A report on an ergodic dichotomy

Abstract

We establish (some directions of) a Ledrappier correspondence between Hölder cocycles, Patterson–Sullivan measures, etc for word-hyperbolic groups with metric-Anosov Mineyev flow. We then study Patterson–Sullivan measures for $\vartheta $-Anosov representations over a local field and show that these are parameterized by the $\vartheta $-critical hypersurface of the representation. We use these Patterson–Sullivan measures to establish a dichotomy concerning directions in the interior of the $\vartheta $-limit cone of the representation in question: if ${\mathsf {u}}$ is such a half-line, then the subset of ${\mathsf {u}}$-conical limit points has either total mass if $|\vartheta |\leq 2$ or zero mass if $|\vartheta |\geq 4.$ The case $|\vartheta |=3$ remains unsettled.

1 Introduction

Let ${\mathsf {G}}$ be the real points of a semi-simple real-algebraic group of the non-compact type. The (Riemannian) globally symmetric space ${\mathsf {X}}$ associated to ${\mathsf {G}}$ is non-positively curved, its visual boundary $\partial _\infty {\mathsf {X}}$ is a union of compact ${\mathsf {G}}$ -orbits, parameterized by directions in a (fixed beforehand) closed Weyl chamber ${\mathfrak a}^+$ of ${\mathfrak g}.$ The ${\mathsf {G}}$ -orbit associated to a direction ${\mathsf {u}}\in \mathbb {P}({\mathfrak a}^+)$ is ${\mathsf {G}}$ -equivariantly identified with the flag space $\mathcal {F}_{\vartheta _{\mathsf {u}}}$ of ${\mathsf {G}},$ where $\vartheta _{\mathsf {u}}$ is the subset of simple roots that do not vanish on ${\mathsf {u}}.$

Let now be a finitely generated group and a representation with discrete image. A fundamental object of study is the limit set $\mathrm {L}_{\rho }$ of on the visual boundary $\partial _\infty {\mathsf {X}},$ defined as the set of accumulation points of an (any) orbit on the natural compactification ${\mathsf {X}}\cup \partial _\infty {\mathsf {X}}.$

When is Zariski-dense, this object has the following topological description by Benoist [7]: the action of on each flag space $\mathcal F_\vartheta $ has a smallest closed invariant set, called the limit set on $\mathcal F_\vartheta $ and denoted by $\mathrm {L}_\rho ^\vartheta $ ; additionally, one has the limit cone $\mathscr L_\rho \subset {\mathfrak a}^+$ of defined as the subset of ${\mathfrak a}^+$ of accumulation points of sequences of the form

$$ \begin{align*}t_na(\rho(\gamma_n)),\end{align*} $$

where $t_n\in \mathbb {R}_+$ converges to goes to infinity, and $a:{\mathsf {G}}\to {\mathfrak a}^+$ is the Cartan projection. It is a convex cone with non-empty interior and the limit set on $\partial _\infty {\mathsf {X}}$ is the ‘fibration’ over $\mathbb {P}(\mathscr L_\rho ),$ whose fiber over a given direction ${\mathsf {u}}\in \mathbb {P}(\mathscr L_\rho )$ is the limit set $\mathrm {L}_\rho ^{\vartheta _{\mathsf {u}}}$ of on $\mathcal F_{\vartheta _{\mathsf {u}}}.$

Inspired by Sullivan’s [68] work on the rank-1 case, one may seek to distinguish the subset of conical points of $\mathrm {L}_\rho $ , that is, points on the limit set that are approached in a uniform manner by elements of the orbit However, the definition of uniform depends on:

• • the type of ${\mathsf {G}}$ -orbit the point lies in—a point $x\in \mathrm {L}_\rho ^\vartheta $ is conical if there exists a (to be called conical) sequence converging to x such that for every $y\in \mathrm {L}^{\operatorname {\mathrm {i}}\vartheta }_\rho $ (here we let $\operatorname {\mathrm {i}}:{\mathfrak a}\to {\mathfrak a}$ be the opposition involution and $\operatorname {\mathrm {i}}\vartheta :=\vartheta \circ \operatorname {\mathrm {i}}$ ) in general position with x, the sequence $\gamma _n^{-1}(y,x)$ has compact closure on the space of pairs of flags in general position $\mathcal F_\vartheta ^{\kern2pt(2)};$

• • the specific direction ${\mathsf {u}}\in \mathbb {P}(\mathscr L_\rho ),$ associated to the given point—fix a norm $\|\,\|$ on ${\mathfrak a}$ and define the tube of size $r>0$ as the r-tubular neighborhood

  $$ \begin{align*}{\mathbb T}_r({\mathsf{u}})=\{v\in{\mathfrak a}:B(v,r)\cap{\mathsf{u}}\neq\emptyset\},\end{align*} $$
  then $x\in \mathrm {L}^{\vartheta _{\mathsf {u}}}_\rho $ is ${\mathsf {u}}$ -conical if there exists $r>0$ and a conical sequence $\gamma _n\to x$ such that for all n, one has
  $$ \begin{align*}a(\rho(\gamma_n))\in{\mathbb T}_r({\mathsf{u}}).\end{align*} $$

A measurable description has been recently established by Burger et al [20] for ${\mathsf {u}}$ -conical points of Zariski-dense subgroups: under some extra assumptions, the Patterson–Sullivan measure associated to the direction ${\mathsf {u}}$ charges totally the subset of ${\mathsf {u}}$ -conical points if and only if ${\mathsf {G}}$ has rank $\leq 3;$ if $\operatorname {\mathrm {rank}}{\mathsf {G}}\geq 4$ , then the subset of ${\mathsf {u}}$ -conical points has zero mass.

In this paper, we will also study a measurable description of ${\mathsf {u}}$ -conical limit points, but for general Anosov representations, a class introduced by Labourie [43] for fundamental groups of closed negatively curved manifolds and generalized by Guichard and Wienhard [35] for arbitrary (finitely generated) word-hyperbolic groups. Thanks to the recent work by Kapovich, Leeb, and Porti [40] (see also those of Bochi, Potrie, and Sambarino [10] and Guéritaud et al [34]), we can define them as follows, see §5.2.

Definition 1.0.1. Let $\vartheta \subset {{\Delta }}$ be a non-empty subset of simple roots and denote by $|\,|$ the word length on for some (fixed) symmetric generating set. A representation is $\vartheta $ -Anosov if there exist positive constants $c,\mu $ such that for all and $\sigma \in \vartheta $ , one has

$$ \begin{align*}\sigma(a(\rho(\gamma)))\geq \mu|\gamma|-c.\end{align*} $$

A key feature of a $\vartheta $ -Anosov representation $\rho $ is that

is necessarily word-hyperbolic and there exist continuous $\rho $ -equivariant limit maps (Proposition 5.2.3) defined on its Gromov boundary,

such that the flags $\xi ^{\operatorname {\mathrm {i}}\vartheta }(x)$ and $\xi ^\vartheta (y)$ are in general position whenever $x\neq y.$

We begin by studying the Patterson–Sullivan theory for these groups. Fix then $\vartheta \subset {{{\Delta }}},$ let

$$ \begin{align*}{\mathfrak a}_\vartheta=\bigcap_{\sigma\in{{{\Delta}}}-\vartheta}\ker\sigma\end{align*} $$

be the center of the associated Levi group and let $p_\vartheta :{\mathfrak a}\to {\mathfrak a}_\vartheta $ be the projection invariant under the subgroup of the Weyl group point-wise fixing ${\mathfrak a}_\vartheta $ (see §4.2). The dual space $({\mathfrak a}_\vartheta )^*$ sits naturally as the subspace of ${\mathfrak a}^*$ of $p_\vartheta $ -invariant linear forms. It is spanned by the fundamental weights of the elements in $\vartheta $ :

$$ \begin{align*}({\mathfrak a}_\vartheta)^*=\langle\{\varpi_\sigma|{\mathfrak a}_\vartheta:\sigma\in\vartheta\}\rangle.\end{align*} $$

Let us write $a_\vartheta $ for the composition $a_\vartheta =p_\vartheta \circ a:{\mathsf {G}}\to {\mathfrak a}_\vartheta .$

Let $\beta :{\mathsf {G}}\times \mathcal F_{{{\Delta }}}\to {\mathfrak a}$ be the Buseman–Iwasawa cocycle of ${\mathsf {G}}$ introduced by Quint [62] (see §4.6). The map $\beta _\vartheta =p_\vartheta \circ \beta $ factors as a cocycle $\beta _\vartheta :{\mathsf {G}}\times \mathcal F_\vartheta \to {\mathfrak a}_\vartheta .$

Definition 1.0.2. A Patterson–Sullivan measure for $\rho $ on $\mathcal F_\vartheta $ is a probability measure $\nu $ on $\mathcal F_\vartheta $ such that there exists $\varphi \in {\mathfrak a}_\vartheta ^*$ with, for every

$$ \begin{align*}\frac{d\rho(\gamma) _*\nu}{\,d\nu}(\cdot)=q^{-\varphi(\,\beta_\vartheta(\rho(\gamma)^{-1},\cdot))}.\end{align*} $$

For $\varphi \in ({\mathfrak a}_\vartheta )^*$ , denote

and, inspired by Quint’s growth indicator [61], consider the $\vartheta $ -critical hypersurface

$$ \begin{align*}\mathcal Q_{\vartheta,\rho} =\{\varphi\in({\mathfrak a}_\vartheta)^*:\delta^\varphi=1\}.\end{align*} $$

Let us define the $\vartheta $ -limit cone of $\rho ,$ denoted by $\mathscr L_{\vartheta ,\rho },$ as the asymptotic cone of the projections

that is, all limits of sequences of the form $t_n a_\vartheta (\rho (\gamma _n)),$ where $\gamma _n\to \infty $ in

and $t_n\to 0$ in $\mathbb {R}_+.$

In the real case, if is Zariski-dense, then Benoist’s aforementioned result implies that $\mathscr L_{\vartheta ,\rho }$ has non-empty interior. However, for arbitrary local fields, this is no longer the case (even assuming Zariski-density and Anosov). We aim to work on this more general context, so let us assume now that ${\mathsf {G}}$ is (the $\mathbb {K}$ -points of) a semi-simple algebraic group over a local field $\mathbb {K}.$ We refer the reader to §4 for the analogous definitions, where ${\mathfrak a}_\vartheta $ is replaced by the real vector space ${\mathsf {E}}_\vartheta ,$ etc.

Let $\mathrm {Ann}(\mathscr L_{\vartheta ,\rho })$ be the annihilator of the $\vartheta $ -limit cone and denote by $\pi ^\vartheta _\rho :({\mathsf {E}}_\vartheta )^*\to ({\mathsf {E}}_\vartheta )^*/\mathrm {Ann}(\mathscr L_{\vartheta ,\rho })$ the quotient projection.

Theorem A. Let be $\vartheta $ -Anosov. Then, $\mathcal Q_{\vartheta ,\rho }$ is a closed co-dimension-one analytic sub-manifold of $({\mathsf {E}}_\vartheta )^*$ that bounds a convex set; moreover, the projection $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ is also a closed co-dimension-one analytic sub-manifold boundary of a strictly convex set. For each $\varphi \in \mathcal Q_{\vartheta ,\rho }$ , there exists a unique Patterson–Sullivan measure $\nu ^\varphi $ with support on The map $\varphi \mapsto \nu ^\varphi $ is an analytic homeomorphism between the projection $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ and the space of Patterson–Sullivan measures on $\mathcal F_\vartheta $ whose support is contained in Such Patterson–Sullivan measures are ergodic and pairwise mutually singular.

We refer the reader to Corollary 5.5.3 and Proposition 5.9.2 for the proofs of the above statements.

The fact that both $\mathcal Q_{\vartheta ,\rho }$ and $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ are closed analytic hypersurfaces is stated by Potrie and Sambarino [57, Proposition 4.11] for $\mathbb {K}=\mathbb {R}$ with essentially the same arguments. The parameterization of Patterson–Sullivan measures by $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ was previously established by Lee and Oh [46, Theorem 1.3] for $\mathbb {K}=\mathbb {R}, \vartheta ={{{\Delta }}}$ , and assuming Zariski-density of Existence and ergodicity was previously established, for $\mathbb {K}=\mathbb {R}$ , by Dey and Kapovich [29, Main Theorem] for $\operatorname {\mathrm {i}}$ -invariant functionals $\varphi \in ({\mathfrak a}^+)^*\cap ({\mathfrak a}_\vartheta )^*$ and $\operatorname {\mathrm {i}}$ -invariant subsets $\vartheta ;$ and by Sambarino [66, Corollary 4.22] for arbitrary functionals but Zariski-dense representations of fundamental groups of negatively curved manifolds. Existence of Patterson–Sullivan measures has also been established by Canary, Zhang, and Zimmer [21] in the real case for relative Anosov representations.

We keep the discussion for $\mathbb {K}=\mathbb {R}$ since this is essential in the following result. Consider $\varphi \in \mathcal Q_{\vartheta ,\rho }$ with associated Patterson–Sullivan measure $\mu ^\varphi .$ Via the duality

$$ \begin{align*}\operatorname{\mathrm{Gr}}_{\dim{\mathfrak a}_\vartheta-1}(({\mathfrak a}_\vartheta)^*)\to\mathbb{P}({\mathfrak a}_\vartheta),\end{align*} $$

the tangent space ${\mathsf {T}}_{\varphi }\mathcal Q_{\vartheta ,\rho }$ gives a direction ${\mathsf {u}}_\varphi $ of $\mathbb {P}({\mathfrak a}_\vartheta )$ contained in the relative interior of the limit cone $\mathscr L_{\vartheta ,\rho }$ (Corollary 5.9.1). We then further investigate the $\mu ^\varphi $ -mass of ${\mathsf {u}}_\varphi $ -conical points on

Since we are dealing with the limit cone on ${\mathfrak a}_\vartheta $ (and not on ${\mathfrak a}$ as before), ${\mathsf {u}}_\varphi $ -conical points are yet to be defined. It is standard that every point

is conical. (This follows from the fact that every point

is conical and the existence of the equivariant limit maps for $\rho $ ). Let us say it is further ${\mathsf {u}}_\varphi $ -conical if there exists a conical sequence (for x) as above and $r>0$ such that $a_\vartheta (\rho (\gamma _n))\in {\mathbb T}_r({\mathsf {u}}_\varphi ).$ Denote by

the subset

Theorem B. (Theorem 5.13.3)

Let $\mathbb {K}=\mathbb {R}$ and assume $\rho $ is $\vartheta $ -Anosov and Zariski-dense. If $|\vartheta |\leq 2$ , then if $|\vartheta |\geq 4$ , then

The case $|\vartheta |=3$ is sadly presently untreated. The missing fact that would make our technique directly apply is an ergodicity result for translation skew-products over metric-Anosov flows where the abelian group is isomorphic to $\mathbb {R}^2=\mathbb {R}^{|\vartheta |-1},$ more precisely, we need equivalence between ergodicity and $\dim V\leq 2$ in Corollary 2.5.5.

When $\vartheta ={{{\Delta }}},$ a stronger version of Theorem B dealing also with the case $|{{{\Delta }}}|=3$ was previously established by Burger et al [20, Theorem 1.6]. It is likely that the combination of their techniques and ours settles the missing $|\vartheta |=3$ case.

1.1 General strategy for Theorem B

Let us briefly explain the proof of Theorem B, which we believe is the main contribution of this work. The main ingredient is a precise description of the $\vartheta $ -parallel sets dynamics of ${\mathsf {G}}.$ If $(x,y)\in \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ are in general position, the associated parallel set is a subset of ${\mathsf {X}} $ consisting of the union of totally geodesic maximal flats ${\mathsf {p}}$ of ${\mathsf {X}}$ whose associated complete flags in the Furstenberg boundary ${\mathsf {p}}(-{\mathfrak a}^+)$ and ${\mathsf {p}}({\mathfrak a}^+)$ contain respectively x and y as a partial flag. This parallel set is a reductive symmetric space and the associated dynamical system consists of moving along its center.

More concisely, if one considers the space $\mathcal F^{(2)}_\vartheta \subset \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ of transverse flags, then the space $\mathcal F^{(2)}_\vartheta \times {\mathfrak a}_\vartheta $ carries a ${\mathsf {G}}$ -action (on the left) given by

$$ \begin{align*}g(x,y,v)=(gx,gy,v-\beta_\vartheta(g,y)),\end{align*} $$

and an ${\mathfrak a}_\vartheta $ -action (on the right) by translation on the last coordinate.

Observe however that the left-action of on $\mathcal F^{(2)}_\vartheta \times {\mathfrak a}_\vartheta $ need not be proper. For $\vartheta $ -Anosov groups though, one finds an ${\mathfrak a}_\vartheta $ -invariant subset which is also -invariant and on which this latter action is proper (§§3.5.2 and 5.3.2). Its quotient by will be denoted, throughout this introduction, by $\mathcal O_{\vartheta ,\rho }.$

For each $\varphi \in \mathcal Q_{\vartheta ,\rho }$ , the space $\mathcal O_{\vartheta ,\rho }$ will carry a $\varphi $ -Bowen–Margulis measure $\Omega ^\varphi $ invariant under the directional flow $\unicode{x3c9} ^\varphi :\mathcal O_{\vartheta ,\rho }\to \mathcal O_{\vartheta ,\rho }$ along $u_\varphi \in {\mathsf {u}}_\varphi ,$ defined by (the induction on the quotient by of)

$$ \begin{align*}(x,y,v)\mapsto(x,y,v-tu_\varphi).\end{align*} $$

The idea generalizing that of Sambarino [66] is that $\unicode{x3c9} ^\varphi $ is conjugated to a skew-product over a metric-Anosov flow $\phi ^\varphi =(\phi _t^\varphi :{\unicode{x3c7} ^{\varphi }}\to {\unicode{x3c7} ^{\varphi }})_{t\in \mathbb {R}}$ on a compact metric space ${\unicode{x3c7} ^{\varphi }}.$ This is established in §5.12 and was previously established by Carvajales [23, Appendix] for $\vartheta ={{{\Delta }}}.$

Remark 1.1.1. The flow $\phi ^\varphi $ plays a central role in this work. We propose to name it the  $\varphi $ -refraction flow of $\rho ,$ because one may think that the projection on the base ${\unicode{x3c7} ^{\varphi }}$ refracts the orbits of $\unicode{x3c9} ^\varphi $ (almost all of them wondering when $|\vartheta |\geq 4$ ) to bind them in a compact space ${\unicode{x3c7} ^{\varphi }}$ and obtain non-trivial dynamical behavior. (In spite of being topologically mixing, these flows are wondering in a measurable sense, that is, almost every point belongs to a subset of positive measure with bounded return times.) Also, the term geodesic flow has too many meanings on this setting (the geodesic flow of the geodesic flow of the locally symmetric space the geodesic flow of a projective-Anosov representation associated to $\rho $ by Plucker embeddings, and so forth).

An ergodicity result for skew-products over metric-Anosov flows (see §2.5) gives an ergodic versus totally dissipative dichotomy for $\unicode{x3c9} ^\varphi $ according to $|\vartheta |\leq 2$ or $|\vartheta |\geq 4.$ Here, the base field $\mathbb {K}=\mathbb {R}$ and Zariski-density of $\rho $ are essential, since Benoist’s [8] density of Jordan projections does not hold for non-Archimedean $\mathbb {K}.$ This dichotomy is reminiscent of Sullivan’s [68] conservative versus totally dissipative dichotomy in rank 1. Observe again the untreated case $|\vartheta |=3.$

These dynamical properties of $\unicode{x3c9} ^\varphi $ imply the following. The set $\mathcal K(\unicode{x3c9} ^\varphi )$ of points in $\mathcal O_{\vartheta ,\rho }$ , whose future orbit returns unboundedly to some open bounded set, has either zero $\Omega ^\varphi $ -mass if $|\vartheta |\geq 4$ or its complement has zero $\Omega ^\varphi $ -mass if $|\vartheta |\leq 2.$

The key feature now is to relate ${\mathsf {u}}_\varphi $ -conical points with the set $\mathcal K(\unicode{x3c9} ^\varphi ).$ This is attained in Lemma 5.13.3 where it is shown that a triple $(x,y,v)\in \mathcal F^{(2)}_\vartheta \times {\mathfrak a}_\vartheta $ projects to $\mathcal K(\unicode{x3c9} ^\varphi )$ if and only if y is ${\mathsf {u}}_\varphi $ -conical. The previous dynamical dichotomy gives then the dichotomy on the $\mu ^\varphi $ -measure on conical points:

(1)

The global strategy of our proof is different from the analog result of Burger et al [20]. While, inspired by them, we also use a mixing result. The use of a Dirichlet–Poincaré series along tubes does not play any role in the proof of Theorem B, nor on the ergodicity dichotomy for directional flows.

Let us end §1 by observing that both Burger et al [20] and Chow and Sarkar [25] prove dynamical statements on (as opposed to ).

1.2 Plan of the paper

In §2, we recall some basic facts about the ergodic theory of metric-Anosov flows, and then study translation cocycles over them. Section 3 deals with a Ledrappier correspondence for word-hyperbolic groups whose Gromov–Mineyev geodesic flow is metric-Anosov. We will mainly apply these results to the Buseman–Iwasawa cocycle of ${\mathsf {G}}.$ For applications to other cocycles, the reader may check the work of Carvajales [22, 23].

We then recall in §4 necessary definitions on semi-simple algebraic groups over a local field and deal with Anosov representations in §5. We explain in this section how the Ledrappier correspondence applies in this setting to give, mainly:

• • uniqueness results on the Patterson–Sullivan measures;

• • precise dynamical information on the directional flows $\unicode{x3c9} ^\varphi .$

The proof of Theorem B can be found in §5.13.

2 Skew-products over metric-Anosov flows

Throughout this section, we let X be a compact metric space and V a finite dimension real vector space.

2.1 Thermodynamic formalism and reparameterizations

Let $\phi =(\phi _t:X \to X)_{t\in \mathbb {R}}$ be a continuous flow without fixed points. The space of $\phi $ -invariant probability measures on X is denoted by $\mathcal M^\phi .$ It is a convex, weakly compact subset of $C^*(X),$ the dual space to the space of continuous functions equipped with the uniform topology. The metric entropy of $m\in \mathcal M^\phi $ will be denoted by $h(\phi ,m).$ Its definition can be found in Aaronson’s book [1]. Via the variational principle, we will define the topological pressure (or just pressure) of a function $f:X\to \mathbb {R}$ as the quantity

(2) $$ \begin{align}P(\phi,f)= \sup_{m\in\mathcal M^{\phi}} \bigg(h(\phi,m)+\int_X f\,dm\bigg).\end{align} $$

A probability measure m realizing the least upper bound is called an equilibrium state of $f.$ An equilibrium state for $f\equiv 0$ is called a measure of maximal entropy, and its entropy is called the topological entropy of $\phi ,$ denoted by $h(\phi ).$

Let $f:X\to \mathbb {R}_{>0}$ be continuous. For every $x\in X$ , the function $k_f:X\times \mathbb {R}\to \mathbb {R},$ defined by $k_f(x,t)=\int _0^tf(\phi _sx)\,ds,$ is an increasing homeomorphism of $\mathbb {R}.$ There is thus a continuous function $\alpha _f:X\times \mathbb {R}\to \mathbb {R}$ such that for all $(x,t)\in X\times \mathbb {R},$

$$ \begin{align*}\alpha_f(x,k_f(x,t))=k_f(x,\alpha_f(x,t))=t.\end{align*} $$

The reparameterization of $\phi $ by $f:X\to \mathbb {R}_{>0}$ is the flow $\phi ^f\!=(\phi ^f_t\!:X \to X)_{t\in \mathbb {R}}$ defined, for all $(x,t)\in X\times \mathbb {R}$ , by

$$ \begin{align*}\phi^f_t\!(x)=\phi_{\alpha_f(x,t)}(x).\end{align*} $$

The Abramov transform of $m\in \mathcal M^\phi $ is the probability measure $m^\#\in \mathcal M^{\phi ^f\!}$ defined by

(3) $$ \begin{align}m^\#=\frac{f\cdot m}{\int f\,dm}.\end{align} $$

One has the following lemma.

Lemma 2.1.1. (Sambarino [65, Lemma 2.4])

Let $f:X\to \mathbb {R}_{>0}$ be a continuous function. Assume the equation

$$ \begin{align*}P(\phi,-sf)=0\quad s\in\mathbb{R}\end{align*} $$

has a finite positive solution $h,$ then h is the topological entropy of $\phi ^f\!.$ Conversely, if $h(\phi ^f\!)$ is finite, then it is a solution to the last equation. In this situation, the Abramov transform induces a bijection between the set of equilibrium states of $-hf$ and the set of probability measures maximizing entropy for $\phi ^f\!.$

Two continuous maps $f,g:X\to V$ are Livšic-cohomologous if there exists a $U:X\to ~V$ of class $\operatorname {\mathrm {C}}^1$ in the direction of the flow (that is, such that if for every $x\in X,$ the map $t\mapsto U(\phi _tx)$ is of class $\operatorname {\mathrm {C}}^1$ and the map $x\mapsto {\partial }/{\partial t}|_{t=0}U(\phi _tx)$ is continuous), such that for all $x\in X$ , one has

$$ \begin{align*}f(x)-g(x)=\frac{\partial}{\partial t}\bigg|_{t=0} U(\phi_tx).\end{align*} $$

Remark 2.1.2. If f and g are real-valued and Livšic-cohomologous, then $P(\phi ,f)=P(\phi ,g).$

2.2 Metric-Anosov flows I: Livšic-cohomology

Metric-Anosov flows are a metric version of what is commonly known as hyperbolic flows. The former are called Smale flows by Pollicott [56], who transferred to this more general setting the classical theory carried out for the latter. We recall here their definition and some well-known facts on their ergodic theory needed in what follows. Throughout this subsection, we will further assume that $\phi $ is Hölder-continuous with an exponent independent of $t,$ that it is transitive, that it has a dense orbit, and that it is metric-Anosov.

For $\varepsilon>0$ , the local stable/unstable set of x are (respectively)

$$ \begin{align*}W^{{\mathrm{s}}}_{\varepsilon}(x) & =\{y\in X:d(\phi_tx,\phi_t y)\leq\varepsilon \text{ for all } t>0\textrm{ and }d(\phi_tx,\phi_t y)\to0\textrm{ as }t\to\infty\},\\ W^{{\mathrm{u}}}_{\varepsilon}(x) & =\{y\in X:d(\phi_{-t}x,\phi_{-t} y)\leq\varepsilon \text{ for all } t>0\textrm{ and }d(\phi_{-t}x,\phi_{-t} y)\to0\textrm{ as }t\to\infty\}.\end{align*} $$

Definition 2.2.1. (Metric-Anosov)

The flow $\phi $ is metric-Anosov if the following hold.

• • (Exponential decay) There exist positive constants $C, \unicode{x3bb} $ and $\varepsilon $ such that for every $x\in ~X,$ every $y\in W^{{\mathrm {s}}}_\varepsilon (x)$ , and every $t>0$ , one has

  $$ \begin{align*}d(\phi_tx,\phi_ty)\leq Ce^{-\unicode{x3bb} t},\end{align*} $$
  and such that for every $y\in W^{{\mathrm {u}}}_\varepsilon (x)$ , one has $d(\phi _{-t}x,\phi _{-t}y)\leq Ce^{-\unicode{x3bb} t}.$

• • (Local product structure) There exist $\delta ,\varepsilon>0$ and a Hölder-continuous map

  $$ \begin{align*}\unicode{x3bd}:\{(x,y)\in X\times X:d(x,y)<\delta\}\to \mathbb{R}\end{align*} $$
  such that $\unicode{x3bd} (x,y)$ is the unique value $\unicode{x3bd} $ such that $W^{{\mathrm {u}}}_\varepsilon (\phi _\unicode{x3bd} x)\cap W^{{\mathrm {s}}}_\varepsilon (y)$ is non-empty, and consists of exactly one point, called $\langle x,y\rangle ;$ and for every $x\in X$ , the map
  $$ \begin{align*}W^{{\mathrm{s}}}_\varepsilon(x)\times W^{{\mathrm{u}}}_\varepsilon(x)\times(-\delta,\delta)\to X,\end{align*} $$
  given by $(y,z,t)\mapsto \phi _t(\langle y,z\rangle ),$ is a Hölder-homeomorphism onto an open neighborhood of $x.$

A translation cocycle over $\phi $ is a map $k: X\times \mathbb {R} \to V$ such that for every $x\in X$ and $t,s\in \mathbb {R},$ one has

$$ \begin{align*}k(x,t+s)=k(\phi_sx,t)+k(x,s),\end{align*} $$

and such that the map $k(\cdot ,t)$ is Hölder-continuous with exponent independent of t and with bounded multiplicative constant when t remains on a bounded set. Two translation cocycles $k_1$ and $k_2$ are Livšic-cohomologous if there exists a continuous map $U:X\to V,$ such that for all $x\in X$ and $t\in \mathbb {R}$ , one has

(4) $$ \begin{align}k_1(x,t)-k_2(x,t)=U(\phi_tx)-U(x).\end{align} $$

If k is a translation cocycle, then the period for k of a periodic orbit $\tau $ is

$$ \begin{align*}\ell_\tau(k)=k(x,p(\tau))\end{align*} $$

for any $x\in \tau .$ The marked spectrum $\tau \mapsto \ell _k(\tau )$ is a cohomological invariant that uniquely determines its class.

Theorem 2.2.2. (Livšic [47])

Let $k:X\times \mathbb {R}\to V$ be a translation cocycle. If $\ell _k(\tau )=0$ for every periodic orbit $\tau ,$ then k is Livšic-cohomologous to $0.$

Observe that if $f:X\to V$ is Hölder-continuous, then the map

$$ \begin{align*}k_f(x,t)=\int_0^tf(\phi_sx)\,ds\end{align*} $$

is a translation cocycle. Two such functions are Livšic-cohomologous if and only if the associated cocycles are, and the period of f on $\tau $ is, for any $x\in \tau ,$

$$ \begin{align*}\ell_\tau(f)=\int_\tau f=k_f(x,p(\tau)).\end{align*} $$

It turns out that every cocycle is Livšic-cohomologous to a cocycle of the form $k_f$ .

Corollary 2.2.3. (Sambarino [66, Lemma 2.6])

If $k:X\times \mathbb {R}\to V$ is a translation cocycle, then there exists a Hölder-continuous $f:X\to V$ such that $ k$ and $k_f$ are Livšic-cohomologous.

Proof. For any $\kappa>0,$ the function $j(x,t)=1/{2\kappa }\int _{-\kappa }^\kappa k(x,t+s)\,ds$ is differentiable on the second variable. Let $f(x)=(\partial /\partial s)|_{s=0}j(x,s).$ Then,

$$ \begin{align*}k_f(x,t)&=\int_0^tf(\phi_ux)\,du =\int_0^t\frac{\partial}{\partial s}\bigg|_{s=0}j(\phi_ux,s)\,du\\ & =\int_0^t\frac{\partial}{\partial s}\bigg|_{s=0}j(x,s+u)\,du = j(x,t)-j(x,0),\end{align*} $$

so the period $k_f(x,p(\tau ))=j(x,p(\tau ))-j(x,0)=k(x,p(\tau )).$ By Theorem 2.2.2, the cocycles k and $k_f$ are thus Livšic-cohomologous.

We record also the following immediate consequence of Livšic’s theorem.

Remark 2.2.4. The space of functions Livšic-cohomologous to a strictly positive function is an (open cone on an) infinite dimensional space.

In this context, much more information can be stated about the pressure function. Recall that the space $\operatorname {\mathrm {Holder}}^\alpha (X)$ of real valued $\alpha $ -Hölder functions is naturally a Banach space when equipped with the norm

$$ \begin{align*}\|f\|_\alpha=\|f\|_\infty+\sup_{x\neq y}\frac{|f(x)-f(y)|}{d(x,y)^\alpha}.\end{align*} $$

Proposition 2.2.5. (Bowen and Ruelle [16] and Parry and Pollicott [53, Proposition 4.10])

The function $P(\phi ,\cdot )$ is analytic on $\operatorname {\mathrm {Holder}}^\alpha (X).$ If $f,g\in \operatorname {\mathrm {Holder}}^\alpha (X),$ then

$$ \begin{align*}\frac{\partial}{\partial t}\bigg|_{t=0}P(\phi,f+tg)=\int gdm_f,\end{align*} $$

where $m_f$ is the equilibrium state of $f,$ and the function $t\mapsto P(\phi ,f+tg)$ is strictly convex unless g is Livšic-cohomologous to a constant. Finally, one also has

(5) $$ \begin{align}P(\phi,f)=\limsup_{t\to\infty}\frac1t\log\sum_{\tau:p(\tau)\leq t}e^{\ell_\tau(f)}.\end{align} $$

Let $f\in \operatorname {\mathrm {Holder}}^\alpha (X)$ have non-negative (and not all vanishing) periods and define its entropy by

$$ \begin{align*}\mathscr h_f=\limsup_{s\to\infty}\frac1s\log\#\bigg\{\tau\textrm{ periodic}:\int_\tau f\leq s\bigg\}\in(0,\infty].\end{align*} $$

Remark 2.2.6. Observe that $\mathscr h_f$ is necessarily $>0$ since f must have a positive maximum and $h(\phi )>0.$

One has the following lemma.

Lemma 2.2.7. (Ledrappier [44, Lemma 1] and Sambarino [65, Lemma 3.8])

Consider a Hölder-continuous function $f:X\to \mathbb {R}$ with non-negative periods. Then the following statements are equivalent:

• • the function f is Livšic-cohomologous to a positive Hölder-continuous function;

• • there exists $\kappa>0$ such that $\int _\tau f>\kappa p(\tau )$ for every periodic orbit $\tau ;$

• • the entropy $\mathscr h_f$ is finite;

• • the function $t\mapsto P(\phi , -tf)$ has a positive zero, in which case is $\mathscr h_f.$

Let us fix an exponent $\alpha $ and consider the cone $\operatorname {\mathrm {Holder}}^\alpha _+(X,\mathbb {R})$ of Hölder-continuous functions that are Livšic-cohomologous to a strictly positive function. The implicit function theorem for Banach spaces (see [2]) and the explicit formula for the derivative of pressure (Proposition 2.2.5) give the following corollary.

Lemma 2.2.8. The entropy map $\mathscr h:\operatorname {\mathrm {Holder}}^\alpha _+(X,\mathbb {R})\to \mathbb {R}_+$ is analytic.

Proof. Indeed, Lemma 2.2.7 gives the equation $P(\phi ,-\mathscr h_ff)=0$ and equation (2.2.5) gives the non-vanishing derivative

$$ \begin{align*}d_{-\mathscr h_ff}P(\phi,f)=\int f\,dm_{-h_ff}>0,\end{align*} $$

so the implicit function theorem completes the result.

2.3 Metric-Anosov flows II: ergodic theory

A fundamental tool for studying the ergodic theory of metric-Anosov systems is the existence of a Markov coding.

Let $\Sigma $ be an irreducible sub-shift of finite type equipped with its shift transformation $\sigma :\Sigma \to \Sigma ,$ and $r:\Sigma \to \mathbb {R}_{>0}$ be Hölder-continuous. Let $\hat r:\Sigma \times \mathbb {R}\to \Sigma \times \mathbb {R}$ be defined as

$$ \begin{align*}\hat r(x,s)=(\sigma x, s-r(x)),\end{align*} $$

and consider the quotient space It is equipped with the flow induced on the quotient by the translation flow.

Definition 2.3.1. (Markov coding)

A triplet $(\Sigma ,\pi ,r)$ is a Markov coding for $\phi $ if $\Sigma $ and r are as above, $\pi :\Sigma \to X$ is Hölder-continuous, and the function $\pi _r:\Sigma \times \mathbb {R}\to X$ defined as

$$ \begin{align*}\pi_r(x,t)=\phi_t\pi(x)\end{align*} $$

verifies the following conditions:

1. (i) $\pi _r$ is Hölder-continuous, surjective, and $\hat r$ -invariant, it passes then to the quotient

2. (ii) is bounded-to-one and injective on a residual set which is of full measure for every ergodic $\unicode{x3c3} ^r$ -invariant measure of total support;

3. (iii) for every $t\in \mathbb {R}$ , one has $\pi _r\unicode{x3c3} ^r_t=\phi _t\pi _r.$

The following result has a long history, see for example the works of Bowen [14, 15], Ratner [63], Pollicott [56], and more recently Constantine, Lafont, and Thompson [27].

Theorem 2.3.2. (Existence of coding)

A transitive metric-Anosov flow admits a Markov coding.

The above is a fundamental tool to obtain the following, see for example the works of Bowen and Ruelle [16], Parry and Pollicott [53], and more recently Giulietti et al [31]. Recall from §2.1 the definition of equilibrium state.

Theorem 2.3.3. (Uniqueness of equilibrium states)

Let $f:X\to \mathbb {R}$ be Hölder-continuous. Then there exists a unique equilibrium state for $f,$ denoted by $m_f;$ it is an ergodic measure. If $g:X\to \mathbb {R}$ is also Hölder, then $m_g\ll m_f$ if and only if $f-g$ is Livšic-cohomologous to a constant function, in which case, $m_g=m_f.$ The function $f\mapsto m_f,$ defined on the space of Hölder-continuous functions with fixed exponent, is analytic. (We emphasize that the space of measures is endowed with the differentiable structure induced by being the dual space of continuous functions.)

A final fact we will require in this setting (introduced by Margulis [48]) is the decomposition of the measure of maximal entropy along the stable/central-unstable sets of $\phi .$

The stable/unstable leaf of x is

$$ \begin{align*} W^{\mathrm{s}}(x) & = \bigcup_{t\in\mathbb{R}_+} \phi_{-t}(W^{{\mathrm{s}}}_\varepsilon(\phi_tx)),\\ W^{{\mathrm{u}}}(x) & = \bigcup_{t\in\mathbb{R}_+} \phi_{t}(W^{{\mathrm{u}}}_\varepsilon(\phi_{-t}x)),\end{align*} $$

and the central stable/unstable leaf is (respectively) the $\phi $ -orbit of $W^{\mathrm {s}}(x)$ (respectively $W^{{\mathrm {u}}}(x)$ ). These sets are independent of (any small enough) $\varepsilon $ (that is smaller than the $\varepsilon $ given by Definition 2.2.1).

One has the following, see for example the works of Margulis [49], Pollicott [55] for a construction via Markov codings or Katok and Hasselblatt’s book [41, §5 of Ch. 20] for the discrete-time case.

Theorem 2.3.4. (Margulis description)

For each $x\in X$ , there exists a measure $\mu _x^{{\mathrm {s}}}$ on the stable leaf $W^{\mathrm {s}}(x)$ and a measure $\mu _x^{\mathrm {cu}}$ on the central unstable leaf such that:

• • for all $t>0$ and all measurable $U\subset W^{{\mathrm {s}}}(x)$ , one has

  (6) $$ \begin{align}\mu_{\phi_tx}^{\mathrm{s}}(\phi_tU)=e^{-h(\phi)t}\mu_{x}^{\mathrm{s}}(U); \end{align} $$

• • the local product structure in Definition 2.2 induces a local isomorphism between the measure $\mu _x^{\mathrm {s}}\otimes \mu ^{\mathrm {cu}}_x$ and the measure of maximal entropy of $\phi .$

The family of measures is unique in the following sense. If $\nu _x^{\mathrm {s}}$ and $\nu _x^{\mathrm {cu}}$ are also a family of measures along the stable and central-unstable leafs of x such that the product $\nu _x^{\mathrm {s}}\otimes \nu _x^{\mathrm {cu}}$ is locally isomorphic to a $\phi $ -invariant measure and $\nu ^{{\mathrm {s}}}_x$ verifies equation (6) with some arbitrary fixed $\delta>0$ (instead of $h(\phi )$ ), then $\delta =h(\phi ), \nu ^{{\mathrm {s}}}_x=\mu ^{{\mathrm {s}}}_x$ , and $\mu ^{\mathrm {cu}}_x=\nu ^{\mathrm {cu}}_x.$

2.4 Skew-products over sub-shifts

Consider now the two-sided subshift $\Sigma $ and let ${\mathsf {K}}:\Sigma \to V$ be Hölder-continuous.

Definition 2.4.1. We say that ${\mathsf {K}}$ is non-arithmetic if the group spanned by the periods of ${\mathsf {K}}$ is dense in $V.$

The skew-product system is defined by $f=f^{\mathsf {K}}:\Sigma \times V\to \Sigma \times V$

(7) $$ \begin{align}f(p,v)=(\sigma(p),v-{\mathsf{K}}(p)).\end{align} $$

If $\nu $ is a $\sigma $ -invariant probability measure on $\Sigma $ , then the measure is f-invariant.

The following proposition seems to be well known but we have not been able to find a specific reference in the literature, for completeness, we added a short proof in Appendix A.

Proposition 2.4.2. Let $\Sigma $ be a two-sided sub-shift, $\nu $ be an equilibrium state of some Hölder potential, and ${\mathsf {K}}:\Sigma \to \mathbb {R}$ a non-arithmetic Hölder-continuous function with $\int {\mathsf {K}}d\nu =0.$ Then the skew-product $f^{\mathsf {K}}:\Sigma \times \mathbb {R}\to \Sigma \times \mathbb {R}$ is ergodic with respect to

We record also the following classical lemma. Let us say that a subset of $\Sigma \times V$ is bounded if it has compact closure, and that it has total mass (with respect to ) if its complement has measure zero. As the space $\Sigma \times V$ is non-compact, it is natural to study the subset of points of $\Sigma \times V$ whose future orbit returns infinitely many times to a fixed open bounded set:

$$ \begin{align*}\mathcal K(f)=\{p\in \Sigma\times V: \text{ there exists } \mathrm{B}\textrm{ open bounded set and }n_k\to\infty\textrm{ with }f^{n_k}(p)\in\mathrm{B}\}.\end{align*} $$

One can be more specific. If $\mathrm {B}_1,\mathrm {B}_2\subset \Sigma \times W$ , we want to understand the measure of

$$ \begin{align*}\mathcal K(\mathrm{B}_1,\mathrm{B}_2)=\{p\in\mathrm{B}_1:\text{ there exists } n_k\to\infty\textrm{ with }f^{n_k}(p)\in\mathrm{B}_2\}.\end{align*} $$

To this end, one considers the sum .

Lemma 2.4.3. If , then However, if $\nu $ has no atoms and f is ergodic with respect to , then $\mathcal K(f)$ has total mass and for every pair $\mathrm {B}_1,\mathrm {B}_2$ , one has $\mathcal K(\mathrm {B}_1,\mathrm {B}_2)$ has total mass on $\mathrm {B}_1.$

Proof. This is a standard argument valid for any measure-preserving transformation. The first assertion follows by looking at the tail of the series in question:

where, for each $k\in \mathbb {N}, E_k =\{p\in \mathrm {B}_1:\text { there exists } N\geq k\textrm { with }f^N(p)\in \mathrm {B}_2\}.$ The second assertion can be found in, for example, Aaronson’s book [1, p. 22].

2.5 An ergodic dichotomy

As in §2.3, let $\unicode{x3c3} ^r$ be the suspension of the shift on $\Sigma $ by the function $r.$ If $\nu $ is a $\sigma $ -invariant probability measure, then $\nu \otimes \,dt/\int rd\nu $ is invariant under the translation flow $\Sigma \times \mathbb {R}$ and induces thus a $\unicode{x3c3} ^r$ -invariant probability measure on denoted by $\hat \nu .$

Remark 2.5.1. It is a classical fact, stated for example by Bowen and Ruelle [16], that if $\nu $ is the equilibrium state of $-h(\unicode{x3c3} ^r)r $ , then $\hat \nu $ realizes the topological entropy of $\unicode{x3c3} _r.$

Let $K:\Sigma \times \mathbb {R} \to V$ be a $\langle\hat r\rangle $ -invariant Hölder-continuous function and consider the flow

Consider the measure on

defined by

.

Theorem 2.5.2. (Sambarino [66, Theorem 3.8])

Assume the group generated by the periods of $(r,K)$ is dense in $\mathbb {R}\times V$ and that $\int Kd\hat \nu =0$ for the equilibrium state $\nu $ of $-h(\unicode{x3c3} ^r)r.$ Then there exists $\kappa>0$ such that given two compactly supported continuous functions

one has

as $t\to \infty .$

We include the main outline of its proof in Appendix B. As it is classical, the above result holds for characteristic functions of open bounded sets whose boundary has measure zero.

Corollary 2.5.3. Under the same assumptions of Theorem 2.5.2, if $\dim V=1$ , then $\psi $ is ergodic with respect to If $\dim V\geq 3$ , then

Proof. The skew-product system $f^{\mathsf {K}}:\Sigma \times V\to \Sigma \times V$ of equation (7), where ${\mathsf {K}}:\Sigma \to V$ is defined by

(8) $$ \begin{align}{\mathsf{K}}(x)=\int_0^{r(x)}K(x,s)\,ds,\end{align} $$

is the first-return map of $\psi $ to its global section $(\Sigma \times \{0\})/_\sim \,\times \, V.$ Consequently, following the flow-lines until reaching the section, one finds a natural (measurable) bijection between $\psi $ -invariant subsets and f-invariant subsets. If $\mu $ is a $\sigma $ -invariant probability measure on $\Sigma ,$ then this bijection preserves the class of invariant-zero-sets between the measures $\hat \mu \otimes \operatorname {\mathrm {Leb}}$ on and $\mu \otimes \operatorname {\mathrm {Leb}}$ on $\Sigma \times V.$ Thus, we can translate ergodicity results from f to the flow $\psi $ and vice versa.

The case $\dim V=1$ is hence settled by Proposition 2.4.2.

For $\dim V\geq 3$ , one considers an open $A\subset \Sigma $ with $\nu (\partial A)=0,$ an open interval $I\subset \mathbb {R}$ with length $<\min r$ , and $B\subset V$ an open ball. Applying Theorem 2.5.2 to

$$ \begin{align*}\bar {\mathrm{B}}=\bar{\mathrm{B}}_1=\bar{\mathrm{B}}_2=A\times I\times B\end{align*} $$

gives a positive C such that for large t, one has

Thus, for a fixed $t_0>0$ , one has that

If $\dim V\geq 3$ , then and Lemma 2.4.3 gives in particular, the system is not ergodic.

Remark 2.5.4. An ergodicity dichotomy for $f^{\mathsf {K}}$ has been previously established by Guivarc’h [36, Corollaire 3 on p. 443] under the stronger assumption that ${\mathsf {K}}$ is aperiodic.

By means of Markov partitions (Theorem 2.3.2), the above corollary immediately translates to the following. As in the previous section, let X be a compact metric space equipped with a topologically transitive, Hölder-continuous, metric-Anosov flow $\phi .$ Let $F:X\to V$ be Hölder-continuous and consider the flow $\Phi =(\Phi _t:X\times V\to X\times V)_{t\in \mathbb {R}}$

$$ \begin{align*}\Phi_t(p,v)=\bigg(\phi_tp,v-\int_0^s F(\phi_sp)\,ds\bigg).\end{align*} $$

It is convenient to call the flow $\Phi $ by the skew product of $\phi $ by F.

Corollary 2.5.5. (Dichotomy)

Assume the group spanned by the periods of $(1,F)$ is dense in $\mathbb {R}\times V$ and that $\int Fdm=0$ for the measure of maximal entropy m of $\phi .$ Then, $\Phi $ is mixing as in Theorem 2.5.2, moreover

$$ \begin{align*}\dim V\leq1 \Rightarrow \Phi\textrm{ is ergodic with respect to }m\otimes\operatorname{\mathrm{Leb}}\Rightarrow \dim V\leq2.\end{align*} $$

If $\dim V\geq 3$ , then $\mathcal K(\Phi )$ has zero measure.

Proof. The proof follows from the corresponding results for subshifts and Remark 2.5.1 describing the measure of maximal entropy of $\phi .$

2.6 The critical hypersurface

We recall here two results of Babillot and Ledrappier [5]. Their paper concerns differentiable Anosov flows but, as one checks the proofs, only the existence of a Markov coding is required for both their results below. We take the liberty to state them in our broader generality and refer the reader to loc. cit. whose proofs work verbatim.

As before, let $F:X\to V$ be Hölder-continuous.

Assumption A. We will assume throughout the remainder of §2 that the closed group $\Delta $ spanned by the periods of F has rank $\dim V,$ (that is, $\Delta \simeq \mathbb {R}^k\times \mathbb {Z}^{\dim V-k}$ for some ) and that, moreover, the group spanned by

$$ \begin{align*}\bigg\{(p(\tau),\int_\tau F):\tau\textrm{ periodic}\bigg\}\end{align*} $$

is isomorphic to $\mathbb {R}\times \Delta .$

The compact convex subset of V,

$$ \begin{align*} \mathcal M^\phi(F)=\bigg\{\int_XFd\mu:\mu\in\mathcal M^\phi\bigg\},\end{align*} $$

has hence non-empty interior. Also, for each $\varphi \in V^*$ , one can consider the pressure of the function $\varphi (F):X\to \mathbb {R}$ :

$$ \begin{align*}\mathbf{P}(\varphi)=P(\phi,-\varphi\circ F).\end{align*} $$

By Assumption A, the function $\mathbf {P}:V^*\to \mathbb {R}$ is analytic and strictly convex (Proposition 2.2.5). Using the formula for the derivative of pressure (Proposition 2.2.5), and the natural identification $\operatorname {\mathrm {Gr}}_{\dim V-1}(V^*)\to \mathbb {P}(V),$ one has, for $\varphi \in V^*$ , that

(9) $$ \begin{align}d_\varphi\mathbf{P}=\int Fdm_{-\varphi(F)},\end{align} $$

where $m_{-\varphi (F)}$ is the equilibrium state of $-\varphi (F).$ One has the following proposition.

Proposition 2.6.1. (Babillot and Ledrappier [5, Proposition 1.1])

The map $\wp :V^*\to V$ defined by $\varphi \mapsto d_\varphi \mathbf {P}$ is a diffeomorphism between $V^*$ and the interior of $\mathcal M^\phi (F).$

Let us denote by $\mathscr L_F=\mathbb {R}_+\cdot \mathcal M^\varphi (F)$ the closed cone generated by the periods of $F.$ If $0$ does not belong to $\mathcal M^\phi (F)$ , then $\mathscr L_F$ is a sharp cone (that is, does not contain a hyperplane of V) and its interior is

$$ \begin{align*}\operatorname{\mathrm{int}}\mathscr L_F=\mathbb{R}_+\cdot\operatorname{\mathrm{int}}(\mathcal M^\phi(F)).\end{align*} $$

One has moreover the following proposition.

Proposition 2.6.2. (Babillot and Ledrappier [5, Proposition 3.1])

Assume $0\notin \mathcal M^\phi (F).$ Then the set $\wp (\{\varphi \in V^*:\mathbf {P}(\varphi )=0\})$ generates the cone $\operatorname {\mathrm {int}}\mathscr L_F.$

Remark 2.6.3. Observe that if $\varphi \in V^*$ is such that $\sum _{\tau \textrm { periodic}}e^{-\ell _\tau (\varphi \circ F)}<\infty $ , then necessarily $\varphi $ is strictly positive on the cone $\mathscr L_F,$ that is, $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_F^*).$ Indeed, if $\sum _{\tau }e^{-\ell _\tau (\varphi \circ F)}$ is convergent, then the formula for pressure on Proposition 2.2.5 gives that $\mathbf {P}(\varphi )\leq 0.$ Since $\mathbf {P}(0)>0$ , there exists $s\in (0,1]$ such that $\mathbf {P}(s\varphi )=0.$ The variational principle (equation (2)) implies that

$$ \begin{align*}\int_\tau\varphi(F)\geq0\end{align*} $$

for every periodic orbit $\tau ,$ and thus Lemma 2.2.7 applies to give that $\varphi (F)$ is Livšic-cohomologous to a strictly positive function and $\mathscr h_{\varphi (F)}\in (0,1].$

We are thus interested in the convergence domain of F

$$ \begin{align*}\mathcal D_F & =\{\varphi\in\operatorname{\mathrm{int}}(\mathscr L_F^*): \mathscr h_{\varphi(F)}\in(0,1)\}\\ & \subset \bigg\{\varphi\in V^*: \sum_{\tau\textrm{ periodic}} e^{-\ell_\tau(\varphi\circ F)}<\infty\bigg\}, \end{align*} $$

and the critical hypersurface (also usually called the entropy-one set or the Manhattan curve)

$$ \begin{align*}\mathcal Q_F & =\{\varphi\in\operatorname{\mathrm{int}}(\mathscr L_F^*): \mathscr h_{\varphi(F)}=1\}\end{align*} $$

whose name is justified in the next corollary.

Corollary 2.6.4. Assume $0\notin \mathcal M^\phi (F).$ Then $\mathcal D_F=\mathbf {P}^{-1}(-\infty ,0)$ and $\mathcal Q_F=\mathbf {P}^{-1}(0)$ . Consequently, $\mathcal D_F$ is a strictly convex set whose boundary coincides with $\mathcal Q_F.$ The latter is a closed analytic co-dimension-one submanifold of $V^*.$ The map

$$ \begin{align*}\varphi\in\mathcal Q_F\mapsto {\mathsf{T}}_\varphi\mathcal Q_F\end{align*} $$

induces a diffeomorphism between $\mathcal Q_F$ and directions in the interior of the cone $\mathscr L_F.$

Proof. We have already shown the inclusions $\mathbf {P}^{-1}(-\infty ,0)\subset \mathcal D_F$ and $\mathbf {P}^{-1}(0)\subset \mathcal Q_F.$ The other ones follow at once from Lemmas 2.2.7 and 2.1.1. Since $0\notin \mathcal M^\varphi (F)$ , Proposition 2.6.1 implies that $\mathbf {P}$ has no critical points, thus $\mathbf {P}^{-1}(0)=\mathcal Q_F$ is an analytic sub-manifold of $V^*.$ Strict convexity follows from that of $\mathbf {P},$ and the last assertion follows by observing that the tangent space ${\mathsf {T}}_\varphi \mathcal Q_F$ equals $\ker d_\varphi \mathbf {P}.$

We now focus on the variation of the critical hypersurface when F varies. To this end, consider the Banach space $\operatorname {\mathrm {Holder}}^\alpha (X,V)$ of V-valued Hölder continuous functions with exponent $\alpha .$ The pressure can be considered as an analytic map $\mathbf {P}:\operatorname {\mathrm {Holder}}^\alpha (X,V)\times V^*\to \mathbb {R}$ defined as $\mathbf {P}(G,\psi ):=P(\phi ,-\psi (G)).$ Its differential at the point $(F,\varphi )$ on the vector $(G,\psi )$ is

$$ \begin{align*}d_{(F,\varphi)}\mathbf{P}(G,\psi)=-\int(\psi(F)+\varphi(G))\,dm_{-\varphi(F)},\end{align*} $$

and vanishes identically only if $(F,\varphi )=(0,0).$ The pre-image $\mathbf {P}^{-1}(0)$ is thus a Banach-manifold.

If $F\in \operatorname {\mathrm {Holder}}^\alpha (X,V)$ is such that $0\notin \mathcal M^\phi (F),$ then its critical hypersurface

$$ \begin{align*}\mathcal Q_F=\{\varphi\in V^*:(F,\varphi)\in\mathbf{P}^{-1}(0)\}\end{align*} $$

is the intersection of $\{F\}\times V^*$ with the level set $\mathbf {P}^{-1}(0).$ This intersection will vary analytically on compact sets with F as long as the tangent space $\ker d_{(F,\varphi )}\mathbf {P}$ (for fixed $(F,\varphi )$ with $\mathbf {P}(F,\varphi )=0$ ) is transverse to the vector space $\{0\}\times V^*.$ Since $\ker d_{(F,\varphi )}\mathbf {P}$ has co-dimension 1, transversality is implied by $\ker d_{(F,\varphi )}\mathbf {P}\cap \{0\}\times V^*$ being co-dimension $1$ on $V^*.$ However, by Corollary 2.6.4, this latter intersection is, as long as F verifies assumption A and $0\notin \mathcal M^\phi (F),$ the tangent space ${\mathsf {T}}_\varphi \mathcal Q_F,$ which has co-dimension $1.$ We have thus established the following corollary.

Corollary 2.6.5. The critical hypersurface $\mathcal Q_F$ varies analytically on compact sets when varying the function F among Hölder functions verifying the hypothesis of Corollary 2.6.4.

2.7 Dynamical intersection and the critical hypersurface

We recall here the concept of dynamical intersection of Bridgeman et al [17]. Similar concepts have been previously studied by Bonahon [11], Burger [19], Croke and Fathi [28], and Knieper [42], among others.

Let $f:X\to \mathbb {R}_+$ be a positive Hölder-continuous function and, for $t>0,$ consider the finite set ${\mathsf {R}}_t(f)=\{\tau \textrm { periodic}:\ell _\tau (f)\leq t\}.$ Let $g:X\to \mathbb {R}$ be Hölder-continuous (but not necessarily positive). Then the dynamical intersection between f and g is defined by

$$ \begin{align*}\mathbf{I}(f,g)=\lim_{t\to\infty}\frac1{\#{\mathsf{R}}_t(f)}\sum_{\tau\in {\mathsf{R}}_t(f)}\frac{\ell_\tau(g)}{\ell_\tau(f)}.\end{align*} $$

Then one has the following proposition.

Proposition 2.7.1. [17, §3.4]

One has

$$ \begin{align*}\mathbf{I}(f,g)=\frac{\int gdm_{-\mathscr h_ff}}{\int f\,dm_{-\mathscr h_ff}},\end{align*} $$

in particular, $\mathbf {I}$ is well defined and varies analytically with f and g among Hölder- continuous functions with fixed Hölder exponent. If g is moreover positive, then one has $\mathbf {I}(f,g)\geq \mathscr h_f/\mathscr h_g.$

We now place ourselves in the context of the previous subsection, that is, we consider a Hölder-continuous $F:X\to V$ and we assume moreover that $0\notin \mathcal M^\phi (F).$ For $\varphi \in \mathcal Q_F$ , we consider the map $\mathbf {I}_{\varphi }:V^*\to \mathbb {R}$ defined by

(10) $$ \begin{align}\mathbf{I}_{\varphi}(\psi):= \mathbf{I}(\varphi(F),\psi(F))=\frac{\int \psi(F)\,dm_{-\varphi(F)}}{\int\varphi(F)\,dm_{-\varphi(F)}},\end{align} $$

where the last equality comes from Proposition 2.7.1 and the fact that $\mathscr h_{\varphi (F)}=1$ . Observe that it is a linear map. We then have the following explicit interpretation of the tangent space to the critical hypersurface purely in terms of periods.

Corollary 2.7.2. Let $F:X\to V$ be as in §2.6 and such that $0\notin \mathcal M^\phi (F).$ Then for $\varphi \in \mathcal Q_F$ , one has ${\mathsf {T}}_\varphi \mathcal Q_F=\ker \mathbf {I}_\varphi .$

Proof. Since by Corollary 2.6.4 one has $\mathcal Q_F=\mathbf {P}^{-1}(0),$ the tangent space ${\mathsf {T}}_\varphi \mathcal Q_F=\ker d_\varphi \mathbf {P}=\ker \mathbf {I}_\varphi ,$ where the last equality comes from the combination of equations (9) and (10).

3 A Ledrappier correspondence

In [44], Ledrappier establishes, for a closed negatively curved manifold $M,$ bijections between Livšic-cohomology classes of pressure zero functions on ${\mathsf {T}}^1M,$ normalized Hölder cocycles for the action of $\pi _1M$ on the visual boundary $\partial _\infty \tilde M$ of the universal cover of $M,$ quasi-invariant measures on $\partial _\infty \tilde M$ , among other objects. The purpose of this section is to establish, in the context of word-hyperbolic groups with metric-Anosov geodesic flow, some of these correspondences. We also extend results of Sambarino [64–66] to this setting. Some ideas of Bridgeman et al [17, 18] and Carvajales [23, Appendix] are used. The reader can also check the work of Paulin, Pollicott, and Schapira [54] for similar results in situations allowing cusps.

Let

be a finitely generated, non-elementary, word-hyperbolic group (see [30] for a definition). Denote by

the Gromov–Mineyev geodesic flow of

(see [33] and [51]). The total space

is the quotient of

by a properly discontinuous co-compact

-action (defined in loc. cit.). This action restricted to

coincides with the induced

-action on its Gromov boundary, and commutes with the $\mathbb {R}$ -action by translations, giving on the quotient the desired flow ${\mathsf {g}}.$ We will save the notation

for the pair consisting on the space

equipped with the above

-action.

Assumption B. We will assume throughout §3 that ${\mathsf {g}}$ is metric-Anosov (recall that, in general, ${\mathsf {g}}$ is transitive (see Remark 3.0.1)) and that the lamination induced on the quotient by is the central-unstable lamination of ${\mathsf {g}}.$

Let us emphasize that, in what follows, the Gromov–Mineyev geodesic flow is merely an auxiliary object. The whole discussion works verbatim replacing ${\mathsf {g}}$ by the following flows known to satisfy Assumption B:

• • the non-wandering set of the geodesic flow of a convex co-compact action of on a $\mathrm {CAT}(-1)$ space (if this is known to exist), see [27];

• • the geodesic flow of a projective-Anosov representation of $\rho $ (again if this is known to exist) as introduced by Bridgeman et al [17], see also §5.1.

Recall that every hyperbolic element (that is, an infinite order element) has two fixed points on the attracting $\gamma _+$ and the repelling $\gamma _-.$ If , then $\gamma ^nx\to \gamma _+$ as $n\to \infty .$ The axis $(\gamma _-,\gamma _+)\times \mathbb {R}$ projects then to a periodic orbit of ${\mathsf {g}},$ denoted by $[\gamma ].$ If $l(\gamma )$ denotes the translation length of $\gamma $ along this axis, then $l(\gamma )$ is an integer multiple of the period of the periodic orbit $[\gamma ].$ Observe that we allow $[\gamma ]$ to tour several times along the orbits it surjects to, so at least formally, we let $[\gamma ^n]$ be the orbit $[\gamma ]$ toured n-times.

Remark 3.0.1. We briefly justify why ${\mathsf {g}}$ is transitive. It suffices to show that given two open sets $U, V$ , there exists $t\in \mathbb {R}$ such that ${\mathsf {g}}_t(U)\cap V\neq \emptyset ,$ so the question is reduced to the same question for the action of on ; the open sets to be considered can be reduced to be of the form $U_1\times U_2$ and $V_1\times V_2,$ where are open and $\overline {U_1}\cap U_2=\overline {V_1}\cap V_2=\emptyset ;$ an element with $\gamma _-\in U_2$ and $\gamma _+\in V_1$ has a positive power such that $\gamma ^n(U_1\times U_2)\cap V_1\times V_2\neq \emptyset .$

3.1 The Ledrappier potential of a Hölder cocycle

Let V be a finite dimensional real vector space. A Hölder cocycle is a function such that:

• • for all , one has $c(\gamma h,(x,y))=c(h,(x,y))+c(\gamma ,h(x,y));$

• • there exists $\alpha \in (0,1]$ such that for every , the map $c(\gamma ,\cdot )$ is $\alpha $ -Hölder continuous.

The period of a Hölder cocycle for a hyperbolic is $\ell _c(\gamma ):=c(\gamma ,(\gamma ^-,\gamma ^+)).$ Two cocycles $c,c'$ are cohomologous if there exists a Hölder-continuous function such that for all and , one has

$$ \begin{align*}c(\gamma,(x,y))-c'(\gamma,(x,y))=U(\gamma(x,y))-U(x,y).\end{align*} $$

Two cohomologous cocycles have the same marked spectrum $\gamma \mapsto \ell _c(\gamma ).$ The following should be compared with [44, Théorème 3].

Proposition 3.1.1. For every Hölder cocycle c, there exists a Hölder-continuous function such that for every hyperbolic , one has

$$ \begin{align*}\int_{[\gamma]}\mathcal J_c=\ell_c(\gamma).\end{align*} $$

Cohomologous cocycles induce Livšic-cohomolgous functions.

Proof. The general case follows from the case $V=\mathbb {R}$ by the Riesz representation theorem. Assume thus $V=\mathbb {R}$ and consider the trivial line bundle equipped with the bundle automorphisms

$$ \begin{align*}\gamma\cdot(p,s):=(\gamma p,e^{-c(\gamma,(x,y))}s),\end{align*} $$

where $p=(x,y,t).$ Denote by the quotient line bundle. It is equipped with a flow $(\hat {\mathsf {g}}_t:{\mathsf {F}}\to {\mathsf {F}})_{t\in \mathbb {R}}$ by bundle automorphisms, induced on the quotient by

$$ \begin{align*}t\cdot(p,s)=({\mathsf{g}}_tp,s).\end{align*} $$

Let $|\,|$ be a Euclidean metric on ${\mathsf {F}}$ and define, for $v\in {\mathsf {F}}_p,$

(11) $$ \begin{align}\mathcal T(p,t)=\log\frac{|\hat{\mathsf{ g}}_tv|}{|v|}.\end{align} $$

It is a translation cocycle over ${\mathsf {g}},$ indeed since ${\mathsf {F}}_p$ is one dimensional, the choice of v does not matter, and since $\hat {\mathsf {g}}$ is a flow, one has

$$ \begin{align*}\log\frac{|\hat{\mathsf{ g}}_{t+s}(v)|}{|v|}= \log\frac{|\hat{\mathsf{g}}_{t}(\hat{\mathsf{g}}_sv)|}{|v|}\frac{|\hat{\mathsf{g}}_s(v)|}{|\hat{\mathsf{g}}_s(v)|}=\log\frac{|\hat{\mathsf{g}}_{t}(\hat{\mathsf{g}}_sv)|}{|\hat{\mathsf{g}}_s(v)|}+\log\frac{|\hat{\mathsf{ g}}_s(v)|}{|v|}.\end{align*} $$

By Corollary 2.2.3, there exists a Hölder-continuous function such that $\mathcal T$ and $k_{\mathcal J_c}$ are Livšic-cohomologous. We end the proof by a period computation. For every hyperbolic , one has, for all $s\in \mathbb {R},$ that $\gamma (\gamma _-,\gamma _+,s)=(\gamma _-,\gamma _+,e^{-\ell _c(\gamma )}s),$ or equivalently, for any and $v\in {\mathsf {F}}_x,$

$$ \begin{align*}\ell_c(\gamma)=\log\frac{|\hat{\mathsf{ g}}_{p(\gamma)}v|}{|v|}=\ell_{[\gamma]}(\mathcal T)=\int_{[\gamma]}\mathcal J_c,\end{align*} $$

where $p(\gamma )$ is the period of $[\gamma ]$ for ${\mathsf {g}}.$ Since cohomologous cocycles have the same marked spectrum, the associated functions have the same periods and are thus Livšic-cohomolgous by Theorem 2.2.2.

Definition 3.1.2. We say that $\mathcal J_c$ is a Ledrappier potential of c over ${\mathsf {g}}.$

3.2 Real cocycles and reparameterizations

Assume now $V=\mathbb {R}$ and consider a cocycle c with non-negative (and not all vanishing) periods. Define its entropy by

Remark 3.2.1. It follows from Proposition 3.1.1 and Remark 2.2.6 that $\mathscr h_c>0.$

For such a cocycle, consider the action of on via c:

(12) $$ \begin{align}\gamma\cdot(x,y,t)=(\gamma x,\gamma y,t-c(\gamma,(x,y))).\end{align} $$

Let us denote by ${\unicode{x3c7} ^{c}}$ the quotient space The following is reported by Sambarino [65] for fundamental groups of closed negatively curved manifolds and by Carvajales [23] for the refraction cocycle of a ${{{\Delta }}}$ -Anosov representation (see Definition 5.3.1).

Theorem 3.2.2. If c is a Hölder cocycle with non-negative periods and finite entropy, then its Ledrappier potential is Livšic-cohomologous to a strictly positive function. Moreover, the above action of on is properly discontinuous and co-compact, and the flow $\phi ^c=(\phi ^c_t:{\unicode{x3c7} ^{c}}\to {\unicode{x3c7} ^{c}})_{t\in \mathbb {R}}$ induced on the quotient by the $\mathbb {R}$ -translation flow is Hölder-conjugated to the reparameterization of ${\mathsf {g}}$ by $\mathcal J_c.$

The topological entropy of $\phi ^c$ is thus $\mathscr h_c.$ (When has torsion elements, this fact requires some work, see the work of Carvajales et al [24, §5] for details.)

Proof. The first assertion follows at once from Lemma 2.2.7. For the remaining statements, we continue as in the proof of the proposition but for the cocycle $-c.$ Observe first that $-\mathcal J_c=\mathcal J_{-c}.$ Since $\mathcal T$ and $k_{\mathcal J_{-c}}$ are Livšic-cohomologous, there exists such that for all , and $v\in {\mathsf {F}}_p$ , one has (recall equation (11))

$$ \begin{align*}\log\frac{|\hat{\mathsf{ g}}_tv|}{|v|}-\int_0^t\mathcal J_{-c}=U({\mathsf{g}}_tp)-U(x).\end{align*} $$

Since $\mathcal J_{-c}=-\mathcal J_c$ is Livšic-cohomologous to a strictly negative function, the above equation implies that the flow $\hat {\mathsf {g}}$ is contracting on ${\mathsf {F}},$ that is, there exist positive C and $\mu $ such that for all $v\in {\mathsf {F}}$ and $t\in \mathbb {R}$ , one has

$$ \begin{align*}|\hat{\mathsf{g}}_tv|\leq Ce^{-\mu t}|v|.\end{align*} $$

A standard procedure (see for example [41] or [17, Lemma 4.3]) provides a Euclidean metric $\|\,\|$ on ${\mathsf {F}}$ such that the constant C equals $1.$ We denote also by $\|\,\|$ the lift of this metric to it is a -invariant family.

Given then , we let $v_{(x,y,t)}\in (\mathbb {R}-\{0\})/\pm $ be such that $\|v_{(x,y,t)}\|=1.$ As in [17, Proposition 4.2], the map

(13)

is -equivariant and an orbit equivalence between the $\mathbb {R}$ -actions.

Definition 3.2.3. The flow $\phi ^c$ will be called the refraction flow of $c.$

3.3 Patterson–Sullivan measures

Let us consider now Hölder cocycles with $V=\mathbb {R}$ and only depending on one variable, that is, Assume moreover that c has non-negative periods and finite entropy. A cocycle is dual to c if for every hyperbolic , one has

$$ \begin{align*}\ell_{\bar c}(\gamma)=\ell_c(\gamma^{-1}).\end{align*} $$

Definition 3.3.1

• • A Patterson–Sullivan measure for c of exponent $\delta \in \mathbb {R}_+$ is a probability measure $\mu $ on such that for every , one has

  (14) $$ \begin{align}\frac{d \gamma_*\mu}{\,d\mu}(\cdot)=e^{-\delta c(\gamma^{-1},\cdot)}.\end{align} $$

• • A Gromov product for the ordered pair $(\bar c, c)$ is a function such that for all and , one has

  $$ \begin{align*}[\gamma x,\gamma y]-[x,y]=-(\bar c(\gamma,x)+c(\gamma,y)).\end{align*} $$

Consider a pair of dual cocycles $(\bar c,c)$ and assume a Patterson–Sullivan measure of exponent $\delta $ exists for each c and $\bar c,$ denoted by $\mu $ and $\bar \mu $ , respectively. Assume moreover that a Gromov product for the pair $(\bar c, c)$ exists. The measure

(15) $$ \begin{align}\widetilde{\mathscr m}= e^{-\delta[\cdot,\cdot]}\bar\mu\otimes \mu\otimes \,dt\end{align} $$

on

is hence

-invariant (the action being via $c,$ as in equation (12) and $\mathbb {R}$ -invariant. Passing to the quotient, one obtains a measure $\mathscr m$ on ${\unicode{x3c7} ^{c}}$ invariant under the flow $\phi ^c.$ Observe that we can write $\widetilde {\mathscr m}$ as

The measure $e^{-\delta t}\,d\mu \,dt$ is

-invariant on

(indeed, if

is continuous, then

$$ \begin{align*}\int f(\gamma(y,t))e^{-\delta t} \,d\mu \,dt,&=\int f(\gamma y,t-c(\gamma,y))e^{-\delta t} \,d\mu \,dt & \\& =\int f(\gamma y,t)e^{-\delta (t-c(\gamma,y))} \,d\mu \,dt \,\quad\qquad\textrm{(by translation invariance)}&\\& =\int f( y,t)e^{-\delta (t-c(\gamma,\gamma^{-1}y))}e^{-\delta c(\gamma^{-1},y)} \,d\mu \,dt \quad\textrm{(by definition of }\mu)&\\& =\int f(y,t)e^{-\delta t}\,d\mu \,dt)& \end{align*} $$

and the family

$$ \begin{align*}m_{(y,t)}=e^{\delta t}e^{-\delta[\cdot,y]}d\bar\mu\end{align*} $$

is

-equivariant.

This decomposition induces then a local decomposition of $\mathscr m$ as a product of measures on the laminations induced on the quotient by and The local product structure induced by permits to transport the measures $m_{(y,t)},$ parallel to the central stable leaf $W^{\mathrm {cu}}[(x,y,t)],$ to the stable leaf of $[(x,y,t)]$ for $\phi ^c$ to obtain a family of measures at each strong stable leaf $\nu ^{{\mathrm {s}}}_p$ such that

(16) $$ \begin{align}\frac{d(\phi^c_{-t})_*\nu^{{\mathrm{s}}}_p}{\,d\nu^{{\mathrm{s}}}_{\phi^c_tp}}=e^{-\delta t}.\end{align} $$

Margulis’s description of the measure maximizing entropy (§2.3.4) then gives that $\delta =\mathscr h_c$ and that $\mathscr m$ maximizes entropy for $\phi ^c.$ Thus, subject to the existence of the Patterson–Sullivan measures and the Gromov product, we summarize the above discussion in the following proposition.

Proposition 3.3.2. The measure $\mathscr m$ on ${\unicode{x3c7} ^{c}}$ maximizes entropy for the flow $\phi ^c.$ The exponent $\delta $ necessarily equals the topological entropy of $\phi ^c, \mathscr h_c,$ if $\nu $ is another Patterson–Sullivan measure for c then $\nu =\mu .$

Let us consider again the measure $\widetilde {\mathscr m}$ on from equation (15) but let us instead study the -action on the $\mathbb {R}$ -coordinate by the Gromov–Mineyev cocycle, so that and the induced flow is ${\mathsf {g}}.$ The quotient measure, $\mathscr a,$ is ${\mathsf {g}}$ -invariant and the orbit equivalence $\unicode{x3be} $ from equation (13) preserves, by the way it is defined, zero flow-invariant sets between $\mathscr a$ and $\mathscr m.$ Since $\unicode{x3be} $ is a conjugation between ${\mathsf {g}}^{\mathcal J_c}$ and $\phi ^c,$ and $\mathscr m$ maximizes entropy for $\phi ^c,$ the measure $\unicode{x3be} _*\mathscr m$ is ${\mathsf {g}}^{\mathcal J_c}$ -invariant, maximizes entropy for ${\mathsf {g}}^{\mathcal J_c}$ , and has the same zero sets as $\mathscr a.$

One concludes that the Abramov transform in equation (3) ${\mathscr a}^\#$ (with respect to $\mathcal J_c$ ) is a measure of maximal entropy of the flow ${\mathsf {g}}^{\mathcal J_c}.$ Lemma 2.1.1 implies then that $\mathscr a/|\mathscr a|$ is the equilibrium state of $-\mathscr h_c\mathcal J_c.$ Let us summarize in the following remark.

Remark 3.3.3. The probability measure $\mathscr a/|\mathscr a|$ on induced by $\widetilde {\mathscr m}$ is the equilibrium state of $-\mathscr h_c\mathcal J_c.$

Since the zero sets of an equilibrium state are uniquely determined by the Livšic-cohomology class of the associated potential up to an additive constant (Theorem 2.3.3), one concludes the following corollary.

Corollary 3.3.4. Let $\bar \kappa ,\kappa $ be a pair of dual cocycles with non-negative periods and finite entropy. Assume Patterson—Sullivan measures, $\nu $ and $\bar \nu ,$ exist for $\kappa $ and $\bar \kappa $ , respectively, together with a Gromov product for the pair. If $\nu $ has the same zero sets as $\mu $ , then the (scaled) Ledrappier potentials $\mathscr h_c\mathcal J_c$ and $\mathscr h_\kappa \mathcal J_\kappa $ are Livšic-cohomologous and $\nu =\mu .$

Proof. A final argument is required, indeed from §2.3.3, there exists a constant K such that $\mathscr h_c\mathcal J_c$ and $\mathscr h_\kappa \mathcal J_\kappa +K$ are Livšic-cohomologous. However, since $\mathscr h_c$ is the topological entropy of ${\mathsf {g}}^{\mathcal J_c},$ Lemma 2.1.1 gives

$$ \begin{align*}0=P({\mathsf{g}},-\mathscr h_c\mathcal J_c)=P({\mathsf{g}},-\mathscr h_\kappa\mathcal J_\kappa+K)=K,\end{align*} $$

where the second equality holds by Remark 2.1.2, and the third equality by the definition of P (equation (2)).

The above corollary was reported by Ledrappier [44] when is the fundamental group of a negatively curved closed manifold. The proof uses also a disintegration argument. One may also check [39] and Babillot’s survey [4] specifically for the Buseman cocycle ( as in Ledrappier’s aforementioned situation), and [23, Appendix] for the refraction cocycle $\unicode{x3b2} _{\varphi }$ (see §5.3 for the definition) of a ${{{\Delta }}}$ -Anosov representation of an arbitrary word-hyperbolic group.

3.4 Vector-valued cocycles I: the critical hypersurface

Let now

be a Hölder cocycle and consider the compact convex set $\mathcal M^{\mathsf {g}}(\mathcal J_c)\subset V.$ Since this set depends on the base flow ${\mathsf {g}}$ , it is natural to consider the limit cone of c:

Remark 3.4.1. Up to Livšic-cohomology, we can assume that $\mathcal J_c$ has values in the vector space $V'=\langle\mathcal M^{\mathsf {g}}(\mathcal J_c)\rangle .$ We can moreover choose a reparameterization ${\mathsf {g}}^f$ of ${\mathsf {g}}$ so that if is the Ledrappier potential for c over ${\mathsf {g}}^f$ , then the flow ${\mathsf {g}}^f$ together with the potential $\mathcal J_c^{\mathsf {g^f}}\!$ verify Assumption A from §2.6.

Proof. By Remark 2.2.7, the space of Livšic-cohomology classes over ${\mathsf {g}}$ is infinite dimensional, so the remark follows.

We will work from now on with flow ${\mathsf {g}}^f$ given by the remark and the Ledrappier potential $\mathcal J_c^{\mathsf {g^f}}.$ We will rename these by ${\mathsf {g}}$ and $\mathcal J_c$ though as to not overcharge the paper with notation and keep in mind that when we restrict the image of $\mathcal J_c$ to $V',$ Assumption A is verified.

Let $(\mathscr L_c)^*=\{\psi \in V^*:\psi |\mathscr L_c\geq 0\}$ be the dual cone of $c.$ For $\psi \in V^*$ , denote and $\mathscr h_\psi =\mathscr h_{c_\psi }.$

Assumption C. There exists $\psi \in (\mathscr L_c)^*$ such that $c_\psi $ has finite entropy.

Lemma 3.4.2. In this case, $0\notin \mathcal M^{\mathsf {g}}(\mathcal J_c), (\mathscr L_c)^*$ has non-empty interior, and $\operatorname {\mathrm {int}}(\mathscr L_c)^*=\{\varphi \in (\mathscr L_c)^*:\mathscr h_\varphi <\infty \}.$

Proof. The lemma follows essentially from §3.1 and Lemma 2.2.7, indeed since $\mathscr h_{\psi }=\mathscr h_{\psi (\mathcal J_c)}<\infty $ , there exists $\kappa>0$ such that for all hyperbolic , it holds $\psi (\ell _c(\gamma )/p(\gamma ))>\kappa ;$ by density of periodic orbits on $\mathcal M^{\mathsf {g}}$ , one has $\inf \{\psi (\mathcal M^{\mathsf {g}}(\mathcal J_c))\}>\kappa >0.$ The remaining statements follow similarly.

Since $0\notin \mathcal M^{\phi ^{c_\psi }}(\mathcal J_c)$ , we can apply Corollary 2.6. Denote by

respectively the critical hypersurface and the convergence domain of $c.$

Since we have not required the cone $\mathscr L_c$ to have non-empty interior, consider its annihilation space

$$ \begin{align*}\mathrm{Ann}(\mathscr L_c)=\{\psi\in V^*:\mathscr L_c\subset\ker\psi\}.\end{align*} $$

If $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_c)^*$ and $\psi \in \mathrm {Ann}(\mathscr L_c)$ , then the potentials $\varphi (\mathcal J_c)$ and $(\varphi +\psi )(\mathcal J_c)$ are Livšic-cohomologous. Let $\pi ^c:V^*\to V^*/\mathrm {Ann}(\mathscr L_c)$ be the quotient projection.

We also import the concept of dynamical intersection of §2.7 to this setting using the Ledrappier potential of $c.$ For $\varphi \in \mathcal Q_c$ , define the dynamical intersection map associated to c by $\mathbf {I}_\varphi =\mathbf {I}^c_\varphi :V^*\to \mathbb {R}$ be defined by

$$ \begin{align*}\mathbf{I}_\varphi(\psi)=\mathbf{I}(\varphi(\mathcal J_c),\psi(\mathcal J_c)).\end{align*} $$

By definition, $\mathbf {I}(\varphi (\mathcal J_c),\psi (\mathcal J_c))$ only depends on the Livšic-cohomology classes of $\varphi (\mathcal J_c)$ and $\psi (\mathcal J_c),$ so we may freely consider $\mathbf {I}$ as defined on $\mathcal Q_c\times V^*$ or on $\pi ^c(\mathcal Q_c)\times V^*/\mathrm {Ann}(\mathscr L_c).$ One has the following corollary.

Corollary 3.4.3. Under Assumption C, one has that $\pi ^c(\mathcal D_c)$ is a strictly convex set whose boundary is $\pi ^c(\mathcal Q_c).$ The latter is an analytic co-dimension-one sub-manifold. The map ${\mathsf {u}}:\pi ^c(\mathcal Q_c)\to \mathbb {P}(\operatorname {\mathrm {span}}\{\mathscr L_c\})$ defined by

$$ \begin{align*}\varphi\mapsto{\mathsf{u}}_\varphi:={\mathsf{T}}_\varphi\pi^c(\mathcal Q_c)=\ker\mathbf{I}_\varphi\end{align*} $$

is an analytic diffeomorphism between $\pi ^c(\mathcal Q_c)$ and directions in the relative interior of $\mathscr L_c.$

Proof. By Remark 3.4.1, we can apply §2.6 to $\mathcal J_c,$ the equality ${\mathsf {T}}_{\varphi }\mathcal Q_c=\ker \mathbf {I}_\varphi $ follows from Corollary 2.7.2.

3.5 Vector-valued cocycles II: skew-product structure

We remain in the situation of §3.4, that is, we keep Assumption C. It follows at once from Theorem 3.2 that the -action

$$ \begin{align*}\gamma(x,y,v)=(\gamma x,\gamma y,v-c(\gamma,y))\end{align*} $$

is properly discontinuous. We aim to give a description of the V-action on the quotient space .

By Lemma 3.4.2 and Theorem 3.2.2, we can, for every $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_c)^*,$ consider the refraction flow $\phi ^{c_\varphi }=(\phi ^{c_\varphi }_t:{\unicode{x3c7} ^{c_\varphi }}\to {\unicode{x3c7} ^{c_\varphi }})_{t\in \mathbb {R}}.$ Such a $\varphi $ is fixed from now on.

Remark 3.5.1. Let us still denote by $\mathcal J_c$ the Ledrappier potential for c over the flow $\phi ^{c_\varphi },$ that is, for every hyperbolic , one has

$$ \begin{align*}\int_{[\gamma]}\mathcal J_c=\ell_c(\gamma)\in V.\end{align*} $$

Let $\mathscr p$ be the probability measure of maximal entropy of $\phi ^{c_\varphi }.$ The growth direction of $\varphi $ is the line of V

(17) $$ \begin{align}{\mathsf{u}}_\varphi={\mathsf{T}}_\varphi\mathcal Q_c= \mathbb{R}\cdot\int_{{\unicode{x3c7}^{c_\varphi}}}\mathcal J_c d\mathscr p\end{align} $$

(the last equality follows from equation (9) and Corollary 2.6.4). Consider also the projection $\pi ^\varphi :V\to \ker \varphi $ parallel to ${\mathsf {u}}_\varphi $ and denote by $\mathcal J_c^\varphi :{\unicode{x3c7} ^{c_\varphi }}\to \ker \varphi $ the composition $\mathcal J_c^\varphi =\pi ^\varphi \circ \mathcal J_c.$ Observe that

(18) $$ \begin{align} \int_{{\unicode{x3c7}^{c_\varphi}}} \mathcal J_c^\varphi d\mathscr p=0.\end{align} $$

Fix $u_\varphi \in {\mathsf {u}}_\varphi $ with $\varphi (u_\varphi )=1$ and define the directional flow by induction on the quotient of

$$ \begin{align*}t\cdot(x,y,v)=(x,y,v-tu_\varphi).\end{align*} $$

Proposition 3.5.2. (Sambarino [66])

There exists a (bi)-Hölder-continuous homeomorphism

commuting with the $\ker \varphi $ action, that conjugates the flow $\unicode{x3c9} ^\varphi $ with $\Phi ^\varphi =(\Phi _t^\varphi :{\unicode{x3c7} ^{c_\varphi }}\times \ker \varphi \to {\unicode{x3c7} ^{c_\varphi }}\times \ker \varphi )_{t\in \mathbb {R}}$

$$ \begin{align*}{\displaystyle\Phi_t^\varphi(p,v):=\bigg(\phi^{c_\varphi}_t(p),v-\int_0^t\mathcal J_c^\varphi(\phi^{c_\varphi}_s p)\,ds\bigg).}\end{align*} $$

Proof. This is the first item of Sambarino [66, Proposition 3.5] when is the fundamental group of a closed negatively curved manifold. The proof adapts mutatis mutandis once §3.2 is established.

Let us moreover place ourselves in the existence assumptions of §3.3 for the cocycle $c_\varphi ,$ that is, assume there exists:

• • a dual cocycle $\bar c_\varphi $ together with a Gromov product $[\cdot ,\cdot ]$ for the pair $(\bar c_\varphi ,c_\varphi );$

• • a Patterson–Sullivan measure for each $c_\varphi $ and $\bar c_\varphi ,$ denoted by $\mu $ and $\bar \mu $ , respectively. Recall from Proposition 3.3.2 that, necessarily, the exponent of both $\mu $ and $\bar \mu $ is $\mathscr h_{c_\varphi },$ the topological entropy of $\phi ^{c_\varphi }.$

By Proposition 3.3.2, the measure $\mathscr m$ maximizes entropy for $\phi ^{c_\varphi }.$ One has then $\mathscr p=\mathscr m/|\mathscr m|.$ Consider the $\varphi $ -Bowen–Margulis measure $\Omega ^\varphi $ on defined as induction on the quotient by

$$ \begin{align*}e^{-\mathscr h_{c_\varphi}[\cdot,\cdot]}\bar\mu\otimes\mu\otimes \operatorname{\mathrm{Leb}}_{V},\end{align*} $$

for a Lebesgue measure on $V.$ One has the following result.

Proposition 3.5.3. (Sambarino [66])

The (bi)-Hölder-continuous homeomorphism from Proposition 3.5.2 is a measurable isomorphism between $\Omega ^\varphi $ and $\mathscr m\otimes \operatorname {\mathrm {Leb}}_{\ker \varphi }\!.$

Proof. This follows again as in [66, Proposition 3.5] once Proposition 3.3.2 is established.

3.6 Vector-valued cocycles III: dynamical consequences

We remain in the situation of §3.4. Let us say that c is non-arithmetic if the periods of c span a dense subgroup in $V.$

One concludes at once the following consequences.

Corollary 3.6.1. (Ergodicity dichotomy)

Assume c is non-arithmetic, then $\unicode{x3c9} ^\varphi $ is mixing as in Theorem 2.5.2, consequently if $\dim V\geq 4$ , then $\mathcal K(\unicode{x3c9} ^\varphi )$ has zero $\Omega ^\varphi $ -measure. If $\dim V\leq 2$ , then the directional flow $\unicode{x3c9} ^\varphi $ is $\Omega ^\varphi $ -ergodic.

Proof. By Proposition 3.5.2, the flow $\unicode{x3c9} ^\varphi $ is Hölder-conjugated to the skew-product of $\phi ^{c_\varphi }$ with the Ledrappier potential $\mathcal J_c^\varphi ,$ the fiber being $\ker \varphi $ and thus $\dim V-1$ -dimensional. Proposition 3.5.3 describes the desired measures in terms of the skew-product structure; non-arithmeticity of c and equation (18) allow us to apply Corollary 2.5.5.

4 Algebraic semi-simple groups over a local field

This section is a collection of necessary language and basic results needed for the following section. Most of the material covered here can be found in, for example, Borel’s book [12] and/or in the book by Benoist and Quint [9].

Let $\mathbb {K}$ be a local field. If $\mathbb {K}$ is non-Archimedean, let us denote by q the cardinality of its residue field, $u\in \mathbb {K}$ a uniformizing element, and choose the norm $|\,|$ on $\mathbb {K}$ so that $|u|=q^{-1}.$ In this case, $\log $ denotes the logarithm on base $q,$ so that $\log q=1.$ If $\mathbb {K}=\mathbb {R}$ or $\mathbb {C}$ , then $|\,|$ is the standard modulus, $q:=e$ , and $\log $ is the usual logarithm.

Let ${\mathsf {G}}$ be the $\mathbb {K}$ -points of a connected semi-simple $\mathbb {K}$ -group, ${\mathsf {A}}$ the $\mathbb {K}$ -points of a maximal $\mathbb {K}$ -split torus, and $\mathbf {X}({\mathsf {A}})$ the group of its $\mathbb {K}^*$ -characters. Consider the real vector space ${\mathsf {E}}^*=\mathbf {X}({\mathsf {A}})\otimes _{\mathbb {Z}}\mathbb {R}$ and ${\mathsf {E}}$ its dual. For every $\chi \in \mathbf X({\mathsf {A}}),$ we denote by $\chi ^\omega $ the corresponding linear form on ${\mathsf {E}}.$

Let ${{{\Phi }}}$ be the set of restricted roots of ${\mathsf {A}}$ in $\mathfrak g,$ the set ${{{\Phi }}}^\omega $ is a root system of ${\mathsf {E}}^*.$ Let $({{{\Phi }}}^\omega )^+$ be a system of positive roots, ${\mathsf {E}}^+$ the associated Weyl chamber, and ${{{\Phi }}}^+$ and ${{{\Delta }}}\subset {{{\Phi }}}$ the corresponding system of positive roots and simple roots, respectively.

Let $\nu :{\mathsf {A}}\to {\mathsf {E}}$ be defined, for $z\in {\mathsf {A}},$ as the unique vector in ${\mathsf {E}}$ such that for every $\chi \in \mathbf X({\mathsf {A}})$ , one has

$$ \begin{align*}\chi^\omega(\nu(z))=\log|\chi(z)|.\end{align*} $$

Denote ${\mathsf {A}}^+=\nu ^{-1}({\mathsf {E}}^+).$

Let $\mathcal W$ be the Weyl group of ${{{\Phi }}}.$ It is isomorphic to the quotient of the normalizer $N_{\mathsf {G}}({\mathsf {A}})$ of ${\mathsf {A}}$ in ${\mathsf {G}}$ by its centralizer $Z_{\mathsf {G}}({\mathsf {A}}).$ Let $\operatorname {\mathrm {i}}:{\mathsf {E}}\to {\mathsf {E}}$ be the opposition involution: if $u:{\mathsf {E}}\to {\mathsf {E}}$ is the unique element in the Weyl group with $u({\mathsf {E}}^+)=-{\mathsf {E}}^+$ , then $\operatorname {\mathrm {i}}=-u.$

4.1 Restricted roots and parabolic groups

Consider $\vartheta \subset {{{\Delta }}}$ and let ${\mathsf {P}}_\vartheta ,$ respectively $\check {\mathsf {P}}_\vartheta ,$ be the normalizers in ${\mathsf {G}}$ of, respectively the Lie algebras

$$ \begin{align*} & \bigoplus_{\alpha\in<{{{\Delta}}}-\vartheta>}{\mathfrak g}_{-\alpha}\oplus \bigoplus_{\alpha\in{{{\Phi}}}^+}{\mathfrak g}_\alpha,\\ & \bigoplus_{\alpha\in<{{{\Delta}}}-\vartheta>}{\mathfrak g}_{\alpha}\oplus \bigoplus_{\alpha\in{{{\Phi}}}^+}{\mathfrak g}_{-\alpha}. \end{align*} $$

The $\vartheta $ -flag space is $\mathcal F_\vartheta ={\mathsf {G}}/{\mathsf {P}}_\vartheta .$ The orbit ${\mathsf {G}}\cdot ([\check {\mathsf {P}}_\vartheta ],[{\mathsf {P}}_\vartheta ])\subset \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ is the unique open orbit on this product space, we will denote it by $\mathcal F^{(2)}_\vartheta $ and say that $(x,y)\in \mathcal F_{\operatorname {\mathrm {i}}\vartheta }\times \mathcal F_\vartheta $ are transverse if in fact $(x,y)\in \mathcal F^{(2)}_\vartheta .$

Denote by $(\cdot ,\cdot )$ a $\mathcal W$ -invariant inner product on ${\mathsf {E}}, (\cdot ,\cdot )$ the induced inner product on ${\mathsf {E}}^*,$ define $\langle,\rangle $ on ${\mathsf {E}}^*$ by

$$ \begin{align*}\langle\chi,\psi\rangle=\frac{2(\chi,\psi) }{(\psi,\psi)},\end{align*} $$

and let $\{\varpi _\alpha \}_{\alpha \in \Pi }$ be the fundamental weights of ${{{\Phi }}},$ defined by the equations $\langle\varpi _\alpha ,\sigma\rangle=d_\alpha \delta _{\alpha \sigma },$ where $d_\alpha =1$ if $2\alpha \notin {{{\Phi }}}$ and $d_\alpha =2$ otherwise.

4.2 The center of the Levi group

We now consider the vector subspace

$$ \begin{align*}{\mathsf{E}}_\vartheta=\bigcap_{\alpha\in{{{\Delta}}}-\vartheta}\ker\alpha^\omega\end{align*} $$

together with the unique projection $p_\vartheta :{\mathsf {E}}\to {\mathsf {E}}_\vartheta $ invariant under the subgroup of the Weyl group $\mathcal W_\vartheta =\{w\in \mathcal W:w|{\mathsf {E}}_\vartheta =\operatorname {\mathrm {id}}\}.$ The dual space $({\mathsf {E}}_\vartheta )^*$ sits naturally as the subspace of ${\mathsf {E}}^*$ of $p_\vartheta $ -invariant linear forms

$$ \begin{align*}({\mathsf{E}}_\vartheta)^*=\{\varphi\in{\mathsf{E}}^*:\varphi\circ p_\vartheta=\varphi\}.\end{align*} $$

It is spanned by the fundamental weights $\{\varpi _\sigma |{\mathsf {E}}_ \vartheta :\sigma \in \vartheta \}.$

4.3 Cartan decomposition

Let ${\mathsf {K}}\subset {\mathsf {G}}$ be a compact group that contains a representative for every element of the Weyl group $\mathcal W,$ this is to say, such that the normalizer $N_{\mathsf {G}}({\mathsf {A}})$ verifies $N_{\mathsf {G}}({\mathsf {A}})=(N_{\mathsf { G}}({\mathsf {A}})\cap {\mathsf {K}}){\mathsf {A}}.$ One has ${\mathsf {G}}={\mathsf {K}}{\mathsf {A}}^+{\mathsf {K}}$ and if $z,w\in {\mathsf {A}}^+$ are such that $z\in {\mathsf {K}}w{\mathsf {K}}$ , then $\nu (z)=\nu (w).$ There exists thus a function

$$ \begin{align*}a:{\mathsf{G}}\to {\mathsf{E}}^+\end{align*} $$

such that for every $g_1,g_2\in {\mathsf {G}}$ , one has $g_1\in {\mathsf {K}}g_2 {\mathsf {K}}$ if and only if $a(g_1)=a(g_2).$ It is called the Cartan projection of ${\mathsf {G}}.$

4.4 Jordan decomposition

Recall that the Jordan decomposition states that every $g\in {\mathsf {G}}$ has a power $g^k$ ( $k=1$ if $\mathbb {K}$ is Archimedean) that can be written as a commuting product $g=g_eg_hg_n,$ where $g_e$ is elliptic, $g_h$ is semi-simple over $\mathbb {K}$ , and $g_n$ is unipotent. The component $g_h$ is conjugate to an element $z_g\in {\mathsf {A}}^+$ and we let

$$ \begin{align*}\unicode{x3bb}(g)=(1/k)\nu(z_g)\in{\mathsf{E}}^+.\end{align*} $$

The map $\unicode{x3bb} :{\mathsf {G}}\to {\mathsf {E}}^+$ is the Jordan projection of ${\mathsf {G}}.$ We will also denote by $\unicode{x3bb} _\vartheta :{\mathsf {G}}\to {\mathsf {E}}_\vartheta $ the composition $p_\vartheta \circ \unicode{x3bb} .$ For ${\mathsf {G}}=\operatorname {\mathrm {{\mathsf {PGL}}}}_d(\mathbb {K})$ , we will denote by $\unicode{x3bb} _1(g)$ the logarithm of the spectral radius of $g.$

4.5 Representations of ${\mathsf {G}}$

Let ${\mathsf {V}}$ be a finite-dimensional $\mathbb {K}$ -vector space and $\unicode{x3c6} :{\mathsf {G}}\to \operatorname {\mathrm {{\mathsf {PGL}}}}({\mathsf {V}})$ be an algebraic irreducible representation. Then the weight space associated to $\chi \in \mathbf X({\mathsf {A}})$ is the vector space

$$ \begin{align*}{\mathsf{V}}_\chi=\{v\in {\mathsf{V}}:\unicode{x3c6}(a) v=\chi(a) v \text{ for all } a\in{\mathsf{A}}\}\end{align*} $$

and if $V_\chi \neq 0$ , then we say that $\chi ^\omega \in {\mathsf {E}}^*$ is a restricted weight of $\unicode{x3c6} .$ Theorem 7.2 of Tits [69] states that the set of weights has a unique maximal element with respect to the order $\chi \geq \psi $ if $\chi -\psi $ is a sum of simple roots with non-negative coefficients. This is called the highest weight of $\unicode{x3c6} $ and denoted by $\chi _\unicode{x3c6} .$

We denote by $\|\,\|_\unicode{x3c6} $ a norm on ${\mathsf {V}}$ invariant under $\unicode{x3c6} {\mathsf {K}}$ and such that $\unicode{x3c6} {\mathsf {A}}$ consists on semi-homotheties (that is, diagonal on an orthonormal basis $\mathcal E$ of $V,$ in the classical sense if $\mathbb {K}$ Archimedean, and such that $\|\sum _{e\in \mathcal E}v_ee\|=\max \{|v_e|\}$ if $\mathbb {K}$ is non-Archimedean). If $\mathbb {K}$ is Archimedean, the existence of such a norm is classical (see for example [9, Lemma 6.33]), if $\mathbb {K}$ is non-Archimedean, then this is the content of [60, Théorème 6.1] due to Quint.

For every $g\in {\mathsf {G}}$ , one has then

(19)

Denote by $W_{\chi _\unicode{x3c6} }$ the $\unicode{x3c6} {\mathsf {A}}$ -invariant complement of ${\mathsf {V}}_{\chi _\unicode{x3c6} }.$ The stabilizer in ${\mathsf {G}}$ of ${\mathsf {W}}_{\chi _\unicode{x3c6} }$ is $\check {\mathsf {P}}_{\vartheta ,\mathbb {K}},$ and thus one has a map of flag spaces

(20) $$ \begin{align}(\Xi_\unicode{x3c6},\Xi^*_\unicode{x3c6}):\mathcal F_{\vartheta_\unicode{x3c6}}^{(2)}({\mathsf{G}})\to \operatorname{\mathrm{Gr}}_{\dim {\mathsf{V}}_{\chi_\unicode{x3c6}}}^{(2)}({\mathsf{V}}),\end{align} $$

where $\vartheta _\unicode{x3c6} =\{\sigma \in {{{\Delta }}}:\chi _\unicode{x3c6} -\sigma \textrm { is a weight of }\unicode{x3c6} \}$ . This is a proper embedding which is a homeomorphism onto its image. Here, $\mathscr G_{\dim {\mathsf {V}}_{\chi _\unicode{x3c6} }}^{(2)}({\mathsf {V}})$ is the open $\operatorname {\mathrm {{\mathsf {PGL}}}} ({\mathsf {V}})$ -orbit in the product of the Grassmannians of $(\dim {\mathsf {V}}_{\chi _\unicode{x3c6} })$ -dimensional and $(\dim {\mathsf {V}}-\dim {\mathsf {V}}_{\chi _\unicode{x3c6} })$ -dimensional subspaces.

One has the following proposition from Tits [69] that guarantees existence of certain representations of ${\mathsf {G}}.$ We say that $\phi $ is proximal if $\dim {\mathsf {V}}_{\chi _\unicode{x3c6} }=1.$

Proposition 4.5.1. (Tits [69])

For every $\sigma \in {{{\Delta }}}$ , there exists an irreducible proximal representation of ${\mathsf {G}}$ whose highest restricted weight is $l_\sigma \varpi _\sigma $ for some $l_\sigma \in \mathbb {Z}_{\geq 1}.$

Definition 4.5.2. We will fix and denote by $\unicode{x3c6} _\sigma :{\mathsf {G}}\to \operatorname {\mathrm {{\mathsf {GL}}}}({\mathsf {V}}_\sigma )$ such a set of representations.

4.6 Buseman–Iwasawa cocycle

The Iwasawa decomposition of ${\mathsf {G}}$ states that every $g\in {\mathsf {G}}$ can be written as a product $lzu$ with $l\in {\mathsf {K}}, z\in {\mathsf {A}}$ , and $u\in {\mathsf {U}}_{{{\Delta }}},$ where ${\mathsf {U}}_{{{\Delta }}}$ is the unipotent radical of ${\mathsf {P}}_{{{{\Delta }}}}.$ When $\mathbb {K}$ is non-Archimedean, the Iwasawa decomposition is not unique; however, if $z_1,z_2\in {\mathsf {A}}$ are such that $z_1\in {\mathsf {K}} z_2{\mathsf {U}}_{{{{\Delta }}}}$ , then $\nu (z_1)=\nu (z_2).$

The Buseman–Iwasawa cocycle of ${\mathsf {G}}, \beta :{\mathsf {G}}\times \mathcal F\to {\mathsf {E}},$ is defined, for all $g\in {\mathsf {G}}$ and $k[{\mathsf {P}}_{{{\Delta }}}]\in \mathcal F,$ if $gk=lzu$ is an Iwasawa decomposition of $gk$ , by $\beta (g,k[{\mathsf {P}}_{{{\Delta }}}])=\nu (z).$ Quint proved the following lemma.

Lemma 4.6.1. (Quint [62, Lemmas 6.1 and 6.2])

The function $\beta _\vartheta =p_\vartheta \circ \beta $ factors as a cocycle $\beta _\vartheta :{\mathsf {G}}\times \mathcal F_\vartheta \to {\mathsf {E}}_\vartheta .$

The Buseman–Iwasawa cocycle can also be read from the representations of ${\mathsf {G}}.$ Indeed, Quint [62, Lemme 6.4] states that for every $g\in {\mathsf {G}}$ and $x\in \mathcal F_\vartheta $ , one has

(21)

where $v\in \Xi _{\unicode{x3c6} _\sigma }(x)\in \mathbb {P}({\mathsf {V}}_\sigma )$ is non-zero.

4.7 Gromov product

As in [66], the Gromov product $\mathscr G_\vartheta :\mathcal F^{(2)}_\vartheta \to {\mathsf {E}}_\vartheta $ is defined such that, for every $(x,y)\in \mathcal F^{(2)}_\vartheta $ and $\sigma \in \vartheta ,$ one has

where $\varphi \in \Xi ^*_{\unicode{x3c6} _\sigma }(x)$ and $v\in \Xi _{\unicode{x3c6} _\sigma }(y)$ are the equivariant maps from equation (20).

Remark 4.7.1. Observe that the limiting situation $l_\sigma \varpi _\sigma (\mathscr G_\vartheta (x,y))=-\infty $ occurs when $v\in \ker \varphi ,$ that is, when x and y are no longer transverse flags, so a statement of the form $\varpi _\sigma \mathscr G_\vartheta (x,y) \geq -\kappa $ for all $\sigma \in \vartheta $ is a quantitative version (that depends on ${\mathsf {K}}$ ) of the transversality between x and $y.$

A straightforward computation ([66, Lemma 4.12]) gives, for all $g{\kern-1.5pt}\in{\kern-1.5pt} {\mathsf {G}}$ and $(x,{\kern-1pt}y){\kern-1.5pt}\in{\kern-1.5pt} \mathcal F^{(2)}_\vartheta ,$

(22) $$ \begin{align}\mathscr G_\vartheta(gx,gy)-\mathscr G_\vartheta(x,y)=-(\operatorname{\mathrm{i}}\beta_{\operatorname{\mathrm{i}}\vartheta}(g,x)+\beta_\vartheta(g,y)).\end{align} $$

4.8 Proximality

Recall that $g\in \operatorname {\mathrm {{\mathsf {PGL}}}}_d({\mathsf {V}})$ is proximal if it has a unique eigenvalue with maximal modulus and that the multiplicity of this eigenvalue in the characteristic polynomial of g is $1.$ The associated eigenline is denoted by $g^+\in \mathbb {P}({\mathsf {V}})$ and $g^-$ is its g-invariant complementary subspace.

We say then that $g\in {\mathsf {G}}$ is $\vartheta $ -proximal if for every $\sigma \in \vartheta $ , one has $\unicode{x3c6} _\sigma (g)$ is proximal. In this situation, there exists a pair $(g^-_\vartheta ,g^+_\vartheta )\in \mathcal F^{(2)}_\vartheta ,$ defined by, for every $\sigma \in \vartheta , \Xi _{\unicode{x3c6} _\sigma }(g^+_\vartheta )=\unicode{x3c6} _\sigma (g)^+,$ and every flag $x\in \mathcal F_\vartheta $ in general position with $g^-_\vartheta $ verifies $g^nx\to g^+_\vartheta .$

It is also useful to consider a quantified version of proximality. Given $r,\varepsilon $ positive, we say that g is $(r,\varepsilon )$ -proximal if it is proximal,

$$ \begin{align*}\varpi_\sigma\mathscr G_\vartheta(g_\vartheta^-,g^+_\vartheta)\geq-r\end{align*} $$

for all $\sigma \in \vartheta $ , and for every $x\in \mathcal F_\vartheta $ with $\min _{\sigma _\in \vartheta }\varpi _\sigma \mathscr G_\vartheta (g_\vartheta ^-,x)\geq - \varepsilon ^{-1}$ , one has $d_{\mathcal F_\vartheta }(gx,g^+_\vartheta )\leq \varepsilon .$ More details on the following can be found in [65, Lemma 5.6].

Proposition 4.8.1. (Benoist [6, Corollaire 6.3])

For every $\delta>0$ , there exist $r,\varepsilon>0$ such that if $g\in {\mathsf {G}}$ is $(r,\varepsilon )$ -proximal, then

4.9 Cartan attractors

Consider $g\in {\mathsf {G}}$ and let $g=k_gz_gl_g$ be a Cartan decomposition. We say that $g\in {\mathsf {G}}$ has a gap at $\vartheta $ if for all $\sigma \in \vartheta $ , one has

$$ \begin{align*}\sigma(a(g))>0.\end{align*} $$

In that case, the Cartan attractor of g in $\mathcal F_\vartheta $

$$ \begin{align*}U_\vartheta(g)=k_g[{\mathsf{P}}_{\vartheta}]\end{align*} $$

is well defined: uniquely defined if $\mathbb {K}$ is Archimedian; defined up to a ball of radius $q^{-\min _{\sigma \in \vartheta }\sigma (g)}$ if $\mathbb {K}$ is non-Archimedean (see [59, Remark 2.4]).

Remark 4.9.1. For every $\sigma \in \vartheta $ , one has ${\displaystyle \Xi _{\unicode{x3c6} _\sigma }(U_\vartheta (g))=U_{1}(\unicode{x3c6} _\sigma (g)).}$

Lemma 4.9.2. (Bochi, Potrie, and Sambarino [10, Lemma A.5])

Consider $g,h\in {\mathsf {G}}$ such that h and $gh$ have gaps at every $\sigma \in \vartheta ,$ then one has

The Cartan basin of g is defined, for $\alpha>0,$ by (recall Remark 4.7.1)

$$ \begin{align*}B_{\vartheta,\alpha}(g)= \{x\in\mathcal F_\vartheta: \varpi_\sigma(\mathscr G_\vartheta(U_{\operatorname{\mathrm{i}}\vartheta}(g^{-1})),x)>-\alpha \text{ for all }\sigma\in\vartheta\}.\end{align*} $$

It is clear from the definition that given $\alpha>0$ , there exists a constant $K_\alpha $ such that if $y\in \mathcal F_{\vartheta }$ belongs to $B_{\vartheta ,\alpha }(g)$ , then one has

(23) $$ \begin{align} \|a_\vartheta(g)-\beta_\vartheta(g,y)\|\leq K_\alpha. \end{align} $$

Lemma 4.9.3. (Quint [62, Lemme 6.6])

For every $g\in {\mathsf {G}}$ , one has $a_\vartheta (gh)-a_\vartheta (h)-\beta _\vartheta (g,U_\vartheta (h))\to 0$ as $\min _{\sigma \in \vartheta }\sigma (a(h))\to \infty .$

4.10 General facts on discrete subgroups

We record here some facts related to the title that we will need in the following work.

Lemma 4.10.1. Let $\Delta \subset {\mathsf {G}}$ be a discrete subgroup. Then for every $\varphi \in {\mathsf {E}}^*$ strictly positive on ${\mathsf {E}}^+$ , the exponential rate

$$ \begin{align*}\delta^\varphi_\Delta=\limsup_{t\to\infty}\frac1t\log\#\{g\in\Delta:\varphi(a(g))\leq t\}\end{align*} $$

is finite.

Proof. The proof follows from a computation of the Haar measure of ${\mathsf {G}},$ to be found in [37] for the Archimedean case and in [50, §3.2.7] for the non-Archimedean case, see [61, §4] for details.

We record also the following theorem from Benoist [8].

Theorem 4.10.2. (Benoist [8])

Assume $\mathbb {K}=\mathbb {R}$ and let $\Delta \subset {\mathsf {G}}$ be a Zariski-dense subgroup. Then, the group spanned by the Jordan projections $\{\unicode{x3bb} (g):g\in \Delta \}$ is dense in ${\mathsf {E}}.$

5 Anosov representations

Anosov representations were introduced by Labourie [43] for fundamental groups of negatively curved manifolds and extended to arbitrary finitely generated hyperbolic groups by Guichard and Wienhard [35]. They originated as a tool to study higher rank Teichmüller theory, and are nowadays considered as the higher-rank analog of what is known in pinched negative curvature as convex co-compact groups.

Notation. If is a representation, we will simplify notation and denote, for by $\gamma _\rho =\rho (\gamma ).$

5.1 Real projective-Anosov representations

We begin by recalling Labourie’s original approach. Let be a finitely generated word-hyperbolic group. If is a representation, then we can consider the natural flat bundle automorphism defined as follows. Consider the flat bundle defined by , where $(p,v)\sim (\gamma p,\gamma _\rho v),$ and define $\hat {\mathsf {g}}=(\hat {\mathsf {g}}_t:{\mathsf {E}}_\rho \to {\mathsf {E}}_\rho )_{t\in \mathbb {R}}$ as the induction on the quotient by $t\cdot (p,v)=(\widetilde {\mathsf {g}}_t p, v).$

Definition 5.1.1. The representation $\rho $ is projective-Anosov if there exists a pair of continuous $\rho $ -equivariant maps

such that:

• • for every , one has $\ker \xi ^{d-1}(x)\oplus \xi ^1(y)=\mathbb {R}^d.$ This induces a $\hat {\mathsf {g}}$ -invariant decomposition $\Xi \oplus \Theta ={\mathsf {E}}_\rho $ ;

• • the decomposition ${\mathsf {E}}_\rho =\Xi \oplus \Theta $ is a dominated splitting for $\hat {\mathsf {g}},$ that is, there exist $c,\alpha $ positive such that for every $v\in \Xi _p$ and $w\in \Theta _p$ , one has

  $$ \begin{align*}\frac{\|\hat{\mathsf{g}}_tv\|}{\|\hat{\mathsf{g}}_tw\|}\leq ce^{-\alpha t}\frac{\|v\|}{\|w\|}.\end{align*} $$

One has the following standard consequences, see for example [35, Lemma 3.1] or [17, Lemma 2.5 and Proposition 2.6]. Recall from §4.8 that $g\in \operatorname {\mathrm {{\mathsf {PGL}}}}_d(\mathbb {R})$ is proximal if the Jordan block associated to the eigenvalues with maximal modulus is one-dimensional.

Lemma 5.1.2. If $\rho $ is projective-Anosov, then for every hyperbolic $\gamma $ , one has $\gamma _\rho $ is proximal with attracting line $\xi ^1(\gamma _+).$ In particular, the entropy

The equivariant maps $\xi ^1$ and $\xi ^{d-1}$ are Hölder-continuous.

Proof. Let us add a word on finiteness of entropy. Recall from Bowditch [13] that, since

is hyperbolic, its action on the space of pairwise distinct triples

is properly discontinuous and co-compact. If

is hyperbolic, one can choose then $\eta \in [\gamma ]$ (the conjugacy class of $\gamma $ ) whose fixed points are far apart by a constant independent of $\gamma .$ Since the image $\gamma _\rho $ is proximal and the equivariant maps are continuous, one has that $\eta _\rho $ is $(r,\varepsilon )$ -proximal for constants $r,\varepsilon $ independent of $[\gamma ].$ By Proposition 4.8.1, one has then $|\log \|\eta _\rho \|-\unicode{x3bb} _1(\eta _\rho )|<K$ for some K independent of $\eta .$ If follows then that for every $t\in \mathbb {R}_+$ ,

Finiteness of entropy then follows from Lemma 4.10.1.

We use the equivariant maps to construct a bundle whose fiber at is

$$ \begin{align*}\widetilde{\mathsf{ F}}_{(x,y)}=\{(\varphi,v)\in\xi^{d-1}(x)\times\xi^1(y):\varphi(v)=1\}/\sim,\end{align*} $$

where $(\varphi ,v)\sim (-\varphi ,-v).$ This bundle is equipped with a -action $\gamma (\varphi ,v)= (\varphi \circ \gamma ^{-1}_\rho ,\gamma _\rho v)$ and an $\mathbb {R}$ -action $(\widetilde {\mathsf {g^\rho }}_t:\widetilde {\mathsf {F}}\to \widetilde {\mathsf {F}})_{t\in \mathbb {R}}$ defined by $\widetilde {\mathsf {g^\rho }}_t\cdot (\varphi ,v)=(e^t\varphi ,e^{-t}v).$ Let and denote by ${\mathsf {g}}^\rho =({\mathsf {g}}^\rho _t:{\mathsf {F}}\to {\mathsf {F}})_{t\in \mathbb {R}}$ the induced flow on the quotient. It is usually called the geodesic flow of $\rho $ .

Theorem 5.1.3. (Bridgeman et al [17])

The above -action is properly discontinuous and co-compact. The flow ${\mathsf {g}}^\rho $ is Hölder-continuous and metric-Anosov with stable/unstable laminations (induced on the quotient by)

$$ \begin{align*}\tilde W^{\mathrm{s}}((x,y,(\varphi,v)) & =\{(x,\cdot,(\varphi,\cdot))\in\widetilde{\mathsf{F}}\},\\ W^{{\mathrm{u}}}((x,y,(\varphi,v)) & =\{(\cdot,y,(\cdot,v))\in\widetilde{\mathsf{F}}\}.\end{align*} $$

It is moreover Hölder-conjugated to the Gromov–Mineyev geodesic flow ${\mathsf {g}}$ of consequently, this latter flow is also metric-Anosov.

Therefore, hyperbolic groups admitting a real projective-Anosov representation verify Assumption B and are thus subject of a Ledrappier correspondence (§3). It is established by Carvajales [22, Appendix] that ${\mathsf {g}}^\rho $ is topologically mixing (regardless of the Zariski closure of $\rho $ ) and thus mixing for any equilibrium state.

5.2 Arbitrary ${\mathsf {G}},$ coarse geometry viewpoint

Let ${\mathsf {G}}$ be as in §4. We use freely the notation introduced there and fix from now on a subset $\vartheta \subset {{{\Delta }}}$ of simple roots.

Let be a finitely generated group and denote, for by $|\gamma |$ the word length with respect to a fixed finite symmetric generating set of

Definition 5.2.1. A representation is $\vartheta $ -Anosov if there exist $c,\mu $ positive such that for all and $\sigma \in \vartheta $ , one has

(24) $$ \begin{align}\sigma(a(\gamma_\rho))\geq \mu|\gamma|-c.\end{align} $$

The constants c and $\mu $ will be referred to as the domination constants of $\rho .$

Theorem 5.2.2 follows from the main result by Kapovich, Leeb, and Porti [40] and the standard facts from representation theory stated in §4.5, a proof was also reported by Bochi, Potrie, and Sambarino [10].

Theorem 5.2.2. If is $\vartheta $ -Anosov, then is word-hyperbolic. If moreover $\mathbb {K}=\mathbb {R}$ , then for every $\sigma \in \vartheta $ , the representation is projective-Anosov (as in §5.1).

The following lemma is essentially a consequence of [10, Lemma 4.9]. See [59, Proposition 3.5] for details concerning the non-Archimedean case. The last assertion is classical.

Proposition 5.2.3. (Bochi, Potrie, and Sambarino [10, Proposition 4.9])

If is $\vartheta $ -Anosov, then for any geodesic ray $\{\alpha _n\}_0^\infty $ with endpoint x, the limits

$$ \begin{align*}\xi^\vartheta_\rho(x):=\lim_{n\to\infty}U_\vartheta(\rho(\alpha_n)),\quad \xi^{\operatorname{\mathrm{i}}\vartheta}_\rho(x):=\lim_{n\to\infty}U_{\operatorname{\mathrm{i}}\vartheta}(\rho(\alpha_n))\end{align*} $$

exist and do not depend on the ray; they define continuous $\rho $ -equivariant transverse maps , . If is hyperbolic, then $\gamma _\rho $ is $\vartheta $ -proximal with attracting point $\xi ^\vartheta (\gamma ^+)=(\gamma _\rho )_\vartheta ^+.$

Proposition 5.2.3 readily implies the following lemma (recall Remark 4.7.1).

Lemma 5.2.4. Let be $\vartheta $ -Anosov, a divergent sequence, and Then, as $n\to \infty ,$ one has

$$ \begin{align*}\gamma_n\to x&\Leftrightarrow U_\vartheta(\rho(\gamma_n))\to\xi^\vartheta(x)\\&\Leftrightarrow\text{ there exists }\sigma\in\vartheta\textrm{ such that }\varpi_\sigma\mathscr G_\vartheta(U_{\operatorname{\mathrm{i}}\vartheta}(\rho(\gamma_n)),\xi^\vartheta(x))\to-\infty.\end{align*} $$

We finally record the following useful lemma.

Lemma 5.2.5. (Pozzetti, Sambarino, and Wienhard [59, Lemma 3.6])

Let

be $\vartheta $ -Anosov, then for every $\varepsilon>0$ , there exists L such that

where $\mathcal N_\varepsilon $ denotes the $\varepsilon $ -tubular-neighborhood.

Remark 5.2.6. Non-Archimedean case

The existence of continuous $\rho $ -equivariant maps implies, when $\mathbb {K}$ is non-Archimedean, that the boundary of is necessarily a Cantor set and thus is virtually free. The Gromov–Mineyev of is thus a suspension of a sub-shift of finite type and is, hence, metric-Anosov.

Setting. A $\vartheta $ -Anosov representation is fixed from now on. By §5.1 for $\mathbb {K}=\mathbb {R}$ or $\mathbb {C},$ and the preceding paragraph for non-Archimedean $\mathbb {K},$ the Gromov–Mineyev flow ${\mathsf {g}}$ of satisfies Assumption B.

5.3 The refraction cocycle verifies Assumption C

By the equivariant boundary maps of $\rho $ , one can pullback the Buseman–Iwasawa cocycle of ${\mathsf {G}}$ to obtain a Hölder-cocycle on the boundary of .

Definition 5.3.1. The refraction cocycle of $\rho $ is

,

The limit cone of $\unicode{x3b2} _{}$ will be denoted by $\mathscr L_{\vartheta ,\rho }$ and referred to as the $\vartheta $ -limit cone of $\rho $ . The period computation below implies it is the smallest closed cone of ${\mathsf {E}}_\vartheta $ that contains the projections We prove moreover that $\unicode{x3b2} _{}$ verifies Assumption C from §3.4.

Lemma 5.3.2. The periods of $\unicode{x3b2} _{}$ are $\unicode{x3b2} _{}(\gamma ,\gamma _+)=\unicode{x3bb} _\vartheta (\gamma _\rho )$ , consequently Assumption C holds for $\unicode{x3b2} _{}$ , in particular, $\operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*=\{\varphi \in (\mathscr L_{\vartheta ,\rho })^*:\mathscr h_\varphi <\infty \}$ .

Proof. The first assertion follows from Proposition 5.2.3. To prove Assumption C holds, one considers any $\sigma \in \vartheta $ and the representation $\unicode{x3c6} _\sigma .$ By Theorem 5.2.2, the composition is projective-Anosov and thus, by equation (19) and Lemma 5.1.2, the form $\varpi _\sigma \in ({\mathsf {E}}_\vartheta )^*$ has finite entropy. The last assertion follows from Lemma 3.4.2.

Lemma 3.4.2 and Theorem 3.2.2 give then the following corollary.

Corollary 5.3.3. There exists a Hölder-continuous function such that for every hyperbolic , one has $\int _{[\gamma ]}\mathcal J_{\vartheta ,\rho }=\unicode{x3bb} _\vartheta (\gamma _\rho ).$ For every $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ , the -action on defined by

(25)

is properly discontinuous and co-compact. The $\mathbb {R}$ -translation flow induces on the quotient a flow $\phi ^\varphi =(\phi ^\varphi _t:{\unicode{x3c7} ^{\varphi }}\to {\unicode{x3c7} ^{\varphi }})_{t\in \mathbb {R}}$ (bi)-Hölder-conjugated to the reparameterization of ${\mathsf {g}}$ by $\varphi \circ \mathcal J_{\vartheta ,\rho }.$

Definition 5.3.4. The function $\mathcal J_{\vartheta ,\rho }$ will be referred to as the Ledrappier potential of $\rho .$ The flow $\phi ^\varphi $ will be called the $\varphi $ -refraction flow of $\rho $ .

5.4 The $\vartheta $ -limit cone

We mimic some celebrated results by Benoist [7] for Zariski-dense subgroups and $\vartheta ={{{\Delta }}}.$

Lemma 5.4.1. (Benoist [6, Proposition 5.1])

For every compact set $L\subset {\mathsf {G}}$ , there exists a compact set $H\subset {\mathsf {E}}$ such that for every $g\in {\mathsf {G}}$ , one has $a(LgL)\subset a(g)+H.$

Let us also denote $a_\vartheta =p_\vartheta \circ a.$

Proposition 5.4.2. Let be a $\vartheta $ -Anosov representation. Then there exists a compact set $D\subset {\mathsf {E}}_\vartheta $ such that

Proof. As

is finitely generated and word-hyperbolic, there exist $\kappa>0$ and two elements

such that for every non-torsion

, there exists $f\in \{u,v\}$ such that $f\gamma $ verifies

As $\rho $ is $\vartheta $ -Anosov, the above equation implies the element $\rho (f\gamma )$ is $(r,\varepsilon )$ -proximal on $\vartheta $ for some r only depending on $\kappa .$ By Proposition 4.8.1, one has

for some K only depending on $\kappa $ and $\rho .$ We consider the compact set H from the lemma above applied to $L=\rho (\{u^{-1},v^{-1}\})$ and we let $D:=p_\vartheta (H)+B(0,K).$

We will mainly use the following direct consequence.

Corollary 5.4.3. If $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ , then the exponential rate

Proof. If $\sigma \in \vartheta $ then, since both intersections $\ker \varpi _\sigma \cap \mathscr L_{\vartheta ,\rho }$ and $\ker \varphi \cap \mathscr L_{\vartheta ,\rho }$ vanish (the first one always does, the second one by the assumption on $\varphi $ ), the function $\varphi /\varpi _\sigma $ is bounded below away from zero on $\mathscr L_{\vartheta ,\rho }.$ By Proposition 5.4.2, there exist positive c and C such that for all hyperbolic , one has

$$ \begin{align*}\varphi(a(\gamma_\rho ))\geq c\varpi_\sigma(a(\gamma_\rho) )-C.\end{align*} $$

Lemma 4.10.1 gives then the desired result.

5.5 Patterson–Sullivan theory along the Anosov roots: existence

In this section, we will construct, for each $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ , a $\unicode{x3b2} _{\varphi }$ -Patterson–Sullivan measure. (The $\vartheta $ -Anosov property is not really used until the uniqueness corollary. The existence presented here works for any discrete group whose limit cone on ${\mathsf {E}}$ does not intersect any wall associated to $\vartheta $ and replacing by

$$ \begin{align*}\bigcap_{n\in\mathbb{N}}\overline{\{U_\vartheta(g):g\in\Delta \textrm{ with }\min_{\sigma\in\vartheta}\sigma(a(g))\geq n\}}.)\end{align*} $$

The procedure is standard and follows the original idea by Patterson.

We begin by considering the Dirichlet series

It is convergent for every $s>\delta ^\varphi $ and divergent for every $s<\delta ^\varphi .$ As it is customary when constructing Patterson–Sullivan measures, we can assume throughout this subsection that $\mathcal P^\varphi (\delta ^\varphi )=\infty ,$ otherwise, one would consider the series

for some real function h defined, for example, as by Quint [62, Lemma 8.5].

For $s>\delta ^\varphi $ , consider the probability measure on $\mathcal F_\vartheta $ defined by

Lemma 5.5.1. For every

, the signed measure

weakly converges to $0$ as $s\searrow \delta ^\varphi .$

This is a standard argument that can be found, for example, in [59, Lemma 5.11].

Proof. It is sufficient to check the convergence for continuous functions. If $f:\mathcal F_\vartheta \to \mathbb {R}$ is continuous, then

By Lemma 4.9.2 and uniform continuity of f, the convergence follows.

Lemma 5.5.2. Let $\nu ^\varphi $ be any weak-star limit of $\nu _s$ when $s\searrow \delta ^\varphi .$ Then, the support of $\nu ^\varphi $ is contained in Moreover, for every , one has

$$ \begin{align*}\frac{d\rho(\eta)_*\nu^\varphi}{\,d\nu^\varphi}(x)=q^{-\varphi(\,\beta_\vartheta(\eta^{-1}_\rho,x))}.\end{align*} $$

Proof. The first statement follows at once from Lemma 5.2.5 since we assumed $\mathcal P^\varphi (\delta ^\varphi )=\infty .$ For the second statement, consider a sequence $s_k\searrow \delta ^\varphi $ such that $\nu _{s_k}\to \nu ^\varphi .$ One then has

where by Quint’s Lemma 4.9.3 and the fact that $\varphi \circ p_\vartheta =\varphi ,$ one has $\varepsilon '(\eta ,\gamma )\to 0$ as $\min _{\sigma \in \vartheta }\sigma (a(\gamma _\rho ))\to \infty .$ Taking the limit as $s_k\searrow \delta ^\varphi $ , one has, since we assumed $\mathcal P^\varphi (\delta ^\varphi )=\infty ,$ that only elements

with arbitrary big $|\gamma |$ count in the sum. Since $\rho $ is $\vartheta $ -Anosov, this is equivalent to considering elements

such that

$$ \begin{align*}\min_{\sigma\in\vartheta}\sigma(a(\gamma_\rho) )\end{align*} $$

is arbitrarily big. The result then follows as $\varepsilon (\eta ,s_k)\to 0$ by Lemma 5.5.2 and $\varepsilon '(\eta ,\gamma )$ is arbitrarily small.

Since Assumption C holds for $\unicode{x3b2} _{}$ (Lemma 5.3.2), §3.3 is applied to give the following corollary.

Corollary 5.5.3. For every $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ , there exists a $\unicode{x3b2} _{\varphi }$ -Patterson–Sullivan measure $\mu ^\varphi :=(\xi ^\vartheta )_*\nu ^\varphi $ of exponent $\delta ^\varphi .$ Such a measure is ergodic and, moreover, one has $\delta ^\varphi =\mathscr h_\varphi .$ If $\psi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ is such that $\mu ^\psi \ll \mu ^\varphi $ , then for every hyperbolic

, one has

and, in particular, $\mu ^\psi =\mu ^\varphi .$

The above corollary was previously established by Dey and Kapovich [29, Main Theorem] for real algebraic groups, $\operatorname {\mathrm {i}}$ -invariant functionals $\varphi \in \operatorname {\mathrm {int}}({\mathfrak a}^+)^*$ , and $\operatorname {\mathrm {i}}$ -invariant subsets $\vartheta .$ The equality $\delta ^\varphi =\mathscr h_\varphi $ , together with more information, can also be found in Glorieux, Monclair, and Tholozan’s work [32, Theorem 2.31(2)] for real groups.

Remark 5.5.4. We conclude by remarking that for $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*,$ the existence assumptions of §3.3 are guaranteed for $\unicode{x3b2} _{\varphi }.$ Indeed, Proposition 5.5.3 states the existence of a Patterson–Sullivan measure $\mu ^\varphi $ for $\unicode{x3b2} _{\varphi }.$ The cocycle

$$ \begin{align*}\bar{\unicode{x3b2}}_{}(\gamma,x)=\operatorname{\mathrm{i}}\beta_{\operatorname{\mathrm{i}}\vartheta}(\gamma_\rho ,\xi^{\operatorname{\mathrm{i}}\vartheta}(x))\end{align*} $$

is dual to $\unicode{x3b2} _{}$ and moreover, from equation (22), the function

(26) $$ \begin{align}[x,y]_\varphi=\varphi(\mathscr G_\vartheta(\xi^{\operatorname{\mathrm{i}}\vartheta}(x),\xi^\vartheta(y)))\end{align} $$

is a Gromov product for the pair $(\bar {\unicode{x3b2} }_{\varphi },\unicode{x3b2} _{\varphi }).$ Finally, exchanging $\vartheta $ with $\operatorname {\mathrm {i}}\vartheta ,$ Proposition 5.5.3 provides a Patterson–Sullivan measure for $\bar {\unicode{x3b2} }_{\varphi }.$ We can thus apply the results from §§3.4 and 3.5.

5.6 Cartan’s basins have controlled overlaps

The job of understanding the overlaps of Cartan’s basins for Anosov representations has been carried out by Pozzetti, Sambarino, and Wienhard [58]. The idea is to compare the Cartan’s basins of elements $\gamma _\rho ,$ for hyperbolic with the coarse cone type of $\gamma .$

Let $c_0,c_1$ be positive and $I\subset \mathbb {Z}$ an interval. Then, a $(c_0,c_1)$ -quasigeodesic is a sequence

such that for every pair $j,l$ in the interval I, one has

The coarse cone type at infinity of

consists of endpoints on

of quasi geodesic rays based at $\gamma ^{-1}$ passing through the identity (see Figure 1):

Figure 1 The coarse cone type at infinity, the black broken lines are $(c_0,c_1)$ -quasi-geodesics.

Pozzetti, Sambarino, and Wienhard [58, Proposition 3.3] together with Bochi, Potrie, and Sambarino [10, Lemma 2.5] (see also [59, Proposition 3.3]) give the following. The last statement can be found in [58, Proposition 3.5].

Proposition 5.6.1. (Pozzetti, Sambarino, and Wienhard [58, Proposition 3.3])

For a given $\alpha>0$ , there exist $c_0,c_1,$ depending on $\alpha $ and the domination constants of $\rho ,$ such that for every hyperbolic , one has

$$ \begin{align*}(\xi^\vartheta)^{-1}(B_{\theta,\alpha}(\gamma_\rho))\subset {\mathcal C}^{c_0,c_1}_\infty(\gamma).\end{align*} $$

Reciprocally, there exists $\alpha ';$ only depending on $c_0,c_1$ and the domination constants of $\rho ,$ such that

$$ \begin{align*}{\mathcal C}^{c_0,c_1}_\infty(\gamma)\subset(\xi^\vartheta)^{-1}(B_{\theta,\alpha'}(\gamma_\rho)).\end{align*} $$

There exists then $N\in \mathbb {N},$ only depending on $c_0,c_1$ and the domination constants of $\rho $ such that, for all $t\in \mathbb {N}$ , the family

$$ \begin{align*}\mathcal U_t=\{\gamma_\rho B_{\vartheta,\alpha}(\gamma_\rho ):t\leq|\gamma|\leq t+1\}\end{align*} $$

is an open covering of and such that every element $\xi ^\vartheta (x)$ belongs to at most N elements of the covering $\mathcal U_t.$

5.7 Sullivan’s shadow lemma

We establish now a version of Sullivan’s shadow lemma.

Lemma 5.7.1. Consider $\varphi \in ({\mathsf {E}}_\vartheta )^* $ and let $\mu $ be a $\unicode{x3b2} _{\varphi }$ -Patterson–Sullivan measure of exponent $\delta .$ Let $\nu =\xi ^{\vartheta }_*\mu .$ Then given $\alpha>0$ , there exist constants $C, C'$ , and $L\in \mathbb {N}$ such that for every with $|\gamma |\geq L$ , one has

$$ \begin{align*}q^{-\delta\cdot\varphi(a(\gamma_\rho))}C' \leq\nu(\gamma_\rho B_{\vartheta,\alpha}(\gamma_\rho ))\leq Cq^{-\delta\cdot\varphi(a(\gamma_\rho))}.\end{align*} $$

Proof. It suffices to establish that there exist $\alpha $ and $\kappa>0$ such that for all large enough , one has $\nu (B_{\vartheta ,\alpha }(\gamma _\rho ))\geq \kappa .$ Indeed, using this fact, the lemma follows from the defining equations (27) and (23).

To establish the desired lower bound, we suppose by contradiction that there exists $\alpha _n\to \infty , \gamma _n\to \infty $ such that $\nu (B_{\vartheta ,\alpha _n}((\gamma _n)_\rho ))\to 0$ as $n\to \infty .$ We can then extract a subsequence $(\gamma _{n_k})$ such that

$$ \begin{align*}U_{\operatorname{\mathrm{i}}\vartheta}((\gamma_{n_k}^{-1})_\rho)\to Y\in\mathcal F_{\operatorname{\mathrm{i}}\vartheta},\ k\to\infty.\end{align*} $$

Moreover, since $\rho $ is $\vartheta $ -Anosov, Lemma 5.2.5 guarantees that $Y=\xi ^{\operatorname {\mathrm {i}}\vartheta }(y)$ for some Also, since $\nu (B_{\vartheta ,\alpha _{n_k}}((\gamma _{n_k})_\rho ))\to 0$ and $\alpha _{n_k}\to \infty $ , we get that the complement

$$ \begin{align*} &(B_{\vartheta,\alpha_{n_k}}((\gamma_{n_k})_\rho))^c\\ & \quad =\{X\in\mathcal F_\vartheta:\text{ there exists }\sigma\in\vartheta\ \textrm{such that}\ \varpi_\sigma\mathscr G_\vartheta(U_{\operatorname{\mathrm{i}}\vartheta}((\gamma_{n_k}^{-1})_\rho),X)\leq-\alpha_n\}\end{align*} $$

converges to the subset of $\mathcal F_\vartheta $

$$ \begin{align*}\{X\in\mathcal F_\vartheta:(X,\xi^{\operatorname{\mathrm{i}}\vartheta}(y))\notin\mathcal F^{(2)}_\vartheta\},\end{align*} $$

and that this subset has total $\nu $ -mass. Since the support of $\nu $ is contained in and the equivariant maps are transverse (Proposition 5.2.3), one has that

$$ \begin{align*}\{\xi^\vartheta(y)\}=\{\xi^\vartheta(x):(\xi^\vartheta(x),\xi^{\operatorname{\mathrm{i}}\vartheta}(y))\notin\mathcal F^{(2)}_\vartheta\}\end{align*} $$

has total $\nu $ -mass. However, considering with $\gamma y\neq y$ , we get, since

(27) $$ \begin{align}\frac{d(\gamma_\rho) _*\nu}{\,d\nu}(\cdot)=q^{-\delta\cdot\varphi(\,\beta_\theta(\gamma_\rho^{-1},\cdot))},\end{align} $$

that $\nu \{\xi ^1(\gamma y)\}>0,$ contradicting that $\{\xi ^\vartheta (y)\}$ has total $\nu $ -mass.

Corollary 5.7.2. For every $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*$ , one has

Proof. We apply Sullivan’s shadow Lemma 5.7.1 to the measure $\nu ^\varphi $ of Lemma 5.5.2. Indeed, considering the coverings of

given by Proposition 5.6.1, one has

for all large enough $t,$ giving divergence of the desired series.

5.8 Patterson–Sullivan theory along the Anosov roots: surjectivity

We prove here surjectivity of the map $\varphi \mapsto \mu ^\varphi $ defined in §5.5.

The following proposition should be compared with [58, Theorem 5.14], where a similar result is obtained for measures on the flag space $\mathcal F_\theta ,$ for $\theta $ not necessarily equal to $\vartheta $ but assuming that $\vartheta \cap \theta \neq \emptyset .$

Proposition 5.8.1. Consider $\varphi \in ({\mathsf {E}}_\vartheta )^*.$ If there exists a $\unicode{x3b2} _{\varphi }$ -Patterson–Sullivan measure $\mu $ of exponent $\delta $ , then $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*, \delta = \delta ^\varphi $ , and $\mu =\mu ^\varphi .$

Proof. We let $\nu =\xi ^\vartheta _*\mu .$ Using Proposition 5.6.1, we get a family of coverings $\mathcal U_t$ with bounded overlap. In combination with Lemma 5.7.1, one has for t large enough that

for some constant $K>0.$ This is to say, there exists $\kappa>0$ such that for all $t\in \mathbb {R}_+$ large, one has $\sum _{\gamma :t\leq |\gamma |\leq t+1}e^{-\delta \varphi (a(\gamma _\rho ))}\leq \kappa ,$ which gives in turn that

$$ \begin{align*}\sum_{\gamma:|\gamma|\leq t}e^{-\delta\varphi(a(\gamma_\rho))}\leq\kappa t.\end{align*} $$

A standard argument (using for example §4.8) permits to replace Cartan projections with Jordan projections giving

$$ \begin{align*}\sum_{[\gamma]:p([\gamma])\leq t}e^{-\delta\varphi(\unicode{x3bb}(\gamma_\rho))}=\sum_{[\gamma]:p([\gamma])\leq t}e^{-\ell_{\delta\mathcal J_{\vartheta,\rho}^\varphi}(\gamma)}\leq\kappa' t,\end{align*} $$

for a suitable $\kappa ',$ where $p([\gamma ])$ is the ${\mathsf {g}}$ -period of the periodic orbit associated to $[\gamma ],$ and $\mathcal J_{\vartheta ,\rho }^\varphi $ is the Ledrappier potential of $\unicode{x3b2} _{\varphi }.$ Equation (5) for the pressure function gives then

$$ \begin{align*}P(-\delta\mathcal J_{\vartheta,\rho}^\varphi)\leq0.\end{align*} $$

Consequently, Lemma 2.2.7 gives that $\mathcal J_{\vartheta ,\rho }^\varphi $ is Livšic-cohomologous to a positive function, this is to say, $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*.$ Finally, since Remark 5.5.4 guarantees the existence assumptions of §3.3 for $\unicode{x3b2} _{\varphi },$ the remaining two equalities in the statement follow from Corollary 3.3.2.

5.9 The critical hypersurface parameterizes Patterson–Sullivan measures

By Lemma 5.3, Assumption C holds for $\unicode{x3b2} _{}$ and thus §3.4 applies. Define the $\vartheta $ -critical hypersurface, respectively $\vartheta $ -convergence domain, of $\rho $ by

Moreover, by Corollary 5.5.3, one has $\delta ^\varphi =\mathscr h_\varphi $ , so one has the equalities

where the last equality comes from Corollary 5.7.2.

For $\varphi \in \mathcal Q_{\vartheta ,\rho }$ , we consider the dynamical intersection map $\mathbf {I}_\varphi =\mathbf {I}_\varphi ^{\unicode{x3b2} _{}}:({\mathsf {E}}_\vartheta )^*\to \mathbb {R},$ associated to the cocycle $\unicode{x3b2} _{}$ as in §3.4 and defined by

where

Let $\mathrm {Ann}(\mathscr L_{\vartheta ,\rho })$ be the annihilator of the $\vartheta $ -limit cone and denote by

$$ \begin{align*}\pi^\vartheta_\rho:({\mathsf{E}}_\vartheta)^*\to({\mathsf{E}}_\vartheta)^*/\mathrm{Ann}(\mathscr L_{\vartheta,\rho})\end{align*} $$

the quotient projection. As before, the map $\mathbf {I}^{\unicode{x3b2} _{}}$ is also well defined on $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })\times ({\mathsf {E}}_\vartheta )^*/\mathrm {Ann}(\mathscr L_{\vartheta ,\rho }).$

Some statements in the following corollary were previously established by Sambarino [64] for $\mathbb {K}=\mathbb {R}$ and Zariski-dense $\vartheta $ -Anosov representations of closed negatively curved manifolds.

Corollary 5.9.1. The sets $\mathcal Q_{\vartheta ,\rho }$ and $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ are closed co-dimension-one analytic sub-manifolds. The latter bounds the strictly convex set $\pi ^\vartheta _\rho (\mathcal D_{\vartheta ,\rho }).$ The map

$$ \begin{align*}\varphi\mapsto {\mathsf{T}}_\varphi\pi^\vartheta_\rho(\mathcal Q_{\vartheta,\rho})=\ker\mathbf{I}_\varphi\end{align*} $$

is an analytic diffeomorphism between $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ and directions in the relative interior of $\mathscr L_{\vartheta ,\rho }.$

We now prove the following proposition.

Proposition 5.9.2. The map $\varphi \mapsto \mu ^\varphi $ is an analytic homeomorphism from the manifold $\pi ^\vartheta _\rho (\mathcal Q_{\vartheta ,\rho })$ to the space of Patterson–Sullivan measures supported on

Proof. By uniqueness in Corollary 5.5.3, the map $\varphi \mapsto \mu ^\varphi $ is well defined and injective. Regularity follows from Remark 3.3.3 and analytic variation of equilibrium states (Theorem 2.3.3). Surjectivity follows from Proposition 5.8.1.

Proposition 5.9.2 was previously established by Lee and Oh [46, Theorem 1.3] for $\mathbb {K}=\mathbb {R}$ and ${{{\Delta }}}$ -Anosov Zariski-dense representations. The convergence domain $\mathcal D_{{{{\Delta }}},\rho }$ is dual to Quint’s growth indicator function [61].

Remark 5.10. Observe that, by definition, a $\unicode{x3b2} _{\varphi }$ -Patterson–Sullivan measure has its support on and thus on when pushed to $\mathcal F_\vartheta .$ One could more generally study measures on $\mathcal F_\vartheta $ verifying

(28) $$ \begin{align}\frac{d(\gamma_\rho) _*\nu}{\,d\nu}(\cdot)=q^{-\delta\cdot\varphi(\,\beta_\theta(\gamma_\rho^{-1},\cdot))},\end{align} $$

without imposing conditions on their support. Such measures exist, for example, the ${\mathsf {K}}$ -invariant measure on $\mathcal F_\vartheta ,$ but their exponent is too large. The question would be totally settled if the following had an affirmative answer: Is the support of a measure verifying equation (28) with $\delta =\delta ^\varphi $ necessarily contained on ?

5.11 Variation of the critical hypersurface

We record the following consequence of Bridgeman et al [17, §6.3]

Corollary 5.11.1. (Bridgeman et al [17])

Let be an analytic family of $\vartheta $ -Anosov representations. Then, Livšic-cohomology class of the Ledrappier potential associated varies analytically with $u.$

Consequently, we can apply Corollary 2.6.5 to obtain the following corollary.

Corollary 5.11.2. Let be an analytic family of $\vartheta $ -Anosov representations. Then the critical hypersurface $\mathcal Q_{\vartheta ,\rho _u}$ varies analytically (on compact sets of ${\mathsf {E}}_\vartheta $ ) with the representation $u.$

5.12 Consequences of the skew-product structure

Consider $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*.$ By Remark 5.5.4, we can freely apply results from §§3.4 and 3.5 to the cocycle $\unicode{x3b2} _{\varphi }.$

Let ${\mathsf {u}}_\varphi ={\mathsf {T}}_{\mathscr h_\varphi \varphi }\mathcal Q_{\vartheta ,\rho }\in \mathbb {P}(\mathscr L_{\vartheta ,\rho })$ be the growth direction of $\varphi $ . By §3.4, the half-line ${\mathsf {u}}_\varphi \cap \mathscr L_{\vartheta ,\rho }$ lies in the relative interior of $\mathscr L_{\vartheta ,\rho }$ (and every direction in this relative interior is obtained in this fashion).

Consider the $\varphi $ -Bowen–Margulis measure $\Omega ^\varphi $ on defined as induction on the quotient by

(29) $$ \begin{align}e^{-\delta^\varphi[\cdot,\cdot]_\varphi}\bar\mu^\varphi\otimes\mu^\varphi\otimes \,d\operatorname{\mathrm{Leb}.}\end{align} $$

Consider $u_\varphi \in {\mathsf {u}}_\varphi $ with $\varphi (u_\varphi )=1$ and denote by the directional flow induced on the quotient of

$$ \begin{align*}t\cdot(x,y,v)=(x,y,v-tu_\varphi).\end{align*} $$

The ergodic dichotomy from §3.6 then gives the following theorem.

Theorem 5.12.1. Assume $\mathbb {K}=\mathbb {R}$ and $\rho $ is Zariski-dense, and let $\varphi \in \operatorname {\mathrm {int}}(\mathscr L_{\vartheta ,\rho })^*.$ If $|\vartheta |\leq 2$ , then the directional flow $\unicode{x3c9} ^\varphi $ is ergodic with respect to $\Omega ^\varphi ,$ in particular, $\mathcal K(\unicode{x3c9} ^\varphi )$ has total mass. If $|\vartheta |\geq 4$ , then $\mathcal K(\unicode{x3c9} ^\varphi )$ has measure $0.$

Proof. The non-arithmeticity assumption for $\unicode{x3b2} _{}$ holds by Benoist’s Theorem 4.10.2 and thus Corollary 3.6.1 applies.

5.13 Directional conical points

The present task is to study the set of points on that are conical in the direction ${\mathsf {u}}_\varphi .$

Consider and a sequence with $\gamma _n\to y.$ Then we say that $\gamma _n$ converges conically to y if for every , the sequence $\gamma _n^{-1}(z,y)$ remains on a compact subset of

Remark 5.13.1. Equivalently, since any compact subset of is contained in a compact subset of the form for a fixed $\kappa ,$ one has that $\gamma _n\to y$ conically if and only if there exists a geodesic ray $\{\alpha _i\}_0^\infty $ on converging to $y,$ such that $\{\gamma _n\}$ is at bounded Hausdorff distance from $\{\alpha _i\}_0^\infty .$ It follows then the existence of constants, $c_0,c_1$ , such that for all n, one has

(30) $$ \begin{align}\gamma_n^{-1}y\in{\mathcal C}^{c_0,c_1}_\infty(\gamma_n).\end{align} $$

Let us fix an (auxiliary) Euclidean norm on ${\mathsf {E}}_\vartheta $ and denote by $B(v,r)$ the associated ball of radius r about $v.$ The tube of size r about ${\mathsf {u}}_\varphi $ is the tubular neighborhood:

$$ \begin{align*}{\mathbb T}_r({\mathsf{u}}_\varphi)=\{v\in{\mathsf{E}}_\vartheta:B(v,r)\cap{\mathsf{u}}_\varphi\neq\emptyset\}.\end{align*} $$

Definition 5.13.2. We say that is $(r,\varphi )$ -conical if there exists a conical sequence converging to y such that for all n,

$$ \begin{align*}a_\vartheta((\gamma_n)_\rho )\in {\mathbb T}_r({\mathsf{u}}_\varphi).\end{align*} $$

We say that y is $\varphi $ -conical if it is $(r,\varphi )$ -conical for some $r.$

Let us denote by the set of $(r,\varphi )$ -conical points and by the set of $\varphi $ -conical points. We now establish the following dichotomy.

Theorem 5.13.3. Assume $\mathbb {K}=\mathbb {R}$ and that $\rho $ is Zariski-dense. If $|\vartheta |\leq 2$ , then if $|\vartheta |\geq 4$ , then

Theorem 5.13.3 follows directly from Theorem 5.12.1 and the following proposition. Let us denote by the quotient projection.

Proposition 5.13.4. A point belongs to if and only if for every pair , one has ${{\mathsf {p}}}(x,y,v)\in \mathcal K({\unicode{x3c9} }^\varphi ).$

Proof. If

is such that

then consider $r{\kern-1pt}>{\kern-1pt}0$ and $\gamma _n{\kern-1pt}\to{\kern-1pt} y$ conically such that $a((\gamma _n)_\rho )\in {\mathbb T}_r({\mathsf {u}}_\varphi ).$ By equation (30), there exists $\varepsilon $ given by Proposition 5.6.1 (only depending on $c_0$ and $c_1$ ) such that for all n,

$$ \begin{align*}\xi^\vartheta(y)\in(\gamma_n)_\rho B_{\vartheta,\varepsilon}((\gamma_n)_\rho).\end{align*} $$

Consequently, equation (23) gives

(31)

By assumption, $a_\vartheta (\rho (\gamma _n))\in {\mathbb T}_r({\mathsf {u}}_\varphi )$ and one finds thus a divergent sequence $t_n\in \mathbb {R}_+$ such that

(32)

for some $K'$ only depending on r and $\varepsilon .$ The sequence

is thus contained in

for some $\kappa $ only depending on

in particular, ${{\mathsf {p}}}(x,y,v)\in \mathcal K({\unicode{x3c9} }^\varphi )$ as desired.

Reciprocally, if

belongs to $\mathcal K({\unicode{x3c9} }^\varphi ),$ let $\mathrm {B}$ be a bounded open set to which the ${\unicode{x3c9} }^\varphi $ -orbit of ${{\mathsf {p}}}(x_0,y_0,v_0)$ returns to unbounded. Considering an accumulation point of the orbit points in $\mathrm {B}$ , we can assume that $\mathrm {B}={{\mathsf {p}}}(\tilde {\mathrm {B}})$ for some $\tilde {\mathrm {B}}$ of the form

We obtain thus divergent sequences

and $\{t_n\}\subset \mathbb {R}^+$ such that for all n,

(33)

Considering subsequences, we can assume that $\gamma _n^{-1}x_0\to x_\infty $ and $\gamma _n^{-1}y_0\to y_\infty .$ Necessarily, $x_\infty \neq y_\infty $ since they are at least $\kappa '$ apart. The sequence $\{\gamma _n\}$ is thus conical, but it is still to be determined whether it converges to $x_0$ or to $y_0.$

Using the last inequality in equation (33), we deduce, since $t_n\to +\infty ,$ that for all $\sigma \in \vartheta $ ,

By definition of $\unicode{x3b2} _{}$ and the interpretation of the Buseman–Iwasawa cocycle via representations (equation (21)), one has $\log (\|\unicode{x3c6} _\sigma \rho (\gamma _n^{-1})v\|/\|v\|)\to -\infty $ for any non-vanishing $v\in \Xi _{\unicode{x3c6} _\sigma }(\xi ^\vartheta (\gamma _n^{-1}y_0)),$ or equivalently, as $n\to \infty $ ,

We now use a standard linear algebra computation to conclude that

Lemma 5.13.5. (Bochi, Potrie, and Sambarino [10, Lemma A.3])

Let $A\in \operatorname {\mathrm {{\mathsf {GL}}}}_d(\mathbb {R})$ have a gap at $\alpha _1,$ then for every $v\in \mathbb {R}^d$ one has

$$ \begin{align*}\frac{\|Av\|}{\|v\|}\geq \|A\|\sin\measuredangle(\mathbb{R}\cdot v,U_{d-1}(A^{-1})).\end{align*} $$

By Lemma 5.2.4, one concludes that

as $n\to \infty $ for all $\sigma \in \vartheta .$ Again by Lemma 5.2.4, one has $\gamma _n\to y_0$ as $n\to \infty $ (in

) and thus, by conicality of $\{\gamma _n\},$ that for all

, it holds $\gamma _n^{-1}z\to x_\infty .$ It follows then that

and, since $\gamma _n^{-1}y_0\to y_\infty \neq x_\infty ,$ that

Since the latter lower bound holds for all $\sigma \in \vartheta $ , one concludes that $\xi ^\vartheta (\gamma _n^{-1}y_0)$ belongs to the Cartan basin $B_{\vartheta ,\kappa "}(\rho (\gamma _n)).$ Thus, as in equation (31), one has

for some K only depending on $\kappa ".$ The latter, together with the second inequality from equation (33), implies that $y_0$ is $\varphi $ -conical, as desired.

Proof of Theorem 5.13.3

Consider a positive $\varepsilon .$ Fix and two neighborhoods $A^-$ and $A^+$ of x and y, respectively, so that for all $(z,w)\in A^-,A^+$ , one has $|[z,w]_\varphi -[x,y]_\varphi |<\varepsilon .$ Pick also an arbitrary $T>0$ so that the quotient projection ${{\mathsf {p}}}$ is injective on $\tilde {\mathrm {B}}=A^-\times A^+\times B(0,T).$ We can thus compute the measure of $\mathrm {B}={{\mathsf {p}}}(\tilde {\mathrm {B}})$ by equation (29).

If we let $\tilde {\mathcal K}({\unicode{x3c9} }^\varphi )={{\mathsf {p}}}^{-1}(\mathcal K({\unicode{x3c9} }^\varphi )),$ then the lemma above asserts that

If $|\vartheta |\leq 2$ , Theorem 5.12.1 states that $\Omega ^\varphi (\tilde {\mathrm {B}})=\Omega ^\varphi (\tilde {\mathcal K}({\unicode{x3c9} }^\varphi )\cap \tilde {\mathrm {B}}),$ which implies, up to $e^{-\delta ^\varphi \varepsilon },$ that

Since $\varepsilon $ is arbitrary, one concludes

However, if $|\vartheta |\geq 4$ , then we have $\Omega ^\varphi (\tilde {\mathcal K}({\unicode{x3c9} }^\varphi ))=0$ , so

and the theorem is proved.