(\mathcal Y^*({\mathcal {G}}_1), \delta _{\iota (1,j)})$ and $(\Gamma ^*({\mathcal {G}}_1), \delta _{\iota (1,j)}^\circ )$ for ${\mathcal {G}}_1$ are equivariantly isometric. By uniqueness of the blow-up construction, $\Gamma ^*$ is equivariantly pretree-isomorphic to $\mathcal Y^*$ ; through this pretree-isomorphism, we can identify $\delta _{\iota (1,j)}^\circ $ with an extension of $\delta _{\iota (1,j)}$ to an $\mathcal F$ -invariant convex pseudometric (in fact, metric) on $\mathcal Y^*$ . Finally, we can lift $\delta _{\iota (1,j)}$ to an $\mathcal F$ -invariant convex pseudometric on $\mathcal T$ since $\mathcal T$ is an equivariant blow-up of $\mathcal Y^*$ .

As j was arbitrary, the $\mathcal F$ -invariant hierarchy $(\delta _i)_{i=1}^n$ normalizes to the factored $\mathcal F$ -invariant convex pseudometric $\Sigma _{j=1}^k \delta _{\iota (j)}$ , where and . We call the associated factored $\mathcal F$ -forest $(\mathcal Y, \Sigma _{j=1}^{k} \delta _{\iota (j)})$ the complete dominating limit forest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ . This proves the existence part of our main theorem:

Theorem 3.10. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\{\mathcal A_j^{dom}[\psi ]\}_{j=1}^k$ a (possibly empty) subset of $\psi _*$ -orbits of dominating attracting laminations for $[\psi ]$ .

Then there is

1. 1. a minimal factored $\mathcal F$ -forest $(\mathcal Y, \Sigma _{j=1}^k \delta _j)$ with trivial arc stabilizers;

2. 2. a unique $\psi $ -equivariant expanding dilation $f \colon (\mathcal Y, \Sigma _{j=1}^k \delta _j) \to (\mathcal Y, \Sigma _{j=1}^k \delta _j)$ ; and

3. 3. for $1 \le j \le k$ , a nontrivial element $x \in \mathcal F$ is $\delta _j$ -loxodromic if and only if its axis in $\mathbb R(\mathcal F)$ weakly $\psi _*$ -limits to $\mathcal A_j^{dom}[\psi ]$ .

Fix a subset $\{\mathcal A_j^{dom}[\psi ]\}_{j=1}^k$ of $\psi _*$ -orbits of dominating attracting laminations for $[\psi ]$ ; a dominating forest for $[\psi ]$ is a factored $\mathcal F$ -forest satisfying the conclusion of the previous theorem with respect to this subset. Finally, we prove universality:

Theorem 3.11. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism and $\{\mathcal A_j^{dom}[\psi ]\}_{j=1}^k$ a (possibly empty) subset of $\psi _*$ -orbits of dominating attracting laminations for $[\psi ]$ . Any dominating forest for $[\psi ]$ with respect to the given subset has a unique equivariant dilation to any corresponding dominating limit forest for $[\psi ]$ .

Proof. Let $(\mathcal Y_i, \delta _i)_{i=1}^n$ be a descending sequence of limit forests for $[\psi ]$ , $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i] \subset \mathbb R({\mathcal {G}}_i, {\mathcal {Z}}_i)$ the stable laminations for $(\mathcal Y_i, \delta _i)$ , $\mathcal L_i^+[\psi ]$ the closure of $\mathcal L_{{\mathcal {Z}}_i}^+[\psi _i]$ in $\mathbb R(\mathcal F)$ , $\{\mathcal L_{\iota (j)}^+[\psi ]\}_{j=1}^k$ a subset of $\psi _*$ -orbits of dominating attracting laminations, $(\mathcal Y^*, \Sigma _{j=1}^k \delta _{\iota (j)})$ the corresponding dominating limit forest for $(\mathcal Y_i, \delta _i)_{i=1}^n$ , and $(\mathcal Y{' }, \Sigma _{j=1}^k \delta _j{' })$ a corresponding dominating forest for $[\psi ]$ . Turn the factored metrics into hierarchies, and consider the pseudoforests $(\mathcal Y^*, (\delta _{\iota (j)})_{j=1}^k)$ and $(\mathcal Y{' }, (\delta _j{' })_{j=1}^k)$ . By Theorem 3.10(3), $\delta _{\iota (1)}$ and $\delta _1{' }$ have the same maximal elliptic subgroup system ${\mathcal {G}}$ .

For induction, assume the ${\mathcal {G}}$ -pseudoforests $(\mathcal Y^*({\mathcal {G}}), (\delta _{\iota (j)})_{j=2}^k)$ and $(\mathcal Y{' }({\mathcal {G}}), (\delta _j{' })_{j=2}^k)$ are equivariantly homothetic. By uniqueness of the blow-up construction, it is enough to show that the associated $\mathcal F$ -forests for $\delta _{\iota (1)}$ and $\delta _1{' }$ (on $\mathcal Y^*$ and $\mathcal Y{' }$ , respectively) are equivariantly homothetic. So we may assume $k=1$ for the rest of the proof. If $\iota (1) = 1$ , then $(\mathcal Y^*, \delta _{\iota (1)})$ and $(\mathcal Y{' }, \delta _1{' })$ are equivariantly homothetic by Lemma 4.5. Otherwise, $\iota (1)> 1$ and, for induction on complexity, we assume $(\mathcal Y^*({\mathcal {G}}_2), \delta _{\iota (1)})$ and $(\mathcal Y{' }({\mathcal {G}}_2), \delta _1{' })$ are equivariantly homothetic. Either 1) $\mathcal L_{\iota (1)}^+[\psi ] \subset \mathcal L_1^+[\psi ]$ and $\lambda _1 < \lambda _{\iota (1)}$ since $\mathcal L_{\iota (1)}^+[\psi ]$ is dominating; or 2) the lower-support $\operatorname {\underline {supp}}[\psi _1; {{\mathcal {Z}}_1}]$ of $\mathcal L_{{\mathcal {Z}}_1}^+[\psi _1]$ is elliptic in $\mathcal Y^*$ and $\mathcal Y{' }$ by Theorem 3.10(3). The $\mathcal F$ -forests $(\mathcal Y^*, \delta _{\iota (1)})$ and $(\mathcal Y{' }, \delta _1{' })$ are equivariantly homothetic by Lemmas 4.9 and 4.7, respectively, and we are done.

Thus, the factored $\mathcal F$ -forest $(\mathcal Y, \Sigma _{j=1}^{k} \delta _{\iota (j)})$ is the complete dominating forest for $[\psi ]$ .

4 Convergence criteria

This chapter adapts then extends Section 7 of Levitt–Lustig’s paper [19]; they, in turn, gave complete details for the proof sketched by Bestvina–Feighn–Handel in [2, Lemma 3.4].

4.1 Proof of Lemma 4.3

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an expanding irreducible train track $\tau \colon \mathcal T \to \mathcal T$ . Let , $(\mathcal Y_\tau , d_\infty )$ be the limit forest for $[\tau ]$ , $\pi \colon (\mathcal T, d_\tau ) \to (\mathcal Y_\tau , d_\infty )$ the constructed equivariant metric PL-map, $\mathcal L^+[\tau ] \subset \mathbb R(\mathcal T)$ the stable lamination for $[\tau ]$ , and $k \ge 1$ the number of components of $\mathcal L^+[\tau ]$ . Suppose $f \colon (\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ is an equivariant PL-map and $\mathcal L^+[\tau ]$ is in the canonically embedded subspace $\mathbb R(\mathcal Y, \delta ) \subset \mathbb R(\mathcal T)$ .

Claim 4.1 (cf. [19, Lemma 7.1]).

There is a sequence $c(f)$ of positive constants $c_i$ indexed by the components $\Lambda _i^+ \subset \mathcal L^+[\tau ]$ such that

$$\begin{align*}\lim_{m \to \infty} \lambda^{-mk} \delta(f(\tau^{mk}(p)), f(\tau^{mk}(q))) = c_i \, d_\infty(\pi(p),\pi(q))\end{align*}$$

for any leaf segment $[p,q]$ of $\Lambda _i^+$ .

Any two equivariant PL-maps $f, g \colon (\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ are a bounded $\delta $ -distance apart, and ${c(f) = c(g)}$ . So we can define ; note that $c(\mathcal Y, s \, \delta ) = s \, c(\mathcal Y, \delta )$ for $s> 0$ . Without loss of generality, rescale the metric $\delta $ so that f is an equivariant metric PL-map.

Proof. Let (resp. ) be the unique positive right (resp. left) eigenvector for the irreducible transition matrix whose sum of entries is 1 (resp. dot product $\langle \nu ^L, \nu ^R \rangle = k$ ). Suppose $[p,q]$ is a leaf segment (of a component $\Lambda _i^+ \subset \mathcal L^+[\tau ]$ ) with endpoints at vertices of $\mathcal T$ and let be the vector counting the occurrences of $[e]$ in $[p,q]$ : $[e]$ is an $\mathcal F$ -orbit of edges in $\mathcal T$ ; the entries of $v = (v_e)$ are indexed by the $\mathcal F$ -orbits $[e]$ ; and $v_e$ is the number of translates of e in $[p,q]$ . The train track property gives us . Then, as $[p,q]$ is a leaf segment, the positive entries of $v^{(mk)}$ are indexed in the same block $\mathcal B_i = \mathcal B(\Lambda _i^+)$ for all $m \ge 0$ . By Perron’s theorem, if $[e]$ is in the block $\mathcal B_i$ , then

$$\begin{align*}\lim_{m \to \infty} \frac{v_e^{(mk)}}{\lambda^{mk} \langle \nu^L, v \rangle} = \nu_e^R. \end{align*}$$

For small $\epsilon> 0$ , fix $m_\epsilon \gg 1$ such that for every edge $e = [p_e, q_e]$ in $\mathcal T$ – we need the assumption $\mathcal L^+[\tau ] \subset \mathbb R(\mathcal Y, \delta )$ for this. The interval $[\tau ^{(m_\epsilon +m)k}(p), \tau ^{(m_\epsilon +m)k}(q)]$ is a union of $v^{(mk)}_e$ -many translates of $\tau ^{m_\epsilon k}(e)$ , as $[e]$ ranges over all the orbits of edges in $\mathcal T$ . In $\mathcal Y$ , we get

$$\begin{align*}\sum_{[e] \subset \mathcal T} v^{(mk)}_e (\delta_e(m_\epsilon)- 2C[f]) \le \delta(f(\tau^{(m_\epsilon+m)k}(p)), f(\tau^{(m_\epsilon+m)k}(q))) \le \sum_{[e] \subset \mathcal T} v^{(mk)}_e \delta_e(m_\epsilon). \end{align*}$$

Divide by $\lambda ^{(m_\epsilon + m)k} d_\infty (\pi (p), \pi (q)) = \lambda ^{(m_\epsilon + m)k} \langle \nu ^L, v \rangle $ , and let $m \to \infty $ :

$$\begin{align*}\begin{aligned} (1 - 2 \epsilon) \sum_{[e] \in \mathcal B_i} \nu^R_{e} \, & \frac{\delta_e(m_\epsilon)}{\lambda^{m_\epsilon k}} \le \liminf_{m \to \infty} \frac{\delta(f(\tau^{m k}(p)), f(\tau^{m k}(q)))}{\lambda^{mk} d_\infty(\pi(p), \pi(q))} \\ &\le \limsup_{m \to \infty} \frac{\delta(f(\tau^{m k}(p)), f(\tau^{m k}(q)))}{\lambda^{mk} d_\infty(\pi(p), \pi(q))} \le \sum_{[e] \in \mathcal B_i} \nu^R_{e} \, \frac{\delta_e(m_\epsilon)}{\lambda^{m_\epsilon k}}. \end{aligned}\end{align*}$$

Since f is a metric map, we have $\lambda ^{-m_\epsilon k}\delta _e(m_\epsilon ) \le \nu ^L_e$ . So the $\liminf $ and $\limsup $ above are real and equal, and they depend only on the block $\mathcal B_i$ for $\Lambda _i^+$ .

If $\epsilon $ is small, then $\epsilon ^{-1}C[f]> 2C[f] + L$ for some $L> 0$ ; by bounded cancellation,

where $\| v^{(m)} \|_1$ is the sum of the entries in $v^{(m)}$ and $[e]$ is in the same block as $[p,q]$ .

We now relax the restriction that $[p,q]$ is an edge-path (i.e, $p, q$ need not be vertices). For $m \ge 0$ , let $[\bar p_m, \bar q_m]$ be the shortest edge-path containing $[\tau ^{mk}(p), \tau ^{mk}(q)]$ ; for $m, m' \ge 0$ ,

$$\begin{align*}\begin{aligned} \frac{\delta(f(\tau^{mk}(\bar p_{m{'}})), f(\tau^{mk}(\bar q_{m{'}}))) - \lambda^{mk}2}{\lambda^{mk} (d_\infty(\pi(\bar p_{m{'}}), \pi(\bar q_{m{'}})) + 2)} &\le \frac{\delta(f(\tau^{(m+m{'})k}(p)), f(\tau^{(m+m{'})k}(q)))}{\lambda^{(m+m{'})k} d_\infty(\pi(p), \pi(q))} \\ &\le \frac{\delta(f(\tau^{mk}(\bar p_{m{'}})), f(\tau^{mk}(\bar q_{m{'}}))) + \lambda^{mk}2}{\lambda^{mk} (d_\infty(\pi(\bar p_{m{'}}), \pi(\bar q_{m{'}})) - 2)}. \end{aligned}\end{align*}$$

Both upper and lower bounds converge to $c_i$ as $m{' }, m \to \infty $ : $[\bar p_{m{' }}, \bar q_{m{' }}]$ is a leaf segment with endpoints at vertices of $\mathcal T$ , so

$$ \begin{align*} \lim_{m{'} \to \infty} \lim_{m \to \infty} & \frac{\delta(f(\tau^{mk}(\bar p_{m{'}})), f(\tau^{mk}(\bar q_{m{'}}))) \mp \lambda^{mk}2}{ \lambda^{mk}( d_\infty(\pi(\bar p_{m{'}}), \pi(\bar q_{m{'}})) \pm 2)} \\ = & \lim_{m{'} \to \infty} \frac{ c_i \, d_\infty(\pi(\bar p_{m{'}}), \pi(\bar q_{m{'}})) \mp 2}{d_\infty(\pi(\bar p_{m{'}}), \pi(\bar q_{m{'}})) \pm 2} = c_i.\\[-42pt] \end{align*} $$

The next step is extending the claim to all intervals $[p,q] \subset \mathcal T$ . Set and let $d_\infty = \oplus _{i=1}^k d_\infty ^{(i)}$ be the factorization indexed by the components $\Lambda _i^+ \subset \mathcal L^+[\tau ]$ . For convenience, replace $\psi $ with its iterate $\psi ^k$ , $\tau $ with $\tau ^k$ , and $\lambda $ with $\lambda ^k$ .

Claim 4.2 (cf. [19, Lemma 7.2]).

For any $p_1, p_2 \in \mathcal T$ ,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} \delta(f(\tau^{m}(p_1)), f(\tau^{m}(p_2))) = \sum_{i=1}^k c_i \, d_\infty^{(i)}(\pi(p_1), \pi(p_2)). \end{align*}$$

Proof. Let $[p_1, p_2]$ be an interval in $\mathcal T$ and $N(p_1,p_2)$ the number of vertices in $(p_1, p_2)$ . Suppose $\pi (p_1) = \pi (p_2)$ (i.e., $d_\infty (\pi (p_1), \pi (p_2)) = 0$ ). Since f is a metric map, we get

$$\begin{align*}0 \le \lambda^{-m} \delta(f(\tau^m(p_1)), f(\tau^m(p_2))) \le \lambda^{-m} d_\tau(\tau^m(p_1), \tau^m(p_2)), \end{align*}$$

and the limit of the middle term (as $m \to \infty $ ) is $0$ . So we may assume $d_\infty (\pi (p_1), \pi (p_2))> 0$ . For a given $m{' } \ge 0$ , let $[\tau ^{m{' }}(p_1), \tau ^{m{' }}(p_2)]$ be a concatenation of $N{' }+1$ leaf segments $[q_j, q_{j+1}]_{j=0}^{N{' }}$ (of $\Lambda _{i(j)}^+ \subset \mathcal L^+[\tau ]$ ) for some nonegative $N{' } \le N(p_1, p_2)$ and $i(j) \in \{1, \ldots , k\}$ , where $q_0 = \tau ^{m{' }}(p_1)$ and $q_{N{' }+1} = \tau ^{m{' }}(p_2)$ . Then, by Claim 4.1,

$$\begin{align*}\begin{aligned} \limsup_{m \to \infty} &\frac{\delta(f(\tau^{m+m{'}}(p_1)), f(\tau^{m+m{'}}(p_2)))}{\lambda^m} \le \lim_{m \to \infty} \sum_{j=0}^{N{'}}\frac{\delta(f(\tau^m(q_j)),f(\tau^m(q_{j+1})))}{\lambda^m} \\ & = \sum_{j=0}^{N{'}} c_{i(j)} d_\infty(\pi(q_j), \pi(q_{j+1})) = \sum_{i=1}^k c_i d_\tau^{(i)}(\tau^{m{'}}(p_1), \tau^{m{'}}(p_2)), \end{aligned}\end{align*}$$

where the last equality comes from $d_\infty (\pi (q_j), \pi (q_{j+1})) = d_\tau ^{(i(j))}(q_j, q_{j+1})$ since $[q_j, q_{j+1}]$ is a leaf segment. Divide by $\lambda ^{m{' }}$ , let $m{' } \to \infty $ , and invoke the definition of $d_\infty ^{(i)}$ to get

$$\begin{align*}\begin{aligned} \limsup_{m + m{'} \to \infty} &\frac{\delta(f(\tau^{m+m{'}}(p_1)), f(\tau^{m+m{'}}(p_2)))}{\lambda^{m+m{'}}} \le \sum_{i=1}^k c_i d_\infty^{(i)}(\pi(p_1), \pi(p_2)). \end{aligned}\end{align*}$$

Using bounded cancellation, we get a lower bound:

$$\begin{align*}\delta(f(\tau^{m+m{'}}(p_1)), f(\tau^{m+m{'}}(p_2))) \ge \sum_{j=0}^{N{'}}\delta(f(\tau^m(q_j)),f(\tau^m(q_{j+1}))) - 2N{'} C[f], \end{align*}$$

which, after dividing by $\lambda ^{m+m{' }}$ and letting $m \to \infty $ then $m{' } \to \infty $ , leads to

$$\begin{align*}\liminf_{m+m{'} \to \infty} \frac{\delta(f(\tau^{m+m{'}}(p_1)), f(\tau^{m+m{'}}(p_2)))}{\lambda^{m+m{'}}} \ge \sum_{i=1}^k c_i d_\infty^{(i)}(\pi(p_1), \pi(p_2)). \\[-48pt] \end{align*}$$

Like in our construction of limit forests (Section 2.1), let $\delta _m^*$ be the pullback of $\lambda ^{-m}\delta $ via $f \circ \tau ^m$ for $m \ge 0$ . Then $\delta _m^*$ is an $\mathcal F$ -invariant pseudometric on $\mathcal T$ whose associated metric space is equivariantly isometric to $(\mathcal Y\psi ^m, \lambda ^{-m}\delta )$ . By Claim 4.2, the (pointwise) limit $\underset {m \to \infty } \lim \delta _m^*$ is the pullback of $\oplus _{i=1}^k c_i \, d_\infty ^{(i)}$ via $\pi $ . In other words, the sequence $(\mathcal Y\psi ^m, \lambda ^{-m} \delta )_{m \ge 0}$ converges to $(\mathcal Y_\tau , \oplus _{i=1}^k c_i \, d_\infty ^{(i)})$ and we are done.

Lemma 4.3 (cf. [2, Lemma 3.4]).

Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $\psi $ , $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\tau ]$ , and .

If $(\mathcal T, d_\tau ) \to (\mathcal Y, \delta )$ is an equivariant PL-map and the k-component lamination $\mathcal L^+[\tau ]$ is in $\mathbb R(\mathcal Y, \delta ) \subset \mathbb R(\mathcal T)$ , then the sequence $(\mathcal Y \psi ^{mk}, \lambda ^{-mk} \delta )_{m \ge 0}$ converges to $(\mathcal Y_\tau , \oplus _{i=1}^k c_i \, d_\infty ^{(i)})$ , where $d_\infty = \oplus _{i=1}^k \, d_\infty ^{(i)}$ and $c_i> 0$ .

4.2 Proof of Lemma 4.5

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an invariant proper free factor system ${\mathcal {Z}}{' }$ and a descending sequence of irreducible train tracks $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ rel. ${\mathcal {Z}}{' }$ with . Let $\mathcal L_{{\mathcal {Z}}}^+[\psi ] \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ be the k-component stable laminations for $[\psi ]$ rel. , $\mathcal T^\circ $ an equivariant blow-up of the free splittings $(\mathcal T_i)_{i=1}^n$ , $\tau ^\circ \colon \mathcal T^\circ \to \mathcal T^\circ $ a topological representative for $[\psi ]$ induced by $[\tau _i]_{i=1}^n$ , $d^\circ $ an $\mathcal F$ -invariant convex metric on $\mathcal T^\circ $ that extends $d_n$ on $\mathcal T_n$ such that $\tau ^\circ $ is $\lambda $ -Lipschitz on $(\mathcal T^\circ , d^\circ )$ , and $\pi ^\circ \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal Y, \delta )$ the equivariant metric map to a limit forest constructed using $\tau ^\circ $ -iteration. We denote by $d_n$ again the $\mathcal F$ -invariant convex pseudometric on $\mathcal T^\circ $ that extends $d_n$ on $\mathcal T_n$ . Recall that the components $\Lambda _j^+ \subset \mathcal L_{{\mathcal {Z}}}^+[\psi ]$ index the factorizations $d_n = \oplus _{j=1}^k d_n^{(j)}$ and $\delta = \oplus _{j=1}^k \delta _j$ . For convenience, set and , and then replace $\psi $ with $\psi ^k$ , $\tau ^\circ $ with $\tau ^{\circ k}$ , and $\lambda $ with $\lambda ^k$ .

Suppose $(\mathcal Y{' }, \delta {' })$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers, ${\mathcal {Z}}$ is $\mathcal Y{' }$ -elliptic, and $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is in $\mathbb R(\mathcal Y{' }, \delta {' }) \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ . Let $(\mathcal Y_n{' }, \delta {' })$ be the characteristic subforest of $(\mathcal Y{' }, \delta {' })$ for $\mathcal F_n$ and $f_n \colon (\mathcal T_n, d_n) \to (\mathcal Y_n{' }, \delta {' })$ an equivariant PL-map. Extend $f_n$ to an equivariant PL-map $f \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal Y{' }, \delta {' })$ . By Claim 4.1, we can set .

Claim 4.4. For any $p_1, p_2 \in \mathcal T^\circ $ ,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} \delta{'}(f(\tau^{\circ m}(p_1)), f(\tau^{\circ m}(p_2))) = \sum_{j=1}^k c_j \, \delta_j(\pi^\circ(p_1), \pi^\circ(p_2)). \end{align*}$$

Proof. Let $[p_1, p_2]$ be an interval in $\mathcal T^\circ $ and assume $\delta (\pi ^\circ (p_1), \pi ^\circ (p_2))> 0$ without loss of generality. Given Claim 4.2, we may assume $n \ge 2$ . For $m{' } \ge 0$ , the interval $[\tau ^{\circ m{' }}(p_1), \tau ^{\circ m{' }}(p_2)]$ is a concatenation of $\alpha (m{' })$ segments that are in $\mathcal F \cdot \mathcal T_n$ or edges from $\mathcal T_i~(i> 1)$ , where $\alpha (m{' })$ is bounded by a polynomial in $m{' }$ of degree $ \le n-2$ . Set M to be the length of the longest edge from $\mathcal T_i~(i> 1)$ in $(\mathcal T^\circ , d^\circ )$ . For $m{' } \gg 0$ , let $[q_{m{' },l}, q_{m{' },l+1}]_{l=0}^{N(m{' })}$ be the nondegenerate $(\mathcal F \cdot \mathcal T_n)$ -segments. As $\tau ^\circ $ and f are $\lambda $ - and L-Lipschitz, respectively,

$$\begin{align*}\begin{aligned} \delta{'}&(f(\tau^{\circ (m+m{'})}(p_1)), f(\tau^{\circ (m+m{'})}(p_2)) \\ &\le \sum_{l=0}^{N(m{'})} \delta{'}(f_n(\tau_n^{m}(q_{m{'},l})), f_n(\tau_n^{m}(q_{m{'},l+1}))) + \alpha(m{'})\lambda^{m}LM. \end{aligned}\end{align*}$$

Divide by $\lambda ^{m+m{' }}$ , let $m \to \infty $ . Then let $m{' } \to \infty $ , and invoke Claim 4.2 and definition of $\delta _j$ :

$$\begin{align*}\begin{aligned} &\limsup_{m + m{'} \to \infty} \frac{\delta{'}(f(\tau^{\circ (m+m{'})}(p_1)), f(\tau^{\circ{m+m{'}}}(p_2)))}{\lambda^{m+m{'}}} \\ &\le \lim_{m{'} \to \infty} \sum_{l = 0}^{N(m{'})} \sum_{j = 1}^k \frac{ c_j \delta_j(\pi^\circ(q_{m{'},l}), \pi^\circ(q_{m{'},l+1}))}{ \lambda^{m{'}} } \\ &\le \lim_{m{'} \to \infty} \sum_{j = 1}^k \frac{ c_j d_n^{(j)}(\tau^{\circ m{'}}(p_1), \tau^{\circ m{'}}(p_2)) }{ \lambda^{m{'}} } = \sum_{j=1}^k c_j \delta_j(\pi^\circ(p_1), \pi^\circ(p_2)), \end{aligned}\end{align*}$$

using the fact $\pi ^\circ $ is a metric map. The intervals $[\pi ^\circ (q_{m{' },l}),\pi ^\circ (q_{m{' },l+1})]$ contribute at least

$$\begin{align*}\lambda^{m{'}} \delta_j(\pi^\circ(p_1), \pi^\circ(p_2)) -\alpha(m{'})\left( M + 2C[\pi^\circ] \right)\end{align*}$$

to the $\delta _j$ -length of $[\pi ^\circ (\tau ^{\circ m{' }}(p_1)), \pi ^\circ (\tau ^{\circ m{' }}(p_2))]$ . As before, bounded cancellation gives us

$$\begin{align*}\begin{aligned} \delta{'}&(f(\tau^{\circ (m+m{'})}(p_1)), f(\tau^{\circ(m+m{'})}(p_2)))\\ &\ge \sum_{l=0}^{N(m{'})}\delta{'}(f_n(\tau_n^{m}(q_{m{'},l})), f_n(\tau_n^{m}(q_{m{'},l+1}))) - 2\, \alpha(m{'}) C[f]. \end{aligned}\end{align*}$$

Divide by $\lambda ^{m+m{' }}$ and let $m \to \infty $ . Then letting $m{' } \to \infty $ yields

$$\begin{align*}\begin{aligned} &\liminf_{m + m{'} \to \infty} \frac{\delta{'}(f(\tau^{\circ (m+m{'})}(p_1)), f(\tau^{\circ{m+m{'}}}(p_2)))}{\lambda^{m+m{'}}} \\ &\ge \lim_{m{'} \to \infty} \sum_{j = 1}^k c_j \sum_{l = 0}^{N(m{'})} \frac{ \delta_j(\pi^\circ(q_{m{'},l}), \pi^\circ(q_{m{'},l+1}))}{ \lambda^{m{'}} } \ge \sum_{j=1}^k c_j \delta_j(\pi^\circ(p_1), \pi^\circ(p_2)), \end{aligned}\end{align*}$$

where the last inequality comes from the contribution inequality above.

The rest of the argument is the same as in the previous section. Let $\delta _m^*$ be pullback of $\lambda ^{-m} \delta {' }$ via $f \circ \tau ^{\circ m}$ for $m \ge 0$ . By Claim 4.4, the limit $\underset {m \to \infty } \lim \delta _m^*$ is the pullback of $\oplus _{j=1}^k c_j \, \delta _j$ via $\pi ^\circ $ and we are done:

Lemma 4.5. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, ${\mathcal {Z}}{' }$ a $[\psi ]$ -invariant proper free factor system, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ rel. ${\mathcal {Z}}{' }$ with , $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ , $(\mathcal Y{' }, \delta {' })$ a minimal $\mathcal F$ -forest with trivial arc stabilizers, and .

If ${\mathcal {Z}}$ is $\mathcal Y{' }$ -elliptic and the k-component lamination $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is in $\mathbb R(\mathcal Y{' }, \delta {' }) \subset \mathbb R(\mathcal F, {\mathcal {Z}})$ , then the limit of $(\mathcal Y{' } \psi ^{mk}, \lambda ^{-mk} \delta {' })_{m \ge 0}$ is $(\mathcal Y, \oplus _{j=1}^k c_j \, \delta _j)$ , where $\delta = \oplus _{j=1}^k \, \delta _j$ and $c_j> 0$ .

4.3 Sketch of Lemma 4.7

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ . Let $(\tau _i \colon \mathcal T_i \to \mathcal T_i)_{i=1}^n$ be a descending sequence of irreducible train tracks for $[\psi ]$ , , $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ the stable lamination for $[\psi ]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ , $(\mathcal Y_1, \delta _1)$ the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ , , $[\psi _{{\mathcal {G}}}]$ the restriction of $[\psi ]$ to ${\mathcal {G}}$ , $(\mathcal Y_{{\mathcal {G}}}, \delta )$ a minimal ${\mathcal {G}}$ -forest with trivial arc stabilizers, and $h_{{\mathcal {G}}} \colon (\mathcal Y_{{\mathcal {G}}}, \delta ) \to (\mathcal Y_{{\mathcal {G}}}, \delta )$ a $\psi _{{\mathcal {G}}}$ -equivariant $\lambda $ -homothety. Construct the equivariant psuedoforest blow-up $(\mathcal Y_1^*, (\delta _1, \delta ))$ of $(\mathcal Y_1, \delta _1)$ rel. $(\mathcal Y_{{\mathcal {G}}}, \delta )$ and expanding homotheties representing $[\psi ]$ and $[\psi _{{\mathcal {G}}}]$ . For this section, we will assume $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are independent: the pseudoleaf segments for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ in $\mathcal Y_1^*$ have 0 $\delta $ -diameter intersections with $\mathcal Y_{{\mathcal {G}}}$ . Set and $[\psi _n]$ to be the restriction of $[\psi ]$ to $\mathcal F_n$ ; the characteristic convex subset $\mathcal Y_1^*(\mathcal F_n) \subset \mathcal Y_1^*$ has a graph of actions decomposition with vertex forests $\widehat {\mathcal Y}_{{\mathcal {G}}}$ and the overlapping classes for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ .

Let the minimal simplicial $\mathcal F_n$ -forest $\mathcal S$ be the skeleton for the graph of actions for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ . By construction, there is a $\psi _n$ -equivariant simplicial automorphism $\sigma \colon \mathcal S \to \mathcal S$ . The lower-support $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is given by stabilizers of vertices in $\mathcal S$ corresponding to overlapping classes. Construct the equivariant blow-up $\mathcal T^\diamond $ of $(\mathcal T_i)$ , $\mathcal S$ and $\mathcal Y_{{\mathcal {G}}}$ ; then extend the metric $\delta $ to an $\mathcal F$ -invariant convex metric $d \oplus \delta $ on $\mathcal T^\diamond $ so that the $\psi $ -equivariant map $\tau _c^\diamond \colon (\mathcal T^\diamond , (c \, d) \oplus \delta ) \to (\mathcal T^\diamond , (c \, d) \oplus \delta )$ induced by $[\tau _i]_{i=1}^{n-1}$ , $\sigma $ and linearly extending $h_{{\mathcal {G}}}$ is $\lambda $ -Lipschitz for any parameter $c \gg 1$ . Let . For $c \gg 1$ , construct using $\tau _c^\diamond $ -iteration an equivariant metric surjection $\pi _c^\diamond \colon (\mathcal T^\diamond , d_c^\diamond ) \to (\mathcal X, \delta )$ that extends the identification of $(\mathcal Y_{{\mathcal {G}}}, \delta )$ and semiconjugates $\tau _c^\diamond $ to a $\psi $ -equivariant $\lambda $ -homothety on $(\mathcal X, \delta )$ .

Suppose $(\mathcal Y{' }, \delta {' })$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers and whose characteristic subforest for ${\mathcal {G}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {G}}}, \delta )$ . So if we also assume $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ is $\mathcal Y{' }$ -elliptic, then there is an equivariant map $f_c \colon (\mathcal T^\diamond , d_c^\diamond ) \to (\mathcal Y{' }, \delta {' })$ that linearly extends the identification of $(\mathcal Y_{{\mathcal {G}}}, \delta )$ ; this is necessarily a Lipschitz map. Pick any free splitting $\mathcal T$ of $\mathcal F$ with trivial $\mathcal F[\mathcal T]$ . Then any equivariant PL-map $\mathcal T \to \mathcal T^\diamond $ is surjective (by minimality) and composes with $f_c$ to give (up to an equivariant homotopy rel. the vertices) an equivariant PL-map with a cancellation constant. So $f_c$ must have a cancellation constant. The proof of the next claim is a variation of Claim 4.4’s proof:

Claim 4.6. For any $p_1, p_2 \in \mathcal T^\diamond $ ,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} \delta{'}(f_c(\tau_c^{\diamond m}(p_1)), f_c(\tau_c^{\diamond m}(p_2))) = \delta(\pi_c^\diamond(p_1), \pi_c^\diamond(p_2)). \end{align*}$$

Sketch of proof.

For $m{' } \ge 0$ , the interval $[\tau _c^{\circ m{' }}(p_1), \tau _c^{\circ m{' }}(p_2)]$ is a concatenation of $\alpha (m{' })$ segments that are in the orbit of $\mathcal Y_{{\mathcal {G}}}$ or edges from $\mathcal T_i~(i \ge 1)$ , where $\alpha (m{' })$ is bounded by a polynomial in $m{' }$ of degree $\le n-1$ . With an almost identical argument, invoke the definition of $\pi _c^\circ $ to conclude

$$\begin{align*}\lim_{m + m{'} \to \infty} \frac{\delta{'}(f_c(\tau_c^{\circ (m+m{'})}(p_1)), f_c(\tau_c^{\circ{m+m{'}}}(p_2)))}{\lambda^{m+m{'}}} = \delta(\pi_c^\circ(p_1), \pi_c^\circ(p_2)). \end{align*}$$

The setup is simpler as $\tau _c^\circ $ (resp. $f_c$ ) is a $\lambda $ -homothety (resp. isometry) on $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ .

As in the previous section, we have proven the following:

Lemma 4.7. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ , , ${\mathcal {G}}$ the nontrivial point stabilizer system for the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ , $[\psi _{{\mathcal {G}}}]$ the $[\psi ]$ -restriction to ${\mathcal {G}}$ , $(\mathcal Y_{{\mathcal {G}}}, \delta )$ a minimal ${\mathcal {G}}$ -forest with trivial arc stabilizers such that $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are independent, $h_{{\mathcal {G}}} \colon (\mathcal Y_{{\mathcal {G}}}, \delta ) \to (\mathcal Y_{{\mathcal {G}}}, \delta )$ a $\psi _{{\mathcal {G}}}$ -equivariant $\lambda $ -homothety, $\mathcal S$ a minimal simplicial $\mathcal F[\mathcal T_{n-1}]$ -forest that is the skeleton for the graph of actions for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ , $\sigma \colon \mathcal S \to \mathcal S$ the corresponding simplicial automorphism, and $(\mathcal X, \delta )$ the limit forest for $[\tau _i]_{i=1}^{n-1}$ , $\sigma $ , and $h_{{\mathcal {G}}}$ .

If $(\mathcal Y{' }, \delta {' })$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers, the characteristic subforest of $(\mathcal Y{' }, \delta {' })$ for ${\mathcal {G}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {G}}}, \delta )$ , and the lower-support $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ is $\mathcal Y{' }$ -elliptic, then the limit of $(\mathcal Y{' } \psi ^{m}, \lambda ^{-m} \delta {' })_{m \ge 0}$ is $(\mathcal X, \delta )$ .

4.4 Sketch of Lemma 4.9

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an invariant proper free factor system ${\mathcal {Z}}$ and a minimal ${\mathcal {Z}}$ -forest $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ with trivial arc stabilizers. Let $(\tau _i \colon \mathcal T_i \to \mathcal T_i)_{i=1}^n$ be a descending sequence of irreducible train tracks for $[\psi ]$ with $\mathcal F[\mathcal T_n] = {\mathcal {Z}}$ , $d_n$ the eigenmetric on $\mathcal T_n$ for $[\tau _n]$ , and $h_{{\mathcal {Z}}} \colon (\mathcal Y_{{\mathcal {Z}}}, \delta ) \to (\mathcal Y_{{\mathcal {Z}}}, \delta )$ a $\psi _{{\mathcal {Z}}}$ -equivariant $\lambda $ -homothety, where $\lambda> \lambda [\tau _n]$ and $[\psi _{{\mathcal {Z}}}]$ is the $[\psi ]$ -restriction to ${\mathcal {Z}}$ . Set and .

Choose an arbitrary equivariant iterated blow-up $\mathcal T^*$ of $(\mathcal T_i)_{i=1}^n$ and let $\tau ^* \colon \mathcal T^* \to \mathcal T^*$ be the $\psi $ -equivariant topological representative induced by $(\tau _i)_{i=1}^n$ . Extend the metric $d_n$ on $\mathcal T_n$ to an $\mathcal F$ -invariant convex metric $d^*$ on $\mathcal T^*$ so that $\tau ^* \colon (\mathcal T^*, d^*) \to (\mathcal T^*, d^*)$ is $\lambda [\tau _n]$ -Lipschitz. Finally, choose an arbitrary equivariant metric blow-up $(\mathcal T^\circ , d^* \oplus \delta )$ of $(\mathcal T^*, d^*)$ rel. $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ . For a parameter $c> 0$ , the topological representative $\tau ^*$ induces a $\psi $ -equivariant map $\tau _c^\circ $ on $\mathcal T^\circ $ that linearly extends the $\lambda $ -homothety $h_{{\mathcal {Z}}}$ with respect to the metric . As $\lambda> \lambda [\tau _n]$ , the map $\tau _c^\circ $ is $\lambda $ -Lipschitz with respect to $d_c^\circ $ for $c \gg 1$ . Let $(\mathcal Y, \delta )$ be the limit forest for $[\tau _c^\circ ]$ and $\pi _c^\circ \colon (\mathcal T^\circ , d_c^\circ ) \to (\mathcal Y, \delta )$ the equivariant metric surjection constructed through $\tau ^\circ $ -iteration.

Suppose $(\mathcal Y{' }, \delta {' })$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers and whose characteristic subforest for ${\mathcal {Z}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ . Let $f_c \colon (\mathcal T^\circ , d_c^\circ ) \to (\mathcal Y', \delta ')$ be an equivariant map that linearly extends the identification of $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ .

Claim 4.8. For any $p_1, p_2 \in \mathcal T^\circ $ ,

$$\begin{align*}\lim_{m \to \infty} \lambda^{-m} \delta'(f_c(\tau_c^{\circ m}(p_1)), f_c(\tau_c^{\circ m}(p_2))) = \delta(\pi_c^\circ(p_1), \pi_c^\circ(p_2)). \end{align*}$$

Sketch of proof.

For $m{' } \ge 0$ , the interval $[\tau _c^{\circ m{' }}(p_1), \tau _c^{\circ m{' }}(p_2)]$ is a concatenation of $\beta (m{' })$ segments that are in the orbit of $\mathcal Y_{{\mathcal {Z}}}$ or edges from $\mathcal T_i~(i \ge 1)$ , where $\beta (m{' })$ has exponential growth rate $\lambda [\tau _n] < \lambda $ . Proceed just as in the proof of Claim 4.6.

Altogether, we have proven the following:

Lemma 4.9. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, ${\mathcal {Z}}$ a $[\psi ]$ -invariant proper free factor system, $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ a minimal ${\mathcal {Z}}$ -forest with trivial arc stabilizers, $\left (\tau _i \colon \mathcal T_i \to \mathcal T_i\right )_{i=1}^n$ a descending sequence of irreducible train tracks for $[\psi ]$ with $\mathcal F[\mathcal T_n] = {\mathcal {Z}}$ , $h_{{\mathcal {Z}}} \colon (\mathcal Y_{{\mathcal {Z}}}, \delta ) \to (\mathcal Y_{{\mathcal {Z}}}, \delta )$ a $\psi _{{\mathcal {Z}}}$ -equivariant $\lambda $ -homothety, and $(\mathcal Y, \delta )$ the limit forest for $[\tau _i]_{i=1}^n$ and $h_{{\mathcal {Z}}}$ , where $\lambda> \lambda [\tau _n]$ and $[\psi _{{\mathcal {Z}}}]$ is the $[\psi ]$ -restriction to ${\mathcal {Z}}$ .

If $(\mathcal Y{' }, \delta {' })$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers and the characteristic subforest of $(\mathcal Y{' }, \delta {' })$ for ${\mathcal {Z}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ , then the limit of $(\mathcal Y{' } \psi ^{m}, \lambda ^{-m} \delta {' })_{m \ge 0}$ is $(\mathcal Y, \delta )$ .

5 Expanding forests

We finally characterize the expanding forests for an automorphism $\psi \colon \mathcal F \to \mathcal F$ – that is, minimal very small $\mathcal F$ -forests that admit $\psi $ -equivariant expanding homotheties. By the last paragraph in the proof of Corollary 2.11, expanding forests have trivial arc stabilizers. We start with a criterion of nonconvergence that complements Lemmas 4.7 and 4.9.

5.1 Nonconvergence criterion

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ with an invariant proper free factor system ${\mathcal {Z}}$ and a minimal ${\mathcal {Z}}$ -forest $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ with trivial arc stabilizers. Let $\tau \colon \mathcal T \to \mathcal T$ be an expanding irreducible train track for $[\psi ]$ with $\mathcal F[\mathcal T] = {\mathcal {Z}}$ , $d_\tau $ the eigenmetric on $\mathcal T$ for $[\tau ]$ , $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ the stable lamination for $[\psi ]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ , $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ , and $h_{{\mathcal {Z}}} \colon (\mathcal Y_{{\mathcal {Z}}}, \delta ) \to (\mathcal Y_{{\mathcal {Z}}}, \delta )$ a $\psi _{{\mathcal {Z}}}$ -equivariant $\lambda $ -homothety, where $1 < \lambda \le \lambda [\tau ]$ and $[\psi _{{\mathcal {Z}}}]$ is the restriction of $[\psi ]$ to ${\mathcal {Z}}$ .

Set , and denote the restriction of $[\psi ]$ to ${\mathcal {G}}$ by $[\psi _{{\mathcal {G}}}]$ . Since $[\psi _{{\mathcal {G}}}]$ is polynomially growing rel. ${\mathcal {Z}}$ , we can equivariantly include $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ in a minimal ${\mathcal {G}}$ -forest $(\mathcal Y_{{\mathcal {G}}}, \delta )$ with trivial arc stabilizers and extend $h_{\mathcal {Z}}$ to a $\psi _{{\mathcal {G}}}$ -equivariant $\lambda $ -homothety $h_{{\mathcal {G}}} \colon (\mathcal Y_{{\mathcal {G}}}, \delta ) \to (\mathcal Y_{{\mathcal {G}}}, \delta )$ . Construct the equivariant psuedoforest blow-up $(\mathcal Y^*, (d_\infty , \delta ))$ of $(\mathcal Y_\tau , d_\infty )$ rel. $(\mathcal Y_{{\mathcal {G}}}, \delta )$ and the expanding homotheties representing $[\psi ]$ and $[\psi _{{\mathcal {G}}}]$ . Finally, suppose $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are dependent (i.e., the pseudoleaf segments for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ in $\mathcal Y^*$ have some positive $\delta $ -diameter intersections with $\mathcal Y_{{\mathcal {G}}}$ ). We are essentially in the case not covered by Lemmas 4.7 and 4.9.

Choose an iterate $[\tau ^{k}]$ that maps all $\mathcal F$ -orbits of branches in $\mathcal T$ to $[\tau ^{k}]$ -fixed orbits. Pick a branch $e^+$ in $\mathcal T$ ; suppose its basepoint $p \in \mathcal T$ is a vertex with a nontrivial stabilizer. Without loss of generality, assume $\tau ^{k}(e^+) = e^+$ . Use the contraction mapping theorem to decide how to equivariantly attach $\tau ^{k}(e^+)$ to $\mathcal F \cdot \mathcal Y_{{\mathcal {Z}}}$ ; then equivariantly attach $e^+$ to the same point. Now suppose the basepoint p has a trivial stabilizer but $\tau ^{k}(p)$ has a nontrivial one. Then there are finitely many directions $e_1^+, \ldots , e_l^+$ at p. We have described how to attach their images $\tau ^{k}(e_1), \ldots , \tau ^{k}(e_l)$ to the $\mathcal F$ -orbit of $\mathcal Y_{{\mathcal {Z}}}$ ; let $C_p \subset \mathcal F \cdot \mathcal Y_{{\mathcal {Z}}}$ be the convex hull of these attaching points. Equivariantly replace $p \in \mathcal T$ with a copy of $(C_p, \lambda ^{-k} \delta )$ and attach $e_j^+$ to the copy of the attaching point for its $\tau ^{k}$ -image. Finally, if $\tau ^{k}(p)$ has a trivial stabilizer, then there is nothing to do. As $[e^+]$ ranges over all $\mathcal F$ -orbits of branches in $\mathcal T$ , this defines a preferred equivariant metric blow-up $(\mathcal T^\circ , d_\tau \oplus \delta )$ of $(\mathcal T, d_\tau )$ rel. $(\mathcal Y_{\mathcal {Z}}, \delta )$ . The train track $\tau $ induces a $\psi $ -equivariant map $\tau ^\circ \colon (\mathcal T^\circ , d_\tau \oplus \delta ) \to (\mathcal T^\circ , d_\tau \oplus \delta )$ that linearly extends the homothety $h_{{\mathcal {Z}}}$ . The preferred construction guarantees $\tau ^\circ $ is a train track in a sense: $\tau ^{\circ m}$ is injective on the edges from $\mathcal T$ for all $m \ge 1$ .

Suppose $(\mathcal Y, \delta )$ is a minimal $\mathcal F$ -forest with trivial arc stabilizers and whose characteristic subforest for ${\mathcal {Z}}$ is equivariantly isometric to $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ .

Claim 5.1. For some element x in $\mathcal F$ , $ \lambda ^{-m} \| \psi ^m(x) \|_{\delta } \to \infty $ as $m \to \infty $ .

Sketch of proof.

A long leaf segment in $\mathcal T$ contains at least three (unoriented) translates $x_i \cdot e~(1\le i\le 3)$ of an edge e. So is $\mathcal T$ -loxodromic for some $i\ \pmod 3$ . Choose a fundamental domain $[p,q]$ of x acting on its axis that is a leaf segment with endpoints at vertices. Set , and let $f \colon (\mathcal T^\circ , d^\circ ) \to (\mathcal Y, \delta )$ be an equivariant map that linearly extends the identification of $(\mathcal Y_{{\mathcal {Z}}}, \delta )$ .

The assumption that $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ were dependent implies the $\tau ^\circ $ -image of some edge e from $\mathcal T$ has a nondegenerate intersection with $\mathcal F \cdot \mathcal Y_{{\mathcal {Z}}}$ . Fix $m{' } \gg 1$ so that, for some $L> 0$ , $\tau ^{\circ m{' }}(e) \cap \mathcal F \cdot \mathcal Y_{{\mathcal {Z}}}$ has a component with $\delta $ -length $\ge 2C[f] + L$ for all edges from $\mathcal T$ . For $m \ge 0$ , let $\beta (m)$ be the number of edges from $\mathcal T$ in $[\tau ^{\circ m}(p), \tau ^{\circ m}(q)]$ ; note that $\beta (m)$ grows exponentially in m with rate $\lambda [\tau ]$ – the growth of $[\psi ]$ rel. ${\mathcal {Z}}$ . By the train track property of $\tau ^\circ $ and bounded cancellation for f, $\lambda ^{-m}\|\psi ^{m+m{' }}(x)\|_{\delta } \ge \sum _{i=0}^m \beta (i)\lambda ^{-i} L$ tends to infinity as $m \to \infty $ since $\lambda \le \lambda [\tau ]$ .

Thus, there is no $\psi $ -equivariant homothety of $(\mathcal Y, \delta )$ :

Lemma 5.2. Let $\psi \colon \mathcal F \to \mathcal F$ be an automorphism, $\tau \colon \mathcal T \to \mathcal T$ an expanding irreducible train track for $[\psi ]$ , , and $(\mathcal Y, \delta )$ an expanding forest for $[\psi ]$ with stretch factor $\lambda $ . If $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are dependent, then $\lambda> \lambda [\tau ]$ .

5.2 Expanding is dominating

Fix an automorphism $\psi \colon \mathcal F \to \mathcal F$ and an expanding forest $(\mathcal Y, \delta )$ for $[\psi ]$ . Our remaining goal is to generalize Corollary 2.11: $(\mathcal Y, \delta )$ must be some dominating forest for $[\psi ]$ .

Let $\tau \colon \mathcal T \to \mathcal T$ be an expanding irreducible train track for $[\psi ]$ , , $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ the stable lamination for $[\psi ]$ in $\mathbb R(\mathcal F, {\mathcal {Z}})$ , $(\mathcal Y_\tau , d_\infty )$ the limit forest for $[\psi ]$ rel. ${\mathcal {Z}}$ , and .

For induction, assume the characteristic subforest of $(\mathcal Y, \delta )$ for ${\mathcal {G}}$ is equivariantly isometric to the dominating forest for the restriction $[\psi _{{\mathcal {G}}}]$ (of $[\psi ]$ to ${\mathcal {G}}$ ) with respect to some orbits $\{ \mathcal A_i^{dom}[\psi _{{\mathcal {G}}}] \}_{i=1}^k$ with the same stretch factor $\lambda> 1$ ; denote the subforest by $(\mathcal Y_{{\mathcal {G}}}, \delta )$ . Suppose $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are dependent. By Lemma 5.2, $\lambda> \lambda [\tau ]$ and each $\mathcal A_i^{dom}[\psi _{{\mathcal {G}}}] $ is actually a $\psi _*$ -orbit $\mathcal A_i^{dom}[\psi ]$ of dominating attracting laminations for $[\psi ]$ . By Lemma 4.9, $(\mathcal Y, \delta )$ is equivariantly isometric to the dominating forest for $[\psi ]$ with respect to $\{ \mathcal A_i^{dom}[\psi ] \}_{i=1}^k$ .

We may now assume $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ are independent. So $\mathcal A_i^{dom}[\psi _{{\mathcal {G}}}] $ is a $\psi _*$ -orbit $\mathcal A_i^{dom}[\psi ]$ of dominating attracting laminations for $[\psi ]$ . Let $\mathcal A_0^{dom}[\psi ] \subset \mathbb R(\mathcal F)$ be the closure of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $(\mathcal Y^*, d_\infty \oplus \delta )$ the unique equivariant metric blow-up of $(\mathcal Y_\tau , d_\infty )$ rel. $(\mathcal Y_{{\mathcal {G}}}, \delta )$ that admits a $\psi $ -equivariant expanding dilation. By construction, the blow-up is equivariantly isometric to the dominating forest for $[\psi ]$ with respect to $\{ \mathcal A_i^{dom}[\psi ] \}_{i=0}^k$ . Recall that independence of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ implies $\mathcal Y^*$ is a graph of actions with vertex forests coming from $\widehat {\mathcal Y}_{{\mathcal {G}}}$ and overlapping classes for $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ – these are ${\mathcal {G}}$ - and $\operatorname {\underline {supp}}[\psi; {\mathcal {Z}}]$ -forests, respectively; let $\mathcal S$ be the skeleton for this graph of actions.

If the lower-support $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ is $\mathcal Y$ -elliptic, then $(\mathcal Y, \delta )$ is equivariantly isometric to the associated $\mathcal F$ -forest for $\delta $ on $\mathcal Y$ by Lemma 4.7; in particular, $(\mathcal Y, \delta )$ is equivariantly isometric to the dominating forest for $[\psi ]$ with respect to $\{ \mathcal A_i^{dom}[\psi ] \}_{i=1}^k$ . Otherwise, $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ is not $\mathcal Y$ -elliptic. Let $\mathcal T' \subset \mathcal T$ be the characteristic convex subset for the upper-support $\operatorname {\overline {supp}} \mathcal L^+[\psi ]$ of $\mathcal L^+[\psi ]$ (defined at the end of Section 3.4) and $[\psi ']$ the restriction of $[\psi ]$ to the upper-support. Independence of $\mathcal L_{{\mathcal {Z}}}^+[\psi ]$ and $\delta $ implies is $\mathcal Y$ -elliptic. So the characteristic subforests of $(\mathcal Y, \delta )$ and $(\mathcal Y_\tau , d_\infty )$ for the upper-support $\operatorname {\overline {supp}} \mathcal L^+[\psi ]$ are expanding forests for $[\psi ']$ rel. ${\mathcal {Z}}'$ ; by Corollary 2.11, they are equivariantly homothetic and $\lambda [\tau ] = \lambda $ . Thus, the characteristic subforests of $(\mathcal Y, \delta )$ and $(\mathcal Y_\tau , c\,d_\infty )$ for $\operatorname {\underline {supp}}[\psi; {{\mathcal {Z}}}]$ are equivariantly isometric for some $c> 0$ . A minor modification of Lemma 4.7 implies $(\mathcal Y, \delta )$ is equivariantly isometric to $(\mathcal Y^*, c\, d_\infty \oplus \delta )$ – details are left to the reader; therefore, $(\mathcal Y, \delta )$ is equivariantly isometric to the dominating forest for $[\psi ]$ with respect to $\{ \mathcal A_i^{dom}[\psi ] \}_{i=0}^k$ .

Generally, $[\psi ]$ has a descending sequence of irreducible train tracks $(\tau _i \colon \mathcal T_i \to \mathcal T_i)_{i=1}^n$ . If $(\mathcal Y, \delta )$ is degenerate, then there is nothing to show. Otherwise, the $\psi $ -expanding homothety on $(\mathcal Y, \delta )$ implies $\lambda [\tau _n]> 1$ . Set and . The preceding discussion proves that the characteristic subforest of $(\mathcal Y, \delta )$ for $\mathcal F_n$ is equivariantly isometric to some dominating forest for the restriction $[\psi _n]$ . Lemma 4.9 implies $(\mathcal Y, \delta )$ is equivariantly isometric to some dominating forest for $[\psi ]$ . Conversely, it follows from Theorem 3.10(2) that the dominating forest for $[\psi ]$ with respect to a subset of $\psi _*$ -orbits of dominating attracting laminations with the same stretch factor is an expanding forest for $[\psi ]$ :

Theorem 5.3. An $\mathcal F$ -forest $(\mathcal Y, \delta )$ is an expanding forest for an automorphism $\psi \colon \mathcal F \to \mathcal F$ if and only if it is equivariantly isometric to the dominating forest for $[\psi ]$ with respect to a subset of $\psi _*$ -orbits of dominating attracting laminations with the same stretch factor.