Almost blenders and parablenders

Abstract

A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds robustly contains an open set. Introduced by Bonatti and Díaz in the 1990s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper we introduce analogous notions in a measurable setting. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of $\mathbb {R}^2$ leaving invariant the continuation of a hyperbolic basic set. When an inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger, and complements previous works on the construction of blenders by Avila, Crovisier, and Wilkinson or by Moreira and Silva. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak, and Urbański, we study families of skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

1 Introduction

1.1 Blenders and almost blenders

Fractal sets have played a central role in the development of differentiable dynamics. Among several examples, a central notion is that of the blender, introduced by Bonatti and Díaz in the 1990s. It was first introduced in the invertible setting in [BD1] to construct robustly transitive non-hyperbolic diffeomorphisms. A blender is a hyperbolic basic set on which the dynamics has a special behavior: its unstable set forms an ‘impenetrable wall’ in the sense that it intersects any perturbation of a submanifold of dimension lower than the stable dimension. In the case of surface endomorphisms, this notion takes the following simpler form,

Definition A. A $C^{r}$ -blender (of saddle type) for a $C^{r}$ -endomorphism $\mathcal {F}$ of a surface $\mathcal {S}$ is a hyperbolic basic set $\mathcal {K}$ of saddle type such that the union of its local unstable manifolds has $C^r$ -robustly a non-empty interior: there exists a non-empty open set $\mathcal {U} \subset \mathcal {S}$ included in the union of the local unstable manifolds of the continuation $\tilde {\mathcal {K}}$ of $\mathcal {K}$ for any map $\tilde {\mathcal {F}}$ which is $C^{r}$ -close to $\mathcal {F}$ .

A hyperbolic basic set for $\mathcal {F}$ is a compact, $\mathcal {F}$ -invariant, hyperbolic, transitive set $ \mathcal {K} $ such that periodic points of $\mathcal {F} | \mathcal {K} $ are dense in $ \mathcal {K} $ (basic notions about hyperbolic sets for endomorphisms are recalled in the Appendix).

Local unstable manifolds of size $\varepsilon>0$ of hyperbolic basic sets for endomorphisms are defined in the Appendix. In the latter definition and also in what follows, we suppose that $\varepsilon $ has been fixed and is implicit.

Blenders turned out to have a huge number of other powerful applications: for example, the density of stable ergodicity [ACW, RRTU], robust homoclinic tangencies [BD2, Bie1, BR] and thus Newhouse phenomena, the existence of generic families displaying robustly infinitely many sinks [Be1], robust bifurcations in complex dynamics [Bie2, Du, Taf], robust transitivity in Hamiltonian dynamics [NP], a robust criterion for the existence of non-hyperbolic measures [BBD], and the fast growth of the number of periodic points [AST, Be2]. Thus the following question is of fundamental interest: when do blenders appear?

In this direction, Berger proposed the following conjecture.

Conjecture A. (Berger [Be3])

Let $\mathcal {F}$ be a $C^r$ -local diffeomorphism of a manifold $\mathcal {M}$ , for $r \ge 2$ . Let $\mathcal {K}$ be a hyperbolic basic set for $\mathcal {F}$ . Suppose that the topological entropy $h_{\mathcal {F}}$ of $\mathcal {F} | \mathcal {K}$ satisfies

$$ \begin{align*}h_{\mathcal{F}}> \mathrm{dim} \mathcal{E}^s \cdot \lvert \log m( D \mathcal{F} ) \rvert \quad\text{with } m( D \mathcal{F} ) := \min \limits_{z \in \mathcal{K}, u \in \mathcal{E}^s_z , \lVert u\rVert =1} \lVert D_z \mathcal{F}(u) \rVert , \end{align*} $$

and $\mathcal {E}^s$ the stable bundle of $\mathcal {K}$ . Then there exist a $C^r$ -neighborhood $\mathcal {U}$ of $\mathcal {F}$ and an infinite-codimensional subset $\mathcal {N} \subset \mathcal {U}$ such that for every $\tilde {\mathcal {F}} \in \mathcal {U} \setminus \mathcal {N}$ , the continuation $\tilde {\mathcal {K}}$ of $\mathcal {K}$ is a $C^r$ -blender.

A very similar question appears in the paper [ACW] of Avila, Crovisier, and Wilkinson (see §3.3). Under the assumption of a slightly different inequality on the entropy, they manage to perform perturbations of an affine horseshoe in order to obtain a blender (Theorem C in their paper). Blenders (in fact a variant called ‘superblenders’) are ultimately used to prove a $C^1$ -version of the stable ergodicity conjecture of Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ -diffeomorphisms, $r> 1$ , the stably ergodic ones are $C^1$ -dense.

A different approach to the same problem of the construction of blenders was introduced by de Araujo Moreira and Silva [MS]. They obtained blenders for perturbations of (even non-affine) horseshoes, when the dimension of the ambient manifold is at least 3. Their construction is based on the recurrent compact criterion of Moreira and Yoccoz [MY].

In the conservative setting, Rodriguez Hertz et al. [RRTU2] proved that each $C^r$ conservative diffeomorphism with a pair of hyperbolic periodic points of co-index $1$ can be $C^1$ -approximated by $C^r$ conservative diffeomorphisms having a blender. In higher dimension, Bonatti and Díaz [BD3] showed that blenders appear near co-index- $1$ heterodimensional cycles.

In this work we study the problem of the prevalence of blenders, and specifically Conjecture A on surfaces, from a measurable point of view and using thermodynamic formalism. This is also linked to a program proposed by Díaz [Di] on the thermodynamical study of blenders.

Note that we cannot hope that $\mathcal {K}$ is itself systematically a $C^r$ -blender under the assumptions of Conjecture A. Here is an easy counterexample.

Counterexample. Let us consider the doubling map $f : x \mapsto 2x \text { mod } 1$ on the circle $\mathbb {S}:= \mathbb {R} / \mathbb {Z}$ . The whole circle is a hyperbolic basic set of repulsive type and f is a $C^\infty $ -local diffeomorphism. The topological entropy $h_f$ is equal to $\log 2>0$ . Let us pick $\lambda <1$ close to 1 such that $h_f> \lvert \log \lambda \rvert $ . The map $\mathcal {F} : (x,y) \in \mathbb {S} \times \mathbb {R} \mapsto (f(x), \lambda y) \in \mathbb {S} \times \mathbb {R}$ leaves invariant the hyperbolic basic set of saddle type $\mathcal {K}:= \mathbb {S} \times \{ 0 \}$ . Moreover, $\mathcal {F}$ is a $C^\infty $ -local diffeomorphism and $h_{\mathcal {F}} = h_f> \lvert \log \lambda \rvert $ . However, the unstable set of $\mathcal {K}$ is included in $\mathbb {S} \times \{0\}$ and thus has empty interior.

In this paper we give an answer to Conjecture A from a measurable point of view. We will define a measurable variant of the notion of blender, called an ‘almost blender’. It is a hyperbolic basic set whose unstable set has positive volume instead of having non-empty interior. Also, this property will be required to be ‘robust’ in a measurable way instead of topological. Let us be precise about this ‘measurable robustness’. We recall that this notion is delicate since there is no canonical measure on the space $C^r(\mathcal {M}, \mathcal {M})$ of $C^r$ -endomorphisms of a manifold $\mathcal {M}$ . We need something analogous to the finite-dimensional notion of ‘Lebesgue almost every’ in an infinite-dimensional setting. Nevertheless, there are several notions of prevalence or typicity which generalize this concept. A panorama has been drawn by Hunt and Kaloshin in [HK], by Ott and Yorke in [OY], and by Ilyashenko and Li in [IL]. Here is one of these notions of prevalence, particularly adapted to our case.

Definition B. [HK, p. 53]

We say that a set E in a Banach space B is finite-dimensionally prevalent if there exists a continuous family $(v_q)_{q \in \mathcal {Q}}$ of vectors $v_q \in B$ , parameterized by a parameter q varying in a neighborhood $\mathcal {Q}$ of $0$ in $\mathbb {R}^m$ with $m>0$ and $v_0 = 0$ , such that for every fixed $v \in B$ , we have that $v + v_q \in E$ for $\mathrm {Leb}_m$ -almost every (a.e.) $q \in \mathcal {Q}$ .

In other words, we require that for some finite-dimensional family of perturbations, if we start at any point in B, then by adding a perturbation randomly chosen with respect to the Lebesgue measure, we are in E with probability 1. A similar notion, simply called ‘prevalence’, has been designed by Sauer, Ott, and Casdagli (see [OY, Definition 3.5], or [HSY, SYC]) for completely metrizable topological vector spaces and with the additive condition that $v_q$ is a linear function of q. In our results we will have this additional linearity, but since we do not need it, we will take inspiration from the above definition. See [HK, Remark 1, p. 53] for details.

We restrict ourselves in this paper to the case where the manifold $\mathcal {M}$ is equal to $\mathbb {R}^2$ endowed with its usual Euclidean metric. The vector space $C^r(\mathbb {R}^2,\mathbb {R}^2)$ of $C^r$ -endomorphisms of $\mathbb {R}^2$ is endowed with the topology given by the uniform $C^r$ -norm:

$$ \begin{align*}\lVert\mathcal{F}\rVert_{C^r} := \sup_{0 \le i \le r, z \in \mathbb{R}^2} \lVert D^i_z \mathcal{F}\rVert\quad \text{when } 0 \le r < \infty. \end{align*} $$

The space of $C^r$ -bounded $C^r$ -endomorphisms endowed with this norm is a Banach space. Since we are interested in properties depending only on perturbations of a map on a compact set, we can restrict ourselves to $C^r$ -bounded $C^r$ -endomorphisms if necessary, and so the above definition of prevalence from [HK] could fit into our setting. Similarly, for $r =\infty $ , we endow $ C^\infty (\mathbb {R}^2,\mathbb {R}^2)$ with the union of uniform $C^s$ -topologies on $K_j$ between integers s and j, for an exhausting sequence of compact sets $K_j$ of $\mathbb {R}^2$ , which endows $ C^\infty (\mathbb {R}^2,\mathbb {R}^2)$ with a complete metrizable topology.

However, we cannot hope that a blender-like property, even in a weak sense, holds true densely in $C^r(\mathbb {R}^2,\mathbb {R}^2)$ , even less in a prevalent way. This is why we introduce the following immediate adaptation.

Definition C. A prevalent $C^r$ -neighborhood $\mathcal {N}$ of a $C^r$ -endomorphism $\mathcal {F}$ of $\mathbb {R}^2$ is a subset of $C^r(\mathbb {R}^2,\mathbb {R}^2)$ such that there exist a $C^r$ -neighborhood $\mathcal {U}$ of $\mathcal {F}$ in $C^r(\mathbb {R}^2,\mathbb {R}^2)$ and a continuous family $(\Sigma _q)_{q \in \mathcal {Q}}$ of $C^r$ -endomorphisms $\Sigma _q$ of $\mathbb {R}^2$ , with q varying in a neighborhood $\mathcal {Q}$ of $0$ in $\mathbb {R}^m$ with $m>0$ and $\Sigma _0 = 0$ , such that for every fixed $\mathcal {G} \in \mathcal {U}$ , the map $\mathcal {G}+\Sigma _q$ belongs to $\mathcal {N}$ for $\mathrm {Leb}_m$ -a.e. $q \in \mathcal {Q}$ .

In particular, a prevalent $C^r$ -neighborhood $\mathcal {N}$ of $\mathcal {F}$ contains arbitrarily small $C^r$ -perturbations of $\mathcal {F}$ .

Here is a new notion, which formalizes a measurable variant of blenders.

Definition D. A hyperbolic basic set $\mathcal {K}$ of saddle type for a $C^r$ -endomorphism $\mathcal {F}$ of $\mathbb {R}^2$ , $r \ge 1$ , is an almost $C^r$ -blender if the union of the local unstable manifolds of the continuation $\tilde {\mathcal {K}}$ of $\mathcal {K}$ has positive measure for any map $\tilde {\mathcal {F}}$ in a prevalent $C^r$ -neighborhood of $\mathcal {F}$ :

$$ \begin{align*}\mathrm{Leb}_{2}( W^u_{\mathrm{loc}}( \tilde{\mathcal{K}}))>0. \end{align*} $$

Our first result gives a positive answer to Conjecture A on surfaces, from a measurable point of view.

Theorem A. Let $\mathcal {F}$ be a $C^r$ -local diffeomorphism of $\mathbb {R}^2$ , with $ 2 \le r \le \infty $ . Let $\mathcal {K}$ be a hyperbolic basic set of saddle type for $\mathcal {F}$ . Suppose that the topological entropy $h_{\mathcal {F}}$ of $\mathcal {F} | \mathcal {K}$ satisfies $h_{\mathcal {F}}> \lvert \log m(D \mathcal {F}) \rvert $ . Then $\mathcal {K}$ is an almost $C^r$ -blender.

Conjecture A seems to be a very difficult problem in its full generality. A related question is the following long-standing open problem.

Conjecture B. [Ho1, Conjecture 3.1]

Let $\mu $ be the self-similar measure associated to an iterated function system (IFS) $\Psi = (\psi ^a)_{a \in \mathcal {A}}$ formed by a finite number of contracting similarities $\psi ^a$ on $\mathbb {R}$ . Suppose that there are no exact overlaps and that the similarity dimension of the IFS is strictly greater than 1. Then $\mu $ is absolutely continuous with respect to $\mathrm {Leb}_1$ .

One can refer to the survey of Hochman [Ho1] for more details.

1.2 Parablenders and almost parablenders

Berger introduced in [Be1] a variant of blenders, and it is defined for families of maps where not only the unstable set of a hyperbolic set but also the set of jets of points inside unstable manifolds contains an open set. Such objects were named parablenders (‘para’ standing for ‘parameter’). They were introduced to prove the existence of generic families displaying robustly infinitely many sinks, thus yielding a counter-example to a conjecture of Pugh and Shub from the 1990s [PS].

Definition E. [BCP, Be1]

A $C^{r}$ -parablender at $p_0 \in \mathcal {P}$ for a $C^{r}$ -family $(\mathcal {F}_p)_{p \in \mathcal {P}}$ of endomorphisms of a surface $\mathcal {S}$ , $r \ge 1$ , parameterized by a parameter p in an open subset $\mathcal {P} \subset \mathbb {R}^d$ , is a family $(\mathcal {K}_p)_{p \in \mathcal {P}}$ of continuations of a hyperbolic basic set $\mathcal {K}_{p_0}$ for $\mathcal {F}_{p_0}$ and such that

• • for every $(\gamma _p)_{p \in \mathcal {P}}$ in a non-empty open set of $C^r$ -families of points $\gamma _p \in \mathcal {S}$ ,

• • for every $C^{r}$ -family $(\tilde {\mathcal {F}}_p)_{p \in \mathcal {P}}$ of endomorphisms $C^r$ -close to $(\mathcal {F}_p)_{p \in \mathcal {P}}$ ,

there exists a $C^r$ -family $(\zeta _p)_{p \in \mathcal {P}}$ of points $\zeta _p \in \mathcal {S}$ such that

• • there is a local unstable manifold of $\mathcal {K}_{p_0}$ whose continuation for $\tilde {\mathcal {F}}_p$ contains $\zeta _p$ , for any $p \in \mathcal {P}$ ,

• • the r-jets of $\zeta _p$ and $\gamma _p$ at $p_0$ are equal,

  $$ \begin{align*}(\zeta_p, \partial_p \zeta_p , \ldots, \partial^r_p \zeta_p )_{| p = p_0} = (\gamma_p, \partial_p \gamma_p, \ldots, \partial^r_p \gamma_p )_{| p = p_0}. \end{align*} $$

In particular, $\mathcal {K}_{p_0}$ is a $C^r$ -blender for $\mathcal {F}_{p_0}$ if $(\mathcal {K}_p)_{p \in \mathcal {P}}$ is a $C^r$ -parablender for $(\mathcal {F}_p)_{p \in \mathcal {P}}$ at $p_0 $ . In a subsequent work [Be2], Berger used parablenders to prove the existence of generic families of maps displaying robustly fast growth of the number of periodic points, solving a problem of Arnold [Ar] in the finitely differentiable case.

From now on, we work with $C^r$ -families $(\mathcal {F}_p)_{p }$ of endomorphisms $\mathcal {F}_p$ of $\mathbb {R}^2$ , with $2 \le r \le \infty $ , parameterized by a parameter p varying in $\mathcal {P} := (-1,1)^d$ for some $1\le d<\infty $ . In fact, we will need to work with families which admit some extension to a larger parameter space. We therefore fix an open set $\mathcal {P}' \subset \mathbb {R}^d$ such that $\mathcal {P} \Subset \mathcal {P}'$ . We then define a $C^r$ -family $(\mathcal {F}_p)_{p }$ of endomorphisms $\mathcal {F}_p$ of $\mathbb {R}^2$ to be an element of $C^r(\mathcal {P}' \times \mathbb {R}^2, \mathbb {R}^2 )$ . We endow this space with the uniform $C^r$ -topology when $0 \le r < \infty $ , and with the union of uniform $C^s$ -topologies on $K_j$ between integers s and j, for an exhaustion of $\mathcal{P}' \times R^2$ by a sequence of compact sets $K_j$ when $r=\infty $ . Note that for simplicity in the following we often denote this family by $(\mathcal {F}_p)_{p \in \mathcal {P}}$ since we are mainly interested in the dynamics when $p \in \mathcal {P}$ but keep in mind that it admits such an extension. Let $(\mathcal {K}_p)_{p \in \mathcal {P}}$ be the (hyperbolic) continuation (extending to $\mathcal {P}'$ ) of a hyperbolic basic set of saddle type $\mathcal {K}_{p_0}$ . Let $\mathcal {E}^s_p$ and $\mathcal {E}^u_p$ be the one-dimensional stable and unstable bundles of $\mathcal {K}_p$ .

Our main result deals with jets of points inside local unstable manifolds of $\mathcal {K}_p$ . Let $(M_p)_p$ be a $C^r$ -curve of points $M_p$ parameterized by p such that the point $M_p$ belongs to the continuation of the same local unstable manifold of $\mathcal {K}_p$ for every p. For any integer $s \le r$ , one can consider the s-jet of $M_p$ at any $p_0 \in \mathcal {P}$ :

$$ \begin{align*}\mathrm{J}_{p_0}^{s} M_p := (M_p, \partial_p M_p , \ldots, \partial^{s}_p M_p)_{ | p = p_0}. \end{align*} $$

An interesting set is then the set $\mathrm {J }_{p_0}^s W^u_{\mathrm {loc}}( \mathcal {K}_p)$ of all the s-jets among such curves $(M_p)_p$ . When this set has robustly a non-empty interior, $(\mathcal {K}_p)_{p \in \mathcal {P}}$ is a $C^{s}$ -parablender at $p_0$ . Let $\delta _{ d, s} $ be the dimension of the set of jets in d variables of order s in one dimension, which is the space $\mathbb {R}_s[X_1, \ldots , X_d]$ of polynomials in d variables of degree at most s. In particular, notice that the space of jets of order s of maps from $\mathcal {P}$ to $\mathbb {R}^2$ is of dimension $2\delta _{d,s} $ .

Here is the counterpart for families of the definition of a prevalent $C^r$ -neighborhood.

Definition F. A prevalent $C^r$ -neighborhood $\mathcal {N}$ of a $C^r$ -family $(\mathcal {F}_p)_{p \in \mathcal {P} }$ of endomorphisms of $\mathbb {R}^2$ is a subset of $C^r(\mathcal {P}' \times \mathbb {R}^2, \mathbb {R}^2 )$ such that there exist a $C^r$ -neighborhood $\mathcal {U}$ of $(\mathcal {F}_p)_{ p \in \mathcal {P} }$ in $C^r(\mathcal {P}' \times \mathbb {R}^2, \mathbb {R}^2 )$ and a continuous family $(\Sigma _{q})_{q \in \mathcal {Q}}$ of $C^r$ -families $\Sigma _q = (\Sigma _{p,q})_{p \in \mathcal {P}}$ of endomorphisms $ \Sigma _{p,q}$ of $\mathbb {R}^2$ , with q in a neighborhood $\mathcal {Q}$ of $0$ in $\mathbb {R}^m$ with $m>0$ and $\Sigma _{p,0} = 0$ for every $p \in \mathcal {P}$ , such that for every fixed family $(\mathcal {G}_p)_p \in \mathcal {U}$ , the family $(\mathcal {G}_p+\Sigma _{p,q})_{p \in \mathcal {P}}$ belongs to $\mathcal {N}$ for $\mathrm {Leb}_m$ -a.e. $q \in \mathcal {Q}$ .

In particular, a prevalent $C^r$ -neighborhood $\mathcal {N}$ of $(\mathcal {F}_p)_{p \in \mathcal {P}}$ contains arbitrarily small $C^r$ -perturbations of $(\mathcal {F}_p)_{p \in \mathcal {P}}$ .

The following is an analogue of $C^s$ -parablenders, from a measurable point of view.

Definition G. The continuation $(\mathcal {K}_p)_{p \in \mathcal {P}}$ of a hyperbolic basic set of saddle type for a $C^r$ -family $(\mathcal {F}_p)_{p \in \mathcal {P}}$ of endomorphisms of $\mathbb {R}^2$ , with $r \ge 1$ , is an almost $C^{r,s}$ -parablender, with s an integer such that $s \le r$ , if for any family $(\tilde {\mathcal {F}}_p)_{p \in \mathcal {P}}$ in a prevalent $C^r$ -neighborhood of $(\mathcal {F}_p)_{p \in \mathcal {P}}$ , the continuation $(\tilde {\mathcal {K}}_p)_{p \in \mathcal {P}}$ of $(\mathcal {K}_p)_{p \in \mathcal {P}}$ satisfies

$$ \begin{align*}\mathrm{Leb}_{ 2\delta_{d,s} }( \mathrm{J }_{p_0}^s W^u_{\mathrm{loc}}( \tilde{\mathcal{K}}_p))>0\quad \text{for } \mathrm{Leb}_{d}\text{-a.e. } p_0 \in \mathcal{P}. \end{align*} $$

Note that if $(\mathcal {K}_p)_{p \in \mathcal {P}}$ is an almost $C^{r,s}$ -parablender and p is a parameter in $\mathcal {P}$ , the set $\mathcal {K}_p$ is an almost $C^r$ -blender.

Here is our second main result, which generalizes Theorem A in terms of jets.

Theorem B. Let $(\mathcal {F}_p)_{p \in \mathcal {P}}$ be a $C^r$ -family of local diffeomorphisms of $\mathbb {R}^2$ , with $2 \le r \le \infty $ . Let $(\mathcal {K}_p)_{p \in \mathcal {P}}$ be the continuation of a hyperbolic basic set of saddle type for $(\mathcal {F}_p)_{p \in \mathcal {P}}$ . Take an integer $s \le r-2$ and suppose that the topological entropy $h_{\mathcal {F}_p}$ of $\mathcal {F}_p | \mathcal {K}_p$ satisfies

$$ \begin{align*}(\star)\quad h_{\mathcal{F}_p }> \delta_{d,s} \cdot\lvert \log m(D \mathcal{F}_p)\rvert \quad \text{for all } p\in \mathcal{P}' .\end{align*} $$

Then $(\mathcal {K}_p)_{p \in \mathcal {P}}$ is an almost $C^{r,s}$ -parablender.

This second result is again in the direction of Conjecture A, in terms of jets. Let us mention that both Theorems A and B still hold true if we assume that the maps involved are local diffeomorphisms only in a neighborhood of the basic sets. We hope to use Theorem B to solve the conjecture of Pugh and Shub [PS] in the smooth $C^\infty $ case which is not handled by [Be1]. Finally, let us mention the following immediate question.

Question. Is it possible to generalize Theorems A and B to the case where $\mathcal {M}$ is any surface (not necessarily equal to $\mathbb {R}^2$ ) and for the alternative notion of prevalence defined by Kaloshin in this context? We recall that the latter is defined as follows: a subset $E \subset C^r(\mathcal {M},\mathcal {M})$ is strictly n-prevalent if there exists an open dense set of n-parameter families $(\mathcal {F}_p)_p$ such that $\mathcal {F}_p \in E$ for a.e. p and if for every $\mathcal {F} \in C^r(\mathcal {M},\mathcal {M})$ there exists such a family with $\mathcal {F}_0 = \mathcal {F}$ . An n-prevalent set is a countable intersection of strictly n-prevalent sets.

1.3 Combinatorics and notation

Let $\mathcal A$ be a finite alphabet of cardinality at least $2$ . Let

$$ \begin{align*}\overrightarrow{ \mathcal A}= \mathcal A^{\mathbb{N}}\text{,}\quad\overleftarrow{ \mathcal A}= \mathcal A^{\mathbb{Z}^*_-},\quad\text{and}\quad \overleftrightarrow{ \mathcal A}= \mathcal A^{\mathbb{Z}}\end{align*} $$

be respectively the sets of infinite forward, backward, and bilateral words with letters in $\mathcal A$ . We consider the left full shift on $\overrightarrow { \mathcal A} $ or $\overleftrightarrow { \mathcal A}$ ,

$$ \begin{align*}\sigma: \alpha= (\alpha_i)_i \in \overrightarrow{ \mathcal A} \sqcup \overleftrightarrow{ \mathcal A} \mapsto \sigma (\alpha) = (\alpha_{i+1})_i \in \overrightarrow{ \mathcal A} \sqcup \overleftrightarrow{ \mathcal A},\end{align*} $$

and the right full shift on $\overleftarrow { \mathcal A}$ ,

$$ \begin{align*}\sigma: \alpha = (\alpha_i)_i \in \overleftarrow{ \mathcal A} \mapsto \sigma (\alpha) = (\alpha_{i-1})_i \in \overleftarrow{ \mathcal A} .\end{align*} $$

In particular, these full shifts are of positive entropy and topologically mixing. We also define $\mathcal A^*$ as the set of finite words with letters in $\mathcal A$ and denote by e the empty word. We endow $\overleftrightarrow { \mathcal A}$ with the distance given by $d_\infty (\alpha ,\beta ) = D^{q}$ for every distinct $\alpha = (\alpha _i)_{i} \in \overleftrightarrow { \mathcal A}$ and $\beta = (\beta _i)_{i} \in \overleftrightarrow { \mathcal A}$ . Here $D \in (0,1)$ is a fixed number and q is the largest integer satisfying the property that $\alpha _i = \beta _i$ for every $\lvert i\rvert < q$ . We endow $\overrightarrow { \mathcal A}$ with a metric defined similarly.

For $\alpha \in \mathcal A^* \cup \overrightarrow { \mathcal A} \cup \overleftarrow { \mathcal A} \cup \overleftrightarrow { \mathcal A} $ , let $\lvert \alpha \rvert \in \mathbb {N} \cup \{+ \infty \}$ be the number of letters in $\alpha $ . When $\lvert \alpha \rvert>n$ for some integer $n>0$ , we call $\alpha _i$ the ith letter of $\alpha $ and denote $\alpha _{| n} :=(\alpha _0, \ldots ,\alpha _{n-1})$ when $\alpha \in \mathcal A^* \cup \overrightarrow { \mathcal A} $ and $\alpha _{| n} :=(\alpha _{-n}, \ldots , \alpha _{-1})$ when $\alpha \in \overleftarrow { \mathcal A} \cup \overleftrightarrow { \mathcal A} $ . Finally, for $\alpha = (\alpha _{-n}, \ldots , \alpha _{-1}) \in \mathcal A^*$ , let $[\alpha ]$ be the corresponding cylinder in $\overleftrightarrow { \mathcal A} $ :

$$ \begin{align*}[\alpha] := \{ \beta \in \overleftrightarrow{ \mathcal A} : \beta_i = \alpha_i \text{ for all } -n \le i \le -1 \}. \end{align*} $$

We similarly define cylinders in $\overleftarrow { \mathcal A}$ and $\overrightarrow { \mathcal A}$ and use the same notation. Greek (respectively, Gothic) letters will be used for finite or backward infinite (respectively, forward infinite) words. For $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $\alpha \in \overleftarrow { \mathcal A}$ , $\beta \in \mathcal A^*$ , we denote by $ \alpha \mathfrak {a}, \beta \mathfrak {a}, \alpha \beta $ their concatenations. The topological closure of a set relative to the Euclidean distance is denoted by an overline ( $\overline {\mathcal {P}}$ , for example). The notation $\preceq $ means that the inequality holds up to an absolute multiplicative constant, and $\asymp $ means that both $\preceq $ and $\succeq $ are true.

2 Example

We give here an application of our results, in which we provide simple examples of almost blenders and parablenders.

Let us consider the segment $X = [-1,1]$ . We pick three integers $n \ge 2$ , $d \ge 1$ , and $s \ge 0$ . We choose $n':=(n+1)^{\lceil \delta _{d,s} \rceil } $ disjoint subsegments $X_j \Subset X$ and $n'$ numbers $0<r_j<1-1/n$ , for $1 \le j \le n'$ . Let $g_j$ be the affine order-preserving map sending $X_j$ onto X. We pick a $C^\infty $ -map $g : \mathbb {R} \rightarrow \mathbb {R}$ which is equal to $g_j$ on a small neighborhood of each interval $X_j$ and also a $C^\infty $ -map $h : \mathbb {R} \rightarrow \mathbb {R}$ which is equal to $r_j$ on a small neighborhood of each interval $X_j$ . The $C^\infty $ -endomorphism

$$ \begin{align*}\mathcal{F} : (x,y) \in \mathbb{R}^2 \mapsto \bigg(g(x), \frac{y}{n} + h(x)\bigg) \in \mathbb{R}^2 \end{align*} $$

is a local diffeomorphism on a small neighborhood of $U:=\bigsqcup _{1 \le j \le n'} X_j \times X$ (see Figure 1). It is easy to verify that the set

(1) $$ \begin{align} \mathcal{K}:= \bigcap_{ n \in \mathbb{Z}} \mathcal{F}^{\kern2pt n}(U) \end{align} $$

is a compact, hyperbolic, invariant, locally maximal set, with stable and unstable dimensions equal to 1. This remains true for any $C^\infty $ -endomorphism $\tilde {\mathcal {F}}$ which is $C^\infty $ -close to $\mathcal {F}$ , with the same formula.

Figure 1 The dynamics of the map $\mathcal{F}$ .

We set $\mathcal {A}:=\{1, \ldots , n'\}$ and call $\mathcal {F}_j$ the restriction of $\mathcal {F}$ on $X_j \times X$ . For $\mathfrak {a} = (\mathfrak {a}_i)_{i \ge 0} \in \overrightarrow {\mathcal A}$ and $\alpha =(\alpha _i)_{i<0} \in \overleftarrow {\mathcal A}$ , the following are local stable and unstable manifolds of $\mathcal {K}$ :

(2) $$ \begin{align} W^{\mathfrak{a}} = \bigcap_{ j \ge 0} \mathrm{Dom} (\mathcal{F}_{\mathfrak{a}_{j}} \circ \cdots \circ \mathcal{F}_{\mathfrak{a}_0}) \quad\text{and}\quad W^{\alpha} = \bigcap_{ j<0} \mathrm{Im} (\mathcal{F}_{\alpha_{j}} \circ \cdots \circ \mathcal{F}_{\alpha_{-1}}) , \end{align} $$

where the domains $ \mathrm {Dom} ( \mathcal {F}_{\mathfrak {a}_{j}} \circ \cdots \circ \mathcal {F}_{\mathfrak {a}_0} ) $ and $ \mathrm {Dom} (\mathcal {F}_{\alpha _{j}} \circ \cdots \circ \mathcal {F}_{\alpha _{-1}})$ of $\mathcal {F}_{\mathfrak {a}_{j}} \circ \cdots \circ \mathcal {F}_{\mathfrak {a}_0}$ and $\mathcal {F}_{\alpha _{j}} \circ \cdots \circ \mathcal {F}_{\alpha _{-1}}$ are $( g_{\mathfrak {a}_{j}} \circ \cdots \circ g_{\mathfrak {a}_0})^{-1}(X) \times X $ and $(g_{\alpha _{j}} \circ \cdots \circ g_{\alpha _{-1}})^{-1}(X) \times X$ .

It is immediate that $W^{\mathfrak {a}}$ is a vertical segment of second coordinate projection X. By hyperbolic continuation, for every $C^\infty $ -endomorphism $\tilde {\mathcal {F}}$ which is $C^\infty $ -close to $\mathcal {F}$ and every $\mathfrak {a} \in \overrightarrow {\mathcal A} $ , we can define a local stable manifold $\tilde { W^{\mathfrak {a}} } $ which is a vertical graph of class $C^\infty $ over $y \in X$ with small slope. We notice that these local stable manifolds are pairwise disjoint. We have analogous properties for local unstable manifolds (except their disjointness) and $\tilde {W}^{\mathfrak {a}}$ and $\tilde {W}^{\alpha } $ intersect in exactly one point.

Let us now consider any d-unfolding $(\mathcal {F}_{p})_{p \in \mathcal {P}}$ of $\mathcal {F}$ , that is, a $C^\infty $ -family of endomorphisms $\mathcal {F}_p$ such that $\mathcal {F}_0=\mathcal {F}$ and $\mathcal {P} = (-1,1)^d$ . Up to restricting and then rescaling the parameter space, this family leaves invariant the continuation $(\mathcal {K}_p)_{p \in \mathcal {P}}$ of the hyperbolic set $\mathcal {K}$ , and $\mathcal {F}_p$ is a local diffeomorphism on a neighborhood of $\mathcal {K}_p$ . We can define families of local stable and unstable manifolds $ W_p^{\mathfrak {a}}$ and $ W_p^{\alpha }$ as above, and we denote by $\phi _p$ the map sending $\beta = \alpha \mathfrak {a} \in \overleftrightarrow {\mathcal {A}}$ (with $\alpha \in \overleftarrow {\mathcal {A}}$ and $ \mathfrak {a} \in \overrightarrow {\mathcal {A}}$ ) to the unique intersection point $\phi _p(\alpha \mathfrak {a}) \in \mathcal {K}_p$ between $ W_p^{\mathfrak {a}}$ and $ W_p^{\alpha }$ . We denote

$$ \begin{align*}\Phi_p : \beta \in \overleftrightarrow{\mathcal{A}} \mapsto ( \phi_p( \sigma^i(\beta) ) )_i \in \overleftrightarrow{\mathcal{K}_p}. \end{align*} $$

We notice that $\Phi _p$ conjugates the full shift $(\overleftrightarrow {\mathcal {A}},\sigma )$ to the dynamics $(\overleftrightarrow {\mathcal {K}_p}, \overleftrightarrow {\mathcal {F}_p})$ on the inverse limit and so periodic points are dense in $\mathcal {K}_p$ and $\mathcal {K}_p$ is transitive, and thus a hyperbolic basic set. The entropy $h_{\mathcal {F}_p}$ of $\mathcal {F}_p | \mathcal {K}_p$ is equal to $\mathrm {log} (n')$ and its stable contraction is close to $1/n$ . We recall that:

$$ \begin{align*}\frac{ \mathrm{log} (n') }{\lvert \mathrm{log} (1/n)\rvert} =\lceil \delta_{d,s} \rceil \cdot \frac{\mathrm{log}(n+1)}{\mathrm{log}(n)}>\delta_{d,s} \ge 1 .\end{align*} $$

Thus $h_{\mathcal {F}}> \lvert \mathrm {log} (1/n)\rvert $ and assumption $(\star )$ holds true for the family $(\mathcal {F}_{p})_{p \in \mathcal {P}}$ . By Theorems A and B, we obtain the following proposition.

Proposition. The set $\mathcal {K}$ is an almost $C^\infty $ -blender and, for any d-unfolding $(\mathcal {F}_{p})_{p \in \mathcal {P}}$ of $\mathcal {F}$ , $(\mathcal {K}_p )_{p \in \mathcal {P}}$ is an almost $C^{\infty ,s}$ -parablender, up to restricting and rescaling $\mathcal {P}$ .

3 Skew-product formalism and strategy

Our method is based on a method introduced by Mihailescu, Simon, Solomyak, and Urbański. Let us give some details. For an IFS without overlaps, the Hausdorff dimension of the limit set is given by Bowen’s formula [Bo2]. In [SiSoU], Simon, Solomyak, and Urbański introduced a method to compute it even in the presence of overlaps. The key ingredient in their proofs (see §4) is a transversality property (see [PeSo, So] for more on transversality, and also [Ho2, SaShSo, ShSo]). This also allows to parameters to be obtained for which the limit set has positive measure, which is what we are interested in. Later these results were extended by Mihailescu and Urbański to the case of hyperbolic and fiberwise conformal skew-products in [MU]. Here we extend these to the setting of families of fiberwise unipotent skew-products.

3.1 Skew-products

We work with (N-dimensional) skew-products acting on $ \overrightarrow { \mathcal A} \times [-1,1]^N$ , where $N>0$ . Here $\mathcal A$ is a fixed finite alphabet of cardinality at least $2$ . For simplicity, we denote $X:= [-1,1]^N$ . We are working with $C^2$ -maps.

Definition 3.1.1. A pre-skew-product is a map F of the form

$$ \begin{align*}F : (\mathfrak{a},x) \in \overrightarrow{ \mathcal A} \times X \mapsto (\sigma (\mathfrak{a}), f_{\mathfrak{a}}(x)) \in \overrightarrow{ \mathcal A} \times X\end{align*} $$

satisfying the requirement that there exists an open set $X' \subset \mathbb {R}^N$ independent of $\mathfrak {a}$ such that $X \Subset X'$ and such that $f_{\mathfrak {a}}: X \rightarrow X$ extends to a $C^2$ -diffeomorphism from $X'$ to $f_{\mathfrak {a}}(X') \Subset X$ for every $\mathfrak {a}$ .

A pre-skew-product F is a skew-product if, moreover, the maps

$$ \begin{align*}\mathfrak{a} \in \overrightarrow{ \mathcal A} \mapsto f_{\mathfrak{a}} \in C^0( X', \mathbb{R}^N)\quad \text{and}\quad \mathfrak{a} \in \overrightarrow{ \mathcal A} \mapsto Df_{\mathfrak{a}} \in C^0( X' , \mathcal{L}(\mathbb{R}^N,\mathbb{R}^N))\end{align*} $$

are Hölder with positive exponent, and the map

$$ \begin{align*}\mathfrak{a} \in \overrightarrow{ \mathcal A} \mapsto D^2f_{\mathfrak{a}} \in C^0(X', \mathcal{L}^2(\mathbb{R}^N,\mathbb{R}^N)) .\end{align*} $$

is continuous.

In the latter definition, $\overrightarrow { \mathcal A}$ is endowed with its distance and the spaces of $C^0$ -maps from $X'$ to $\mathbb {R}^N$ , to the space $\mathcal {L}(\mathbb {R}^N,\mathbb {R}^N)$ of linear maps from $\mathbb {R}^N$ to $\mathbb {R}^N$ and to the space $\mathcal {L}^2(\mathbb {R}^N,\mathbb {R}^N)$ of bilinear maps from $\mathbb {R}^N \times \mathbb {R}^N$ to $\mathbb {R}^N$ endowed with the uniform $C^0$ -metric. In the following, we suppose that the extensions of the maps $f_{\mathfrak {a}}$ are fixed.

We are even more interested in families $(F_p)_{p}$ of (pre-)skew-products, indexed by p varying in $ [-1,1]^d$ , for $1 \le d< \infty $ . We define such a family as a family of maps

$$ \begin{align*}F_p : (\mathfrak{a},x) \in \overrightarrow{ \mathcal A} \times X \mapsto (\sigma (\mathfrak{a}), f_{p,\mathfrak{a}}(x)) \in \overrightarrow{ \mathcal A} \times X \end{align*} $$

such that $\hat {F}: (\mathfrak {a}, (p,x)) \mapsto (\sigma (\mathfrak {a}), (p,f_{p,\mathfrak {a}}(x)))$ is an ( $N+d$ -dimensional) (pre-)skew- product. In particular, there exist open neighborhoods $X'$ and $\mathcal {P}'$ of X and $\overline {\mathcal {P}}$ in $\mathbb {R}^N$ and $\mathbb {R}^d$ such that $(p,x) \mapsto (p,f_{p,\mathfrak {a}}(x))$ extends to a diffeomorphism from $\mathcal {P}' \times X'$ into $ \mathcal {P} \times X$ for each $\mathfrak {a}$ and the map $F_p$ is an (N-dimensional) (pre-)skew-product for every $p \in \mathcal {P}'$ . Again, we denote this family by $(F_p)_{p \in \mathcal {P}}$ since we are mainly interested in the dynamics when $p \in \mathcal {P}$ but still keep in mind that it admits such an extension.

We will say that such a family $(F_p)_{p \in \mathcal P}$ of (pre-)skew-products satisfies the unipotent assumption (U) when, for any $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , and $x \in X'$ , the differential $D f_{p,\mathfrak {a}}(x)$ is inferior unipotent, that is a lower triangular matrix such that all its diagonal coefficients are equal, and its unique eigenvalue is strictly bounded between 0 and 1 in modulus.

We will adopt the following formalism. For every $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , $n>0$ , and $\alpha = (\alpha _{-n}, \ldots , \alpha _{-1}) \in \mathcal {A}^*$ , we set,

$$ \begin{align*}\text{for all } x \in X',\quad \psi_{ p, \mathfrak{a} }^\alpha(x):= f_{ p, \alpha_{-1} \mathfrak{a} } \circ \cdots \circ f_{ p, \alpha_{-n} \cdots \alpha_{-1} \mathfrak{a} } (x). \end{align*} $$

We show below that a consequence of assumption (U) is that $ \psi _{ p, \mathfrak {a} }^\alpha $ is a $C^2$ -contraction from $X'$ to $ \psi _{ p, \mathfrak {a} }^\alpha (X') \Subset X$ when $\lvert \alpha \rvert $ is large enough. If we now take an infinite backward sequence $\alpha = (\ldots , \alpha _{-n}, \ldots , \alpha _{-1}) \in \overleftarrow { \mathcal A}$ , we see that the points $\psi _{p, \mathfrak {a} }^{\alpha _{ |n} }(0)$ converge to a point $ \pi _{p, \mathfrak {a}} (\alpha ) \in X$ . This defines a $C^0$ -map $\pi _{p,\mathfrak {a}} : \overleftarrow { \mathcal A} \rightarrow X$ .

Definition 3.1.2. The limit set $K_{p,\mathfrak {a}}$ of the skew-product $F_p$ inside the $\mathfrak {a} $ -fiber is

$$ \begin{align*}K_{p,\mathfrak{a}}:= \pi_{p,\mathfrak{a}} (\overleftarrow{ \mathcal A} ) .\end{align*} $$

We will give conditions under which this set has positive measure.

For a $C^1$ -map $f : X \rightarrow X$ , let $m(Df)$ and $M(Df)$ be the respective minimum and maximum of $\lVert Df(x) \cdot u \rVert $ among $x \in X$ and $u \in \mathbb {R}^N$ such that $\lVert u\rVert =1$ . We need the following thermodynamical formalism.

Definition 3.1.3. The pressure at the parameter p in the $\mathfrak {a} $ -fiber is the map

$$ \begin{align*} \Pi_{p, \mathfrak{a}}: s \in \mathbb{R}_+ \mapsto \lim_{n \rightarrow + \infty} \frac{1}{n} \mathrm{log} \sum_{\alpha \in \mathcal A^{n} } M(D \psi_{p,\mathfrak{a} }^\alpha)^s. \end{align*} $$

When $ \Pi _{p,\mathfrak {a}} $ has a unique zero, we call it the similarity dimension inside the $\mathfrak {a} $ -fiber.

In Proposition 5.1.3, we show that both the pressure and the similarity dimension are well defined and independent of $\mathfrak {a}$ . In particular, we denote them by $\Pi _{p}$ and $\Delta (p)$ .

We adopt the following terminology to denote perturbations with special properties.

Definition 3.1.4. Let $(F_p)_p$ be a family of (pre-)skew-products and let us fix neighborhoods $X'$ and $\mathcal {P}'$ of X and $\overline {\mathcal {P}}$ such that $(p,x) \mapsto (p,f_{p,\mathfrak {a}}(x))$ extends to a diffeomorphism from $\mathcal {P}' \times X'$ into $ \mathcal {P} \times X$ for every $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ . For any $\vartheta>0$ , a $\vartheta $ -perturbation of $(F_p)_p$ is a family of pre-skew-products $(\tilde {F}_p)_p$ such that the map $(p,x) \mapsto (p,\tilde {f}_{p,\mathfrak {a}}(x))$ extends to a diffeomorphism from $\mathcal {P}' \times X'$ into $ \mathcal {P} \times X$ for every $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ , and

$$ \begin{align*}\sup_{ \mathfrak{a} \in \overrightarrow{\mathcal{A}} } \lVert (p,x) \in X' \times \mathcal{P}' \mapsto (f_{p,\mathfrak{a}} - \tilde{f}_{p,\mathfrak{a}})(x) \rVert_{C^2} < \vartheta. \end{align*} $$

The family $(\tilde {F}_p)_p$ is a $\vartheta $ - $\mathrm {U}$ -perturbation when it satisfies assumption (U).

For $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta $ , we will show that for any $\alpha \in \overleftarrow {\mathcal {A}}$ , the points

$$ \begin{align*}\tilde{f}_{ p, \alpha_{-1} \mathfrak{a} } \circ \cdots \circ \tilde{f}_{ p, \alpha_{-n} \cdots \alpha_{-1} \mathfrak{a} }(0) \end{align*} $$

converge to $ \tilde {\pi }_{p, \mathfrak {a}} (\alpha ) \in X$ such that $p \mapsto \tilde {\pi }_{p, \mathfrak {a}} (\alpha )$ is a $C^2$ -map $C^2$ -close to $p \mapsto \pi _{p, \mathfrak {a}} (\alpha )$ .

We will consider also parameterized families of $\vartheta $ -perturbations $(\tilde {F}_{t,p})_p$ :

$$ \begin{align*}\mathbb{F}:= ((\tilde{F}_{t,p})_p)_{t \in \mathcal{T}} .\end{align*} $$

Here t varies in $\mathcal {T} :=(-1,1)^\tau $ with $\tau>0$ and $(t,p,x) \mapsto \tilde {f}_{t, p,\mathfrak {a}} (x)$ is $C^2$ for every $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ . When each $(\tilde {F}_{t,p})_p$ is a $\vartheta $ - $\mathrm {U}$ -perturbation, we say that $ \mathbb {F}$ is a parameterized family of $\vartheta $ - $\mathrm {U}$ -perturbations. When $\vartheta $ is small enough, we will denote by $\tilde {\pi }_{t,p, \mathfrak {a}} (\alpha )$ the limit point corresponding to any $\alpha \in \overleftarrow {\mathcal {A}}$ and $\tilde {K}_{t,p,\mathfrak {a}}:= \tilde {\pi }_{t,p,\mathfrak {a}} (\overleftarrow { \mathcal A} )$ , and then the $C^2$ -maps $p \mapsto \tilde {\pi }_{t,p, \mathfrak {a}} (\alpha ) $ will be $C^2$ -close to $p \mapsto \pi _{p, \mathfrak {a}} (\alpha )$ , uniformly in $(t,\alpha )$ . We will set conditions under which $\tilde {K}_{t,p,\mathfrak {a}}$ has positive Lebesgue measure for a.e. $t \in \mathcal {T}$ .

3.2 Strategy and organization of the paper

The strategy will be to focus on the dynamics restricted to the local stable manifolds, which allows us to reduce the dynamics to a skew-product.

Hence we forget for the time being families of endomorphisms and we work with families of (pre-)skew-products. We say that such a family of (pre-)skew-products $(F_p)_{p \in \mathcal P}$ satisfying assumption (U) also satisfies the transversality assumption (T) when there exists $C>0$ such that for all sequences $\mathfrak {a} \in \overrightarrow { \mathcal A}$ and $\alpha , \beta \in \overleftarrow { \mathcal A}$ satisfying $\alpha _{-1} \neq \beta _{-1}$ , and for every $r>0$ , we have

$$ \begin{align*}\mathrm{Leb}_d \{ p \in \mathcal{P} : \lVert \pi_{p,\mathfrak{a}}(\alpha) - \pi_{p,\mathfrak{a}}(\beta) \rVert < r \} \le Cr^N, \end{align*} $$

and, moreover, for every small $\vartheta>0$ and every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations, for any $t \in \mathcal {T}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $\alpha , \beta \in \overleftarrow { \mathcal A}$ such that $\alpha _{-1} \neq \beta _{-1}$ and $r>0$ , we have

$$ \begin{align*}\mathrm{Leb}_d \{ p \in \mathcal{P} : \lVert\tilde{\pi}_{t,p,\mathfrak{a}}(\alpha) - \tilde{\pi}_{t,p,\mathfrak{a}}(\beta)\rVert < r \} \le C r^N. \end{align*} $$

We next give the main technical result in the proof of Theorem B. This sets conditions under which a given family of skew-products intersects its fibers into a set of positive measure, up to perturbations.

Theorem C. Let $(F_p)_{p \in \mathcal {P}}$ be a family of skew-products satisfying (U), (T), and $\Delta (p)>N$ for any $p \in \overline {\mathcal {P}}$ . Then, for every $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , we have

$$ \begin{align*}\mathrm{Leb}_N (K_{p,\mathfrak{a}})>0\quad \text{for } \mathrm{Leb}_{d}\text{-a.e. } p \in \mathcal{P} ,\end{align*} $$

and for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations of $(F_p)_{p \in \mathcal {P}}$ with small $\vartheta $ , we have

$$ \begin{align*}\mathrm{Leb}_N (\tilde{K}_{t,p,\mathfrak{a}})>0\quad \text{for } \mathrm{Leb}_{d}\text{-a.e. } p \in \mathcal{P}\text{ and } \mathrm{Leb}_{\tau}\text{-a.e. } t \in \mathcal{T} .\end{align*} $$

Here is the strategy for proving Theorem C. For every parameter p and $\mathfrak {a}$ -fiber, we define a probability measure $\nu _{p,\mathfrak {a}}$ supported on the limit set $K_{p,\mathfrak {a}}$ . To show that $K_{p,\mathfrak {a}}$ has positive measure, it is enough to show that $\nu _{p,\mathfrak {a}}$ is absolutely continuous relative to the N-dimensional Lebesgue measure, and so to prove that its density is finite almost everywhere. We compute the integral of the density relative to the parameter and the phase space. The trick is to use the Fubini theorem to integrate first relative to p. The finiteness of the integral is implied by the transversality assumption (T) and the inequality $\Delta (p)>N$ . The same method will give the same results for families of $\vartheta $ - $\mathrm {U}$ -perturbations, with additional integration relative to t.

To prove Theorem B, we go back to $C^r$ -families $(\mathcal {F}_p)_{p}$ of endomorphisms and we restrict the dynamics to the local stable manifolds $W^{\mathfrak {a}}_p$ , which are tagged in exponent by infinite forward sequences $\mathfrak {a}$ in letters in an alphabet $\mathcal {A}$ . Since $\mathrm {dim} \text { } \mathcal {E}_p^s=1$ , we are led to study families $(F_p)_{p}$ of skew-products acting on fibers which are segments, and such that the restriction on each fiber is $C^r$ .

We then look at the action induced by $(F_p)_{p}$ on s-jets (with $s \le r-2$ ), and this gives a new family of skew-products $(G_{p_0})_{p_0}$ acting on fibers of dimension $\delta _{d,s}$ . This new family satisfies assumption (U) (the assumption $\mathrm {dim} \text { } \mathcal {E}_p^s=1$ is used here); its similarity dimension is larger than $\delta _{d,s}$ by $(\star )$ . We extend $(\mathcal {F}_p)_{p}$ into a larger $C^r$ -family of endomorphisms so that the associated extended family of skew-products acting on s-jets satisfies the transversality assumption (T).

To conclude, we pick a family $(\Gamma _{t,p})_{t,p}$ of parallel segments $\Gamma _{t,p}$ close to a local stable manifold $W^{\mathfrak {a}}_p$ and such that the set of s-jets at any $p_0 \in \mathcal {P}$ of the projection of $\Gamma _{t,p}$ on a fixed direction transverse to $\Gamma _{t,p}$ when varying t has positive measure. The local unstable set intersects each segment $\Gamma _{t,p} $ in a set which is the limit set of a $\vartheta $ - $\mathrm {U}$ -perturbation with small $\vartheta $ , at every parameter p. We then apply the second part of Theorem C to get positive sets of s-jets for the intersection points between the local unstable set and $\Gamma _{t,p}$ for a.e. t at a.e. $p_0$ : in other words, we get positive sets of s-jets in the direction of $\Gamma _{t,p}$ for these values of t and $p_0$ . To conclude, we apply the Fubini theorem to find positive sets of bidimensional s-jets for points inside local unstable manifolds at a.e. $p_0$ . The same extension scheme works for $(\mathcal {G}_p)_{p}$ close to $(\mathcal {F}_p)_{p}$ , which proves that $(\mathcal {K}_p)_{p}$ is an almost $C^{r,s}$ -parablender.

Finally, Theorem A is an immediate consequence of Theorem B together with Remark 6.0.4, by taking the constant family $(\mathcal {F})_{p \in (-1,1)}$ and the order s of the jets equal to 0 (remark, in particular, that $\delta _{1,0}=1$ ).

In §4 we study a model given by families of IFSs of affine maps on an interval. We simplify the proof of Simon, Solomyak, and Urbański [SiSoU] in this context and introduce the strategy for the proof of Theorem C. In §5 we prove Theorem C. In each fiber the behavior of the dynamics looks like the model. Finally, we prove Theorem B in §6.

4 Model: Iterated function systems of affine maps on the interval

4.1 Setting and results

In this section we simplify the proof of a result of Simon, Solomyak, and Urbański about IFSs on an interval. This can be seen as a model for the behavior of the dynamics inside the fibers of a skew-product, as we will see in §5.

Let us fix $X:=[-1,1]$ and $\mathcal {P} :=(-1,1)$ . We consider families $(\Psi _p)_{p \in \mathcal {P}}$ , where, for every $p \in \mathcal {P}$ , the IFS $\Psi _p$ is a finite family $\Psi _p = (\psi _p^a)_{ a \in \mathcal A}$ of affine contractions $\psi _p^a : X \rightarrow X$ such that $\psi _p^a(X) \Subset X$ . The absolute value of the linear coefficient of $\psi _p^a$ is denoted by $\Lambda _{p,a}$ . We suppose that for every $a \in \mathcal {A}$ , the map $\psi _p^a$ depends continuously on p. In fact, we even suppose that the affine contraction $\psi _p^a$ is still defined for p in some open neighborhood of $[-1,1]$ and still depends continuously on p.

For every $ p \in \overline {\mathcal {P}}$ and $\alpha = (\alpha _{-n}, \ldots , \alpha _{-1}) \in \mathcal A^*$ , we denote the corresponding composition by $\psi _p^\alpha = \psi _p^{\alpha _{-1}} \circ \cdots \circ \psi _p^{\alpha _{-n}}$ , which is an affine contraction of the segment X into its own interior. When $\alpha =e$ , the map $\psi _p^e$ is simply the identity. The absolute value of the linear coefficient of $\psi _p^\alpha $ is denoted by $\Lambda _{p,\alpha }$ .

By continuity of the derivative of $\psi _p^a$ relative to $p \in \overline {\mathcal {P}}$ , there exist $0< \gamma ' < \gamma <1$ such that $\gamma ' < \Lambda _{p,a}<\gamma $ for every $p \in \overline {\mathcal {P}}$ and $a \in \mathcal A$ . Then $\Lambda _{p,\alpha } \le \gamma ^{\lvert \alpha \rvert }$ for every $\alpha \in \mathcal A^*$ . If we now take $\alpha \in \overleftarrow { \mathcal A}$ , the sequence of points $\psi _p^{\alpha _{| n}} (0)$ tends to a point $ \pi _p(\alpha )$ . This defines a $C^0$ -map $\pi _p : \overleftarrow { \mathcal A} \rightarrow X$ for every $p \in \overline {\mathcal {P}}$ . Since this convergence is uniform in p, the map $p \mapsto \pi _p(\alpha )$ is $C^2$ for any $\alpha $ . Moreover, the map $p \mapsto \pi _p$ is continuous, the set of $C^0$ -maps from $\overleftarrow { \mathcal A}$ to $\mathbb {R}$ endowed with the uniform $C^0$ -norm. We set

$$ \begin{align*}K_p :=\pi_p(\overleftarrow{ \mathcal A}). \end{align*} $$

We also suppose that the following assumption, which we call $(\mathrm {T}_{\mathrm {aff}})$ , is satisfied by $(\Psi _p)_{p \in \mathcal {P}}$ . There exists $C>0$ such that for every $\alpha , \beta \in \overleftarrow { \mathcal A}$ satisfying $\alpha _{-1} \neq \beta _{-1}$ , we have

$$ \begin{align*}\mathrm{Leb}_1 \{ p \in \mathcal{P} : \lvert \pi_p(\alpha) - \pi_p(\beta) \rvert < r \} \le Cr\quad \text{for any } r>0. \end{align*} $$

It is immediate that for any $p \in \overline {\mathcal {P}}$ , there exists exactly one number $\Delta (p) \ge 0$ such that

$$ \begin{align*}\sum_{a \in \mathcal{A}} \Lambda_{p,a}^{\Delta(p)} = 1 .\end{align*} $$

This is the similarity dimension of the IFS $\Psi _p$ . We are now in position to state the following result, which is a direct consequence of Theorem 3.1 of [SiSoU].

Theorem D. (Simon, Solomyak, and Urbański)

Let $(\Psi _p)_{p \in \mathcal {P}}$ be a family of IFSs of affine contractions satisfying $(\mathrm {T}_{\mathrm {aff}})$ and $\Delta (p)>1$ for any $p \in \overline {\mathcal {P}}$ . Then

$$ \begin{align*}\mathrm{Leb}_{1} (K_p)>0\quad \text{for } \mathrm{Leb}_{1}\text{-a.e. } p \in \mathcal{P}. \end{align*} $$

4.2 Proof of Theorem D

Proof of Theorem D

Let $p_0 \in \mathcal {P} $ such that $\Delta (p_0)>1+\epsilon $ for a small $\epsilon>0$ . To prove the result, it is enough to show that there exists $\delta>0$ such that $\mathcal {B}:=(p_0-\delta ,p_0+\delta )$ is included in $\mathcal {P}$ and $\mathrm {Leb}_1(K_p)>0$ for $\mathrm {Leb}_1$ -a.e. $p \in \mathcal {B}$ .

We define a probability measure $\mu $ on $\overleftarrow { \mathcal A}$ by setting $\mu [\alpha ] = \Lambda _{p_0,\alpha }^{\Delta (p_0)}$ for every cylinder defined by $\alpha \in \mathcal A^*$ . For any $p \in \mathcal {P}$ , let $\nu _p$ be the pushforward of $\mu $ by $\pi _p$ , which is supported on $K_p$ . To conclude, it is enough to show that there exists $\delta>0$ such that for $\mathrm {Leb}_1$ -a.e. $p \in \mathcal {B}$ , the measure $\nu _p$ is absolutely continuous relative to $\mathrm {Leb}_1$ . We set

$$ \begin{align*}D(\nu_p,x) := \liminf_{r \rightarrow 0} \frac{\nu_p( x-r,x+r) }{2 r}\end{align*} $$

for every $p \in \mathcal {P}$ and $x \in \mathbb {R}$ , which is the lower density of the measure $\nu _p$ at x.

Lemma 4.2.1. The map $(p,x) \in \mathcal {P} \times \mathbb {R} \mapsto D(\nu _p,x)$ is Borel measurable.

Proof. Since $p \in \mathcal {P} \mapsto \pi _p$ is continuous, it is also the case for $p \in \mathcal {P} \mapsto \nu _p$ (the set of probability measures endowed with the weak- $*$ topology). Since $x \in \mathbb {R} \mapsto \nu _p(x-r, x+r)$ is Borel measurable for every $p \in \mathcal {P}$ and $r>0$ , the map $(p,x) \in \mathcal {P} \times \mathbb {R} \mapsto \nu _p(x-r,x+r)$ is Borel measurable for every $r>0$ . Since $r \mapsto \nu _p( x-r,x+r)$ is monotonic and $r \mapsto 2r$ continuous, the lower limit $D(\nu _p,x)$ does not change if r is restricted to positive rationals. Thus the measurability of $(p,x) \in \mathcal {P} \times \mathbb {R} \mapsto D(\nu _p,x) $ reduces to that of the lower limit of countably many measurable maps.

We will shortly prove the following proposition.

Proposition 4.2.2. There exists $\delta>0$ such that $\mathcal {B} \subset \mathcal {P}$ and the following expression is finite:

$$ \begin{align*}\mathcal{I} := \int_{p \in \mathcal{B}} \int_{x \in \mathbb{R}} D(\nu_p,x)\,d\nu_p\, d \mathrm{Leb}_1<+ \infty .\end{align*} $$

This is enough to show that for $\mathrm {Leb}_1$ -a.e. $p \in \mathcal {B}$ , $\nu _p$ is absolutely continuous relative to $\mathrm {Leb}_1$ . Indeed, then, for $\mathrm {Leb}_1$ -a.e. $p \in \mathcal {B}$ , we will have $D(\nu _p,x)<+\infty $ for $\nu _p$ -a.e. $x \in \mathbb {R}$ and we apply the following result from the third item of Lemma 2.12 in [Ma].

Proposition 4.2.3. Let $\nu $ be a Radon measure on $\mathbb {R}^n$ , where $n>0$ , such that the density $D(\nu ,x)$ of $\nu $ relative to $\mathrm {Leb}_n$ is finite for $\nu $ -a.e. $x \in \mathbb {R}^n$ . Then $\nu $ is absolutely continuous relative to $\mathrm {Leb}_n$ .

This concludes the proof of Theorem D.

Proof of Proposition 4.2.2

For $\delta $ small enough, the interval $\mathcal {B}:=(p_0-\delta ,p_0+\delta )$ is included in $\mathcal {P}$ . If necessary, we reduce $\delta $ so that $ \Lambda _{p_1,a}^{1+\epsilon /2} \le \Lambda _{p_2,a}$ for every $p_1 ,p_2 \in \mathcal {B}$ and $a \in \mathcal {A}$ . In particular, this implies that

(3) $$ \begin{align} \text{for all } p_1 ,p_2 \in \mathcal{B}, \text{ for all } \alpha \in\mathcal{A}^*, \quad \Lambda_{p_1, \alpha }^{1+ {\epsilon}/{2}} \le \Lambda_{p_2,\alpha}. \end{align} $$

The strategy is to bound $\mathcal {I}$ by a new integral which will be easily shown to be finite using $(\mathrm {T}_{\mathrm {aff}})$ , for this specific choice of $\delta $ . First, by Fatou’s lemma, we have

(4) $$ \begin{align} \mathcal{I} \le \liminf_{r \rightarrow 0} \frac{1}{2 r} \int_{p \in \mathcal{B}} \int_{x \in \mathbb{R}} \nu_p( x-r,x+r)\, d\nu_p\, d \mathrm{Leb}_1. \end{align} $$

We can write $\nu _p( x-r,x+r)= \int _{y \in \mathbb {R}} 1_{ \{\lvert x-y \rvert < r \} }\, d\nu _p$ as the integral of the indicator function $1_{ \{\lvert x-y\rvert < r \} }$ , equal to 1 if $\lvert x-y\rvert < r$ , and 0 if not. Using this and then the definition of $\nu _p$ as the pushforward of $\mu $ by $\pi _p$ , we have

(5) $$ \begin{align} \int_{x \in \mathbb{R}} \nu_p( x-r,x+r)\, d\nu_p = \int_{(\alpha, \beta) \in \overleftarrow{ \mathcal A} \times \overleftarrow{ \mathcal A}} 1_{ \{ \lvert \pi_p(\alpha)-\pi_p(\beta) \rvert < r \} }\, d \mu \times \mu , \end{align} $$

where $1_{ \{ \lvert \pi _p(\alpha )-\pi _p(\beta ) \rvert < r \} } $ is equal to 1 if $ \lvert \pi _p(\alpha )-\pi _p(\beta ) \rvert < r $ and 0 if not. Then we substitute equation (5) into inequality (4) and use Fubini’s theorem to change the order of integration:

(6) $$ \begin{align} \mathcal{I} \le \liminf_{r \rightarrow 0} \frac{1}{2 r} \int_{(\alpha, \beta) \in \overleftarrow{ \mathcal A} \times \overleftarrow{ \mathcal A}} \mathrm{Leb}_1 \{ p \in \mathcal{B} : \lvert \pi_p(\alpha)-\pi_p(\beta) \rvert < r \}\, d \mu \times \mu. \end{align} $$

We write the latter integral as a sum whose terms are all easier to bound. For every finite word $\rho \in \mathcal A^*$ , we denote by $\mathcal C_\rho $ the set of pairs $(\alpha , \beta ) \in \overleftarrow { \mathcal A} \times \overleftarrow { \mathcal A}$ such that $\alpha _{-\lvert \rho \rvert } \cdots \alpha _{-1} = \beta _{-\lvert \rho \rvert } \cdots \beta _{-1} = \rho $ but $\alpha _{- \lvert \rho \rvert -1} \neq \beta _{- \lvert \rho \rvert -1}$ . We notice that $\overleftarrow { \mathcal A} \times \overleftarrow { \mathcal A} = \bigsqcup _{n \ge 0} \bigsqcup _{\rho \in \mathcal {A}^n} \mathcal C_\rho $ , and so by inequality (6) we have

(7) $$ \begin{align} \mathcal{I} \le \liminf_{r \rightarrow 0} \frac{1}{2r} \sum_{n\ge 0} \sum_{\rho \in \mathcal{A}^n } \int_{(\alpha, \beta) \in \mathcal C_\rho} \mathrm{Leb}_1 \{ p \in \mathcal{B} : \lvert \pi_p(\alpha)-\pi_p(\beta) \rvert < r \}\, d \mu \times \mu. \end{align} $$

We show below that the following lemma is a consequence of the transversality assumption $(\mathrm {T}_{\mathrm {aff}})$ .

Lemma 4.2.4. For every $n \ge 0$ , $\rho \in \mathcal {A}^n$ , and $(\alpha , \beta ) \in \mathcal C_\rho $ , we have

$$ \begin{align*}\mathrm{Leb}_1 \{ p \in \mathcal{B} : \lvert \pi_p(\alpha)-\pi_p(\beta) \rvert < r \} \preceq r \cdot \Lambda_{p_0,\rho}^{ -1- \epsilon/2}. \end{align*} $$

We can substitute the bound of Lemma 4.2.4 into inequality (7):

(8) $$ \begin{align} \mathcal{I} \preceq \sum_{n\ge 0} \sum_{\rho \in \mathcal{A}^n } \int_{(\alpha, \beta) \in \mathcal C_\rho} \Lambda_{p_0,\rho}^{ - 1 - \epsilon/2 }\, d \mu \times \mu. \end{align} $$

We use the equality $\mu [\rho ] = \Lambda _{p_0,\rho }^{\Delta (p_0)}$ , the inequality $(1+ {\epsilon }/{2}) /\Delta (p_0) < (1+ {\epsilon }/{2})/ (1+\epsilon ) < 1- {\epsilon }/{3}$ , and finally the inequality $\gamma ' < \Lambda _{p,a}<\gamma $ to get

(9) $$ \begin{align} \Lambda_{p_0,\rho}^{ -1 - \epsilon/2 } \asymp \mu[\rho]^{({-1-\epsilon/2})/{\Delta(p_0)}} \le \mu[\rho]^{ -(1- {\epsilon}/{3}) } \preceq \gamma^{{n\epsilon}/{3} } \cdot \mu [\rho]^{ -1 }. \end{align} $$

We now substitute this bound into inequality (8) to find that

$$ \begin{align*}\mathcal{I} \preceq \sum_{n\ge 0} \gamma^{ {n\epsilon}/{3} } \sum_{\rho \in \mathcal{A}^n } \frac{( \mu \times \mu) (\mathcal{C}_\rho ) }{ \mu [\rho] } \le \sum_{n\ge 0} \gamma^{ {n\epsilon}/{3} } \sum_{\rho \in \mathcal{A}^n } \mu [\rho] = \sum_{n\ge 0} \gamma^{ {n\epsilon}/{3 } } < + \infty , \end{align*} $$

where we used the inequality $( \mu \times \mu ) (\mathcal {C}_\rho ) \le \mu [\rho ]^2$ (coming from $\mathcal {C}_\rho \subset [\rho ]^2$ ) to prove the second inequality. This concludes the proof of Proposition 4.2.2.

Proof of Lemma 4.2.4

For every $p \in \mathcal {P}$ , $n \ge 0$ , $\rho \in \mathcal {A}^n$ , and $(\alpha , \beta ) \in \mathcal C_\rho $ , we have

(10) $$ \begin{align} \lvert \pi_p(\alpha) - \pi_p(\beta) \rvert = \Lambda_{p,\rho} \cdot \lvert \pi_p(\sigma^n(\alpha)) - \pi_p(\sigma^n(\beta))\rvert. \end{align} $$

Indeed, the points $\pi _p(\alpha )$ and $\pi _p(\beta )$ are the respective images of $ \pi _p(\sigma ^n(\alpha ))$ and $\pi _p(\sigma ^n(\beta ))$ by the map $\psi _p^\rho $ which is an affine contraction, and the absolute value of the linear coefficient of $\psi _p^\rho $ is $\Lambda _{p,\rho }$ . Thus, using equation (3), we have

$$ \begin{align*} \mathrm{Leb}_1 \{ p \in \mathcal{B} : \lvert\pi_p(\alpha)-\pi_p(\beta)\rvert < r \} &= \mathrm{Leb}_1 \bigg\{ p \in \mathcal{B} : \lvert \pi_p(\sigma^n (\alpha)) -\pi_p(\sigma^n(\beta)) \rvert < \frac{ r}{ \Lambda_{p,\rho}} \bigg\} \\ &\le \mathrm{Leb}_1 \bigg\{ p \in \mathcal{B} : \lvert \pi_p(\sigma^n (\alpha))-\pi_p(\sigma^n (\beta)) \rvert < \frac{r}{ \Lambda_{p_0,\rho}^{ 1+ {\epsilon}/{2} } } \bigg\}. \end{align*} $$

To conclude, by $(\mathrm {T}_{\mathrm {aff}})$ and since $\mathcal {B} \subset \mathcal {P}$ , the right-hand term of the latter is smaller than $ C \cdot r \cdot \Lambda _{p_0,\rho }^{ -1- \epsilon /2 }$ .

Example. Let us give a simple application of Theorem D. Let $n \ge 2$ be an integer. We set $\mathcal {A}:=\{0,1,2, \ldots , n\}$ , $X:=[0,1]$ , and $\mathcal {P}:=(1/n, 1-1/n)$ . Let $c<1/n$ be a real number close to $1/n$ . For $a \in \mathcal {A}$ , we put $\psi ^a_p(x):= c x + \tfrac {1}{2} (1/n-c) + a/n $ if $0 \le a < n$ and $\psi ^{n}_p(x):= c x + p$ if $a= n$ . Condition $(\mathrm {T}_{\mathrm {aff}})$ is clearly satisfied when n is large. Moreover, trivial computations show that the similarity dimension is $\Delta (p)= -\mathrm {log}(n+1) / \mathrm {log}(c)>1$ for any $p \in \overline {\mathcal {P}}$ . By Theorem D, $K_p$ has positive one-dimensional Lebesgue measure for a.e. $p \in \mathcal {P}$ .

5 The unipotent case: proof of Theorem C

We now extend Theorem D to the case of families of fiberwise unipotent skew-products. The fibers are indexed by $\mathfrak {a} \in \overrightarrow { \mathcal A}$ and the dynamics on each fiber will look like that of the model previously introduced. Here are some differences.

• • We will not restrict ourselves to fibers of dimension 1 and we will not suppose that the dynamics on each fiber is conformal but we will suppose that its differentials are unipotent with contracting eigenvalue (assumption (U)).

• • We will need distortion results (Lemmas 5.2.1– 5.2.4) since the dynamics will not be supposed to be affine this time.

5.1 Notation and immediate facts

We henceforth adopt the following formalism in order to prove Theorem C. Let $(F_p)_{p \in \mathcal {P}}$ be a family of skew-products satisfying assumptions (U) and (T). We recall that there exist open neighborhoods $X'$ and $\mathcal {P}'$ of X and $\overline {\mathcal {P}}$ in $\mathbb {R}^N$ and $\mathbb {R}^d$ such that each map $(p,x) \mapsto (p,f_{p,\mathfrak {a}}(x))$ extends to a diffeomorphism from $X' \times \mathcal {P}' $ into $X \times \mathcal {P} $ and the map $F_p$ is an (N-dimensional) skew-product for every $p \in \mathcal {P}'$ . We set,

(11) $$ \begin{align} \text{for all } p \in \mathcal{P}' , \mathfrak{a} \in \overrightarrow{ \mathcal A} , a \in \mathcal{A}, x \in X', \quad \psi_{ p, \mathfrak{a} }^a(x) := f_{p, a \mathfrak{a}}(x) \end{align} $$

and notice that $\psi _{ p, \mathfrak {a} }^a : X' \rightarrow X$ is a $C^2$ -map depending continuously on $(p, \mathfrak {a})$ . The $C^2$ -norm of $\psi _{ p, \mathfrak {a} }^a$ on X is then bounded independently of $p \in \overline {\mathcal {P}}$ , $ \mathfrak {a} \in \overrightarrow { \mathcal A}$ , and $a \in \mathcal {A}$ . We now define for any $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , $n>0$ , and $\alpha = (\alpha _{-n}, \ldots , \alpha _{-1}) \in \mathcal {A}^*$ :

$$ \begin{align*}\text{for all } x \in X', \quad \psi_{ p, \mathfrak{a} }^\alpha(x) := \psi_{ p, \mathfrak{a} }^{\alpha_{-1}} \circ \cdots \circ \psi_{ p, \alpha_{-n+1} \cdots \alpha_{-1} \mathfrak{a}}^ {\alpha_{-n}} (x)\! = f_{ p, \alpha_{-1} \mathfrak{a} } \circ \cdots \circ f_{ p, \alpha_{-n} \cdots \alpha_{-1} \mathfrak{a} } (x). \end{align*} $$

In particular, assumption (U) implies that for every $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , $\alpha \in \mathcal {A}^*$ , and $x \in X'$ , the differential $D\psi _{ p, \mathfrak {a} }^\alpha (x)$ is unipotent inferior and thus has a unique eigenvalue. This motivates the definition of the following contraction rate.

Definition 5.1.1. For any $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , $\alpha \in \mathcal {A}^*$ , and $x \in X'$ , let $\lambda _{p,\mathfrak {a},\alpha }(x)$ be the absolute value of the unique eigenvalue of the differential $D\psi _{ p, \mathfrak {a} }^\alpha (x)$ and

(12) $$ \begin{align} \Lambda_{p,\mathfrak{a},\alpha}:=\mathrm{max}_{x \in X} \lambda_{ p,\mathfrak{a},\alpha}(x). \end{align} $$

For any $p \in \mathcal {P}'$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , $\alpha =(\alpha _{-n}, \ldots , \alpha _{-1}) \in \mathcal {A}^*$ , and $x \in X'$ , we then have

(13) $$ \begin{align} \lambda_{p,\mathfrak{a},\alpha}(x) = \prod_{k=1}^{n} \lambda_{p, \mathfrak{a}_k, \alpha_{-k}} ( \psi_{p, \mathfrak{a}_{k+1} }^{\alpha_{-k-1}} \circ \cdots \circ \psi_{p, \mathfrak{a}_n }^{\alpha_{-n}}(x) )\quad \text{with } \mathfrak{a}_k:=\alpha_{| k-1} \mathfrak{a}. \end{align} $$

By continuity of $D\psi _{ p, \mathfrak {a} }^a(x)$ relative to $p \in \overline {\mathcal {P}}$ , $ \mathfrak {a} \in \overrightarrow { \mathcal A}$ , and $x \in X$ and by compactness of $\overline {\mathcal {P}}$ , $ \overrightarrow { \mathcal A}$ , and X, there exist $0< \gamma ' < \gamma <1$ so that for any p, $\mathfrak {a}$ , and a,

(14) $$ \begin{align} \text{for all } x \in X, \quad \gamma' < \lambda_{p,\mathfrak{a},a}(x) < \gamma. \end{align} $$

In particular, for every $\alpha \in \mathcal {A}^*$ , we have

(15) $$ \begin{align} \gamma^{\prime \lvert \alpha \rvert} < \Lambda_{p,\mathfrak{a},\alpha} < \gamma^{\lvert \alpha\rvert}. \end{align} $$

We will later need the following lemma, whose proof is in the Appendix.

Lemma 5.1.2. There exists a real polynomial P positive on $\mathbb {R}_+$ such that for any $p \in \overline {\mathcal {P}} $ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $\alpha \in \mathcal {A}^*$ , $x \in X$ , and $(i,j) \in \{1, \ldots , N\}^2$ with $i>j$ , the modulus of the $(i,j)$ th coefficient of the differential $D\psi _{p,\mathfrak {a}}^\alpha (x)$ is smaller than $P(\lvert \alpha \rvert ) \cdot \lambda _{p,\mathfrak {a},\alpha }(x)$ .

As an immediate consequence of Lemma 5.1.2, the pressure function $\Pi _{p,\mathfrak {a}}$ defined in Definition 3.1.3 is equal to

(16) $$ \begin{align} \Pi_{p,\mathfrak{a}}(s)=\lim_{n \rightarrow + \infty} \frac{1}{n} \mathrm{log} \sum_{\alpha \in \mathcal A^n} \Lambda_{p,\mathfrak{a} ,\alpha}^s\quad \text{for any } s \ge 0. \end{align} $$

Moreover, this map satisfies the following nice properties.

Proposition 5.1.3. The map $s \in \mathbb {R}_+ \mapsto \Pi _{p,\mathfrak {a}}(s) \in \mathbb {R}$ is well defined, strictly decreasing, continuous, independent of $\mathfrak {a}$ , with $\Pi _{p,\mathfrak {a}}(0)>0$ and $ \mathrm {lim}_{ s \rightarrow + \infty } \Pi _{p,\mathfrak {a}}(s) = -\infty $ . In particular, it has exactly one zero denoted by $\Delta (p)$ , depending continuously on p.

From now on we suppose that $\Delta (p)>N$ for any $p \in \overline {\mathcal {P}}$ .

Lemma 5.1.2 also implies that for any $p \in \overline {\mathcal {P}}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , and $\alpha \in \mathcal {A}^*$ of length sufficiently large, the map $ \psi _{p,\mathfrak {a}}^{\alpha }$ is a contraction. For $p \in \overline {\mathcal {P}}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , and $\alpha \in \overleftarrow { \mathcal A}$ , the diameter of $ \psi _{p,\mathfrak {a}}^{\alpha _{| n}} (X)$ is then small when n is large. Thus the sequence of points $\psi _{p,\mathfrak {a}}^{\alpha _{| n}} (0)$ converges to $ \pi _{p, \mathfrak {a}} (\alpha ) \in X$ . This defines, for every $p \in \overline {\mathcal {P}}$ and $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , a $C^0$ -map $\pi _{p,\mathfrak {a}} : \overleftarrow { \mathcal A} \rightarrow X$ . The map $(p, \mathfrak {a}) \in \overline {\mathcal {P}} \times \overrightarrow { \mathcal A} \mapsto \pi _{p,\mathfrak {a}}$ is then continuous, the set of $C^0$ -maps from $\overleftarrow { \mathcal A}$ to $\mathbb {R}^N$ being endowed with the uniform $C^0$ -norm. We set

$$ \begin{align*}K_{p,\mathfrak{a}} :=\pi_{p,\mathfrak{a}}( \overleftarrow{ \mathcal A}). \end{align*} $$

For any family $\mathbb {F} := ((\tilde {F}_{t,p})_p)_{t \in \mathcal {T}}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta>0$ , the map $\psi _{t, p, \mathfrak {a} }^\alpha := \tilde {f}_{ t, p, \alpha _{-1} \mathfrak {a} } \circ \cdots \circ \tilde {f}_{t, p, \alpha _{-n} \cdots \alpha _{-1} \mathfrak {a} }$ is also a contraction when $\lvert \alpha \rvert $ is large and so the sequence of points $\psi _{t,p,\mathfrak {a}}^{\alpha _{| n}} (0)$ still converges to $ \tilde {\pi }_{t, p, \mathfrak {a}} (\alpha ) \in X$ . This allows us to define a $C^0$ -map $\tilde {\pi }_{t,p,\mathfrak {a}} : \overleftarrow { \mathcal A} \rightarrow X$ for any t and p, and its limit set $\tilde {K}_{t,p,\mathfrak {a}} :=\tilde {\pi }_{t,p,\mathfrak {a}}( \overleftarrow { \mathcal A})$ .

Lemma 5.1.4. The map $p \mapsto \pi _{p,\mathfrak {a}}(\alpha )$ is continuous for every $\mathfrak {a} \in \overrightarrow { \mathcal A}$ and $\alpha \in \overleftarrow { \mathcal A}$ . Moreover, for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta>0$ , the map $(t,p) \mapsto \tilde {\pi }_{t,p,\mathfrak {a}}(\alpha )$ is continuous and $p \mapsto \tilde {\pi }_{t,p,\mathfrak {a}}(\alpha )$ is $C^0$ -close to $p \mapsto \pi _{p,\mathfrak {a}}(\alpha )$ uniformly in $t \in \mathcal {T}$ .

Proof. We notice that the point $\pi _{p,\mathfrak {a}}(\alpha )$ is the image of $\pi _{p, \alpha _{| n} \mathfrak {a}}(\sigma ^n(\alpha ))$ by the map $\psi _{p,\mathfrak {a}}^{\alpha _{| n}}$ (we recall that $\sigma $ denotes the shift map). The distance between $0 \in X$ and $\pi _{p, \alpha _{| n} \mathfrak {a}}(\sigma ^n(\alpha )) \in X$ is bounded by $\mathrm {diam}(X)$ . Since the map $\psi _{p,\mathfrak {a}}^{\alpha _{| n}}$ contracts by a factor independent of $(p,\mathfrak {a})$ and exponentially small in n (by equation (15) and Lemma 5.1.2), it follows that the map $p \mapsto \pi _{p,\mathfrak {a}}(\alpha )$ is the uniform limit (when $n \rightarrow + \infty $ ) of the sequence of continuous maps $p \mapsto \psi _{p,\mathfrak {a}}^{\alpha _{| n}} (0)$ and hence is continuous. The same argument works for small $\vartheta $ - $\mathrm {U}$ -perturbations.

5.2 Distortion lemmas

We now state distortion results, whose proofs are given in the Appendix.

Lemma 5.2.1. (Bounded distortion with respect to x)

There exists $D_1>1$ such that for every $p \in \overline {\mathcal {P}}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $\alpha \in \mathcal A^*$ , and $x,y \in X$ , we have

$$ \begin{align*}1/D_1 < \frac{ \lambda_{p,\mathfrak{a},\alpha}(x) }{\lambda_{p,\mathfrak{a},\alpha} (y) } < D_1 .\end{align*} $$

Lemma 5.2.2. (Distortion with respect to p)

For every $\eta>0$ , there exist $\delta (\eta )>0$ and $D_2 =D_2(\eta )>1$ such that for every $p_1, p_2 \in \overline {\mathcal {P}}$ and $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , we have

$$ \begin{align*}\lVert p_1-p_2 \rVert \le \delta(\eta) \implies \text{ for all } \alpha \in \mathcal A^*, \text{ } D_2^{-1} e^{-\eta \lvert\alpha\rvert } < \frac{ \Lambda_{p_1,\mathfrak{a},\alpha} }{\Lambda_{p_2,\mathfrak{a},\alpha}} < D_2 e^{\eta \lvert\alpha\rvert } .\end{align*} $$

Lemma 5.2.3. (Bounded distortion with respect to $\mathfrak {a}$ )

There exists $D_3>1$ such that for any $p \in \overline {\mathcal {P}}$ , $\mathfrak {a}, \mathfrak {a}' \in \overrightarrow { \mathcal A}$ , and $\alpha \in \mathcal A^*$ , we have

$$\begin{align*} 1/D_3 < \frac{ \Lambda_{p,\mathfrak{a},\alpha} }{ \Lambda_{p,\mathfrak{a}',\alpha}}< D_3. \end{align*}$$

Lemma 5.2.4. (Distortion with respect to $\vartheta $ -perturbations)

For every $\epsilon '>1$ , there exists $D_4>1$ such that for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with $\vartheta $ small enough, we have, for every $t \in \mathcal {T}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $p \in \overline {\mathcal {P}}$ , and $\alpha \in \mathcal {A}^*$ ,

$$ \begin{align*}\Lambda_{p,\mathfrak{a},\alpha}^{\epsilon'}/D_4 < \tilde{\Lambda}_{t,p,\mathfrak{a},\alpha} < D_4 \Lambda_{p,\mathfrak{a},\alpha}^{1/\epsilon'},\end{align*} $$

where $ \tilde {\Lambda }_{t,p,\mathfrak {a},\alpha } $ is the maximum among $x \in X$ of the absolute value $ \tilde { \lambda }_{t,p,\mathfrak {a},\alpha }(x) $ of the unique eigenvalue of the differential $D \psi ^\alpha _{t,p,\mathfrak {a}}(x)$ .

5.3 Choice of a probability measure $\mu $ on $\overleftarrow {\mathcal {A}}$

We will first need the following result of Bowen [Bo1, Theorem 1.4, p. 7 and its proof, p. 19] about the existence of a Gibbs measure. We recall that $\sigma $ is a full shift. We state the result in this case but it remains true for topologically mixing subshifts of finite type. We suppose that a parameter $p_0$ has been fixed (the precise choice will be made in the next subsection). We fix an arbitrary $\mathfrak {a}_0 \in \overrightarrow { \mathcal A}$ .

Theorem E. (Bowen)

Let $\phi : \overleftrightarrow { \mathcal A} \rightarrow \mathbb {R}$ be a Hölder map with positive exponent (the set $\overleftrightarrow { \mathcal A}$ being endowed with the distance $d_\infty $ ). Then there exists a unique $\sigma $ -invariant measure $\mu $ on $ \overleftrightarrow { \mathcal A}$ such that for every $A \in \overleftrightarrow { \mathcal A}$ , we have

$$ \begin{align*}\mu [A_{| n}] \asymp \mathrm{exp} \bigg(-\Pi n + \sum_{k=0}^{n-1} \phi (\sigma^k (A)) \bigg), \end{align*} $$

where $\Pi = \Pi (\phi ) = \mathrm {lim}_{ n \rightarrow \infty } ({1}/{n}) \mathrm {log} Z_n(\phi )$ , with

$$ \begin{align*}Z_n(\phi):= \sum_{x \in \mathcal{A}^n } \mathrm{exp} (S_x)\quad\text{and}\quad S_x= \mathrm{sup} \bigg\{ \sum_{k=0}^{n-1} \phi (\sigma^k (y)) : y \in [ x ] \bigg\}. \end{align*} $$

Writing $A = \alpha \mathfrak {a} \in \overleftrightarrow { \mathcal A} $ as the concatenation of $\alpha \in \overleftarrow { \mathcal A} $ and $\mathfrak {a} \in \overrightarrow { \mathcal A} $ , we can apply the previous result with the map

$$ \begin{align*} \phi : A \in \overleftrightarrow{ \mathcal A} \mapsto \Delta(p_0) \cdot \mathrm{log} \lambda_{p_0, \mathfrak{a}, \alpha_{-1}} ( \pi_{p_0, \alpha_{-1} \mathfrak{a} }(\sigma (\alpha)) ). \end{align*} $$

We prove the following lemma in the Appendix.

Lemma 5.3.1. The map $\phi $ is Hölder with positive exponent.

Using Lemmas 5.2.1 and 5.2.3, we note that $\Pi = \Pi (\phi )$ coincides with $ \Pi _{p_0}(\Delta (p_0))$ and thus vanishes by definition of $\Delta (p_0) $ (see Proposition 5.1.3). Moreover, by equation (13), for any $A \in \overleftrightarrow { \mathcal A} $ the sum $\sum \nolimits _{k=0}^{n-1} \phi (\sigma ^k(A))$ is equal to

$$\begin{align*} \Delta(p_0) \cdot \mathrm{log} \prod_{k=1}^n \lambda_{p_0, \mathfrak{a}_k, \alpha_{-k}} ( \pi_{p_0, \mathfrak{a}_{k+1} }(\sigma^k (\alpha) )) = \Delta(p_0) \cdot \mathrm{log} \lambda_{ p_0, \mathfrak{a}, \alpha_{| n}} ( \pi_{p_0, \alpha_{| n} \mathfrak{a} }(\sigma^n (\alpha) ) ) \end{align*}$$

with $ \mathfrak {a}_k:= \alpha _{| k-1} \mathfrak {a}$ . By Theorem E, this gives us a $\sigma $ -invariant measure $\mu $ on $ \overleftrightarrow { \mathcal A} $ such that for every $A = \alpha \mathfrak {a} \in \overleftrightarrow { \mathcal A} $ , we have

$$ \begin{align*}\mu [A_{| n}] \asymp \lambda_{ p_0, \mathfrak{a}, \alpha_{| n}}^{\Delta(p_0)} ( \pi_{p_0, \alpha_{| n} \mathfrak{a} }(\sigma^n( \alpha)) )\quad \text{when } n \rightarrow + \infty .\end{align*} $$

Using successively Lemmas 5.2.1 and 5.2.3, for any $A = \alpha \mathfrak {a} \in \overleftrightarrow { \mathcal A} $ , we have

(17) $$ \begin{align} \mu [A_{| n}] \asymp \Lambda_{p_0,\mathfrak{a} , \alpha_{| n} } ^{\Delta(p_0)} \asymp \Lambda_{p_0, \mathfrak{a}_0 , \alpha_{| n} } ^{\Delta(p_0)} \quad\text{when } n \rightarrow + \infty. \end{align} $$

We then define a $\sigma $ -invariant probability measure on $\overleftarrow { \mathcal A}$ , still denoted $\mu $ , by giving each cylinder in $\overleftarrow { \mathcal A} $ the same measure as the corresponding one in $\overleftrightarrow { \mathcal A}$ . Then

(18) $$ \begin{align} \mu [\rho] \asymp \Lambda_{p_0, \mathfrak{a}_0 , \rho } ^{\Delta(p_0)}\quad \text{when } \rho \in \mathcal{A}^n \text{ and } n \rightarrow + \infty. \end{align} $$

Remark 5.3.2. We will not need the $\sigma $ -invariance property of $\mu $ in the following, only the estimation from equation (18).

5.4 Proof of Theorem C

The strategy is the same as for the proof of Theorem D. Let us consider $p_0 \in \mathcal {P} $ and $\mathfrak {a} \in \overrightarrow { \mathcal A}$ . We have $\Delta (p_0)>N+\epsilon $ , where $\epsilon := \tfrac {1}{2}( \min _{p \in \overline {\mathcal {P}} }\Delta (p) - N)>0$ . To prove the result, we show that there exists $\delta>0$ such that the d-dimensional ball $\mathcal {B}$ of center $p_0$ of radius $\delta $ is included in $\mathcal {P}$ with $\mathrm {Leb}_{N}(K_{p,\mathfrak {a}})>0$ and $\mathrm {Leb}_{N}( \tilde {K}_{t,p, \mathfrak {a}})~>~0$ for $\mathrm {Leb}_d$ -a.e. $p \in \mathcal {B}$ and $\mathrm {Leb}_{\tau }\text {-a.e. } t \in \mathcal {T}$ , for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta $ .

We endow $\overleftarrow { \mathcal A}$ with the probability measure $\mu $ defined in §5.3. For any $p \in \mathcal {P}$ and $ t \in \mathcal {T}$ , let $\nu _{p,\mathfrak {a}}$ and $\nu _{t,p,\mathfrak {a}}$ be the images of $\mu $ by the maps $\pi _{p,\mathfrak {a}}$ and $\tilde {\pi }_{t,p,\mathfrak {a}}$ .

As Proposition 4.2.2 implies Theorem D, Theorem C is a consequence of the following result.

Proposition 5.4.1. There exists $\delta>0$ such that the ball $\mathcal {B}$ of center $p_0$ and radius $\delta $ is included in $\mathcal {P}$ and the following two integrals are finite:

$$ \begin{align*} \mathcal{I} &:= \int_{p \in \mathcal{B}} \int_{x \in \mathbb{R}^{N}} \liminf_{r \rightarrow 0} \frac{\nu_{p,\mathfrak{a}}( x + B(r)) } {c_{N} r^{N}}\, d\nu_{p,\mathfrak{a}} \, d \mathrm{Leb}_d<+ \infty , \\ \mathcal{I}' &:= \int_{p \in \mathcal{B}} \int_{t \in \mathcal{T}} \int_{x \in \mathbb{R}^{N}} \liminf_{r \rightarrow 0} \frac{\nu_{t,p,\mathfrak{a}}( x + B(r)) } {c_{N} r^{N}}\, d\nu_{t,p,\mathfrak{a}} \, d \mathrm{Leb}_\tau \, d \mathrm{Leb}_d <+ \infty , \end{align*} $$

for any family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta $ , where $B(r) \subset \mathbb {R}^{N}$ is the ball of center 0 and radius r and the constant $c_{N}$ is defined by $c_{N} r^{N} := \mathrm {Leb}_{N}(B(r))$ .

Proof. Let us take a small $\delta $ so that $\mathcal {B} \subset \mathcal {P}$ . The radius $\delta $ will be reduced once so that $\mathcal {I}$ and $\mathcal {I}'$ are finite. We first begin by bounding $\mathcal {I}$ . The proof is similar to that of Proposition 4.2.2: we begin by using Fatou’s lemma, the definition of $\nu _{p,\mathfrak {a}}$ , and the Fubini–Tonelli theorem to find the following bound:

$$ \begin{align*} \mathcal{I} \le \liminf_{r \rightarrow 0} \frac{1}{c_{N} r^{N} } \int_{(\alpha,\beta) \in \overleftarrow{ \mathcal A} \times \overleftarrow{ \mathcal A} } \mathrm{Leb}_d \{ p \in \mathcal{B} : \lVert \pi_{p,\mathfrak{a}}(\alpha) - \pi_{p,\mathfrak{a}}(\beta) \rVert < r \}\, d \mu \times \mu. \end{align*} $$

We write $\overleftarrow { \mathcal A} \times \overleftarrow { \mathcal A} = \bigsqcup _{n \ge 0} \bigsqcup _{\rho \in \mathcal {A}^n} \mathcal C_\rho $ where $\mathcal C_\rho $ is the set of pairs $(\alpha , \beta ) \in \overleftarrow { \mathcal A} \times \overleftarrow { \mathcal A}$ such that $\alpha _{-\lvert \rho \rvert } \cdots \alpha _{-1} = \beta _{-\lvert \rho \rvert } \cdots \beta _{-1} = \rho $ but $\alpha _{-\lvert \rho \rvert -1} \neq \beta _{-\lvert \rho \rvert -1}$ . Thus $\mathcal {I}$

(19) $$ \begin{align} \mathcal{I}< \liminf_{r \rightarrow 0} \frac{1}{c_{N} r^{N} } \sum_{n\ge 0} \sum_{\rho \in \mathcal{A}^n } \int_{ (\alpha,\beta) \in \mathcal C_\rho } \mathrm{Leb}_d \{ p \in \mathcal{B} : \lVert \pi_{p,\mathfrak{a}}(\alpha) - \pi_{p,\mathfrak{a}}(\beta) \rVert < r \}\, d \mu \times \mu. \end{align} $$

We show below that the following lemma is a consequence of the transversality assumption (T).

Lemma 5.4.2. We fix $\eta := {-\epsilon \mathrm {log} \gamma }/({2N+\epsilon })$ and reduce $\delta $ if necessary so that $\delta < \delta (\eta )$ (where $\delta (\eta )$ is defined in Lemma 5.2.2). Then for any $n \ge 0$ , $\rho \in \mathcal {A}^n$ , $(\alpha , \beta ) \in \mathcal C_\rho $ , we have

$$ \begin{align*} \mathrm{Leb}_d \{ p \in \mathcal{B} : \lVert \pi_{p,\mathfrak{a}}(\alpha) - \pi_{p,\mathfrak{a}}(\beta) \rVert < r \} \preceq r^N \cdot \Lambda_{p_0, \mathfrak{a}_0,\rho}^{ -N- 2\epsilon/3 }. \end{align*} $$

For any family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations of $(F_p)_p$ with small $\vartheta $ , for any $t \in \mathcal {T}$ , we have

$$ \begin{align*} \mathrm{Leb}_d \{ p \in \mathcal{B} : \lVert\tilde{\pi}_{t,p,\mathfrak{a}}(\alpha) - \tilde{\pi}_{t,p,\mathfrak{a}}(\beta) \rVert < r \} \preceq r^N \cdot \Lambda_{p_0, \mathfrak{a}_0,\rho}^{ -N- 2\epsilon/3 } .\end{align*} $$

Notice that when the similarity dimension is close to the dimension N of the fibers (and so $\epsilon $ is small), we need to work with a ball of small radius $\eta $ . We can inject the first bound of Lemma 5.4.2 into inequality (19):

(20) $$ \begin{align} \mathcal{I} \preceq \sum_{n\ge 0} \sum_{\rho \in \mathcal{A}^n } \int_{(\alpha, \beta) \in \mathcal C_\rho} \Lambda_{p_0, \mathfrak{a}_0,\rho}^{ -N - 2\epsilon/3 } \, d \mu \times \mu. \end{align} $$

We use successively equation (18), the inequality $( N+ {2\epsilon }/{3}) /\Delta (p_0) < ( N+ {2\epsilon }/{3})/ (N+\epsilon ) < 1- {\epsilon }/{4 N} $ , and finally inequality (15) (which gives $\mu [\rho ] \preceq \gamma ^n$ ) to get

(21) $$ \begin{align} \Lambda_{p_0,\mathfrak{a}_0 ,\rho}^{ - N-2\epsilon/3 } \asymp \mu[\rho]^{ ({- N-2\epsilon/3})/{\Delta(p_0)}} \le \mu[\rho]^{ -(1- {\epsilon}/{4N} ) } \preceq \gamma^{{n\epsilon}/{4 N} } \cdot \mu [\rho]^{ -1 }. \end{align} $$

We now substitute this bound into inequality (20) to find that

$$ \begin{align*} \mathcal{I} \preceq \sum_{n\ge 0} \gamma^{{n\epsilon}/{4 N} } \sum_{\rho \in \mathcal{A}^n } \frac{( \mu \times \mu) (\mathcal{C}_\rho ) }{ \mu [\rho] } \le \sum_{n\ge 0} \gamma^{ {n\epsilon}/{4 N}} \sum_{\rho \in \mathcal{A}^n } \mu [\rho] = \sum_{n\ge 0} \gamma^{{n\epsilon}/{4 N } } < + \infty ,\end{align*} $$

where we used the inequality $( \mu \times \mu ) (\mathcal {C}_\rho ) \le \mu [\rho ]^2$ (coming from $\mathcal {C}_\rho \subset [\rho ]^2$ ) to prove the second inequality. To bound $\mathcal {I}'$ for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations of $(F_p)_p$ with small $\vartheta $ , we just remark that the same proof works when $\vartheta $ is small enough, with an additional integration relative to $t \in \mathcal {T}$ .

Proof of Lemma 5.4.2

Let us begin with the following distortion lemma.

Lemma 5.4.3. There exists $D_5>0$ such that for every $p_1 ,p_2 \in \overline {\mathcal {P}}$ , $\mathfrak {a}_1 ,\mathfrak {a}_2 \in \overrightarrow { \mathcal A}$ , and $\rho \in \mathcal {A}^*$ , the following holds true:

$$ \begin{align*} \lVert p_1 - p_2\rVert < \delta \implies \Lambda_{p_1,\mathfrak{a}_{1}, \rho}^{1+ {\epsilon}/{2 N}} \le D_5 \cdot \Lambda_{p_2, \mathfrak{a}_2,\rho}. \end{align*} $$

Proof. If $\lVert p_1 - p_2 \rVert < \delta $ , then $\lVert p_1 - p_2 \rVert < \delta (\eta )$ . By Lemma 5.2.2, we have

$$ \begin{align*} \Lambda_{p_1, \mathfrak{a}_{1 }, \rho}^{1+\epsilon_0} \le D_2^{1+\epsilon_0} e^{\lvert\rho\rvert \eta ( 1 +\epsilon_0)} \Lambda_{p_2, \mathfrak{a}_{1}, \rho}^{ 1 +\epsilon_0 } \le D_2^{1+\epsilon_0} e^{\lvert\rho\rvert \eta (1 +\epsilon_0)} \gamma^{\lvert\rho\rvert \epsilon_0} \Lambda_{p_2, \mathfrak{a}_{1}, \rho} ,\end{align*} $$

with $\epsilon _0 := \epsilon / (2N)$ . By Lemma 5.2.3, $ \Lambda _{p_2, \mathfrak {a}_{1},\rho } \asymp \Lambda _{p_2, \mathfrak {a}_2,\rho } $ when $\lvert \rho \rvert \rightarrow + \infty $ , with bounds independent of $p_2$ . The result follows since, by definition of $\eta $ ,

$$ \begin{align*} e^{\lvert\rho\rvert \eta ( 1+\epsilon_0)} \gamma^{\lvert\rho\rvert \epsilon_0} = e^{\lvert\rho\rvert (\eta ( 1 +\epsilon_0 )+\epsilon_0 \mathrm{log} \gamma ) }\quad\text{with } \eta ( 1 +\epsilon_0 )+\epsilon_0 \mathrm{log} \gamma = 0 . \end{align*} $$

Lemma 5.4.4. There exists a real polynomial R positive on $\mathbb {R}_+$ such that for every $p \in \overline {\mathcal {P}}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $n \ge 0$ , $\rho \in \mathcal {A}^n$ , and $(\alpha , \beta ) \in \mathcal C_\rho $ , we have

(22) $$ \begin{align} \lVert \pi_{p,\mathfrak{a}} (\alpha) - \pi_{p,\mathfrak{a}} (\beta) \rVert \ge \frac{\Lambda_{p,\mathfrak{a},\rho}}{R(n)} \cdot \lVert \pi_{p,\rho \mathfrak{a}} (\sigma^n(\alpha)) - \pi_{p,\rho \mathfrak{a}}(\sigma^n(\beta)) \rVert. \end{align} $$

Moreover, for any $\epsilon '>1$ , for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations of $(F_p)_p$ with $\vartheta>0$ small enough we have for every $t \in \mathcal {T}$ , $p \in \overline {\mathcal {P}}$ , $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $n \ge 0$ , $\rho \in \mathcal {A}^n$ , and $(\alpha , \beta ) \in \mathcal C_\rho $ that

(23) $$ \begin{align} \lVert \tilde{\pi}_{t,p,\mathfrak{a}} (\alpha) - \tilde{\pi}_{t,p,\mathfrak{a}} (\beta) \rVert \ge \frac{\Lambda^{\epsilon'}_{p,\mathfrak{a},\rho}}{R(n)} \cdot \lVert \tilde{\pi}_{t,p,\rho \mathfrak{a}} (\sigma^n(\alpha)) - \tilde{\pi}_{t,p,\rho \mathfrak{a}}(\sigma^n(\beta)) \rVert. \end{align} $$

The proof is in the Appendix.

Given $\mathfrak {a} \in \overrightarrow { \mathcal A}$ , $n \ge 0$ , $\rho \in \mathcal {A}^n$ , $(\alpha , \beta ) \in \mathcal C_\rho $ , using inequality (22) and then Lemma 5.4.3 together with the fact that $\mathcal {B}$ is the ball of center $p_0$ and radius $\delta $ , we have

$$ \begin{align*} &\mathrm{Leb}_d \{ p \in \mathcal{B}:\lVert \pi_{p,\mathfrak{a}} (\alpha) - \pi_{p,\mathfrak{a}} (\beta) \rVert < r \} \\ &\ \le \mathrm{Leb}_d \bigg\{ p \in \mathcal{B} : \lVert \pi_{p, \rho \mathfrak{a}} (\sigma^n (\alpha)) - \pi_{p,\rho \mathfrak{a}} (\sigma^n (\beta)) \rVert < \frac{R(n) r}{ \Lambda_{p,\mathfrak{a} ,\rho}} \bigg\} \\ & \ \le \mathrm{Leb}_d \bigg \{ p \in \mathcal{B} : \lVert \pi_{p,\rho \mathfrak{a}} (\sigma^n(\alpha))- \pi_{p,\rho \mathfrak{a}} (\sigma^n(\beta)) \rVert < \frac{D_5 R(n) r}{ \Lambda_{p_0, \mathfrak{a}_0 ,\rho}^{1+ {\epsilon}/{2 N}} } \bigg\}. \end{align*} $$

To conclude, by assumption (T) and since $\mathcal {B} \subset \mathcal {P}$ , the last term of the latter inequality is smaller than

$$ \begin{align*} r^N \cdot Q(n) \cdot \Lambda_{p_0, \mathfrak{a}_0 ,\rho}^{ - N- \epsilon/2 }\quad \text{with } Q(n):= C \cdot D^{ N}_5 \cdot (R(n))^N. \end{align*} $$

The result follows since $ \Lambda _{p_0, \mathfrak {a}_0 ,\rho }$ decreases exponentially with n, and $\epsilon /2<2\epsilon /3$ . The proof of the second item is similar, by taking $\epsilon '$ close to 1 in inequality (23).

6 Jets: proof of Theorem B

Proof of Theorem B. The strategy is to study the dynamics of the family $(\mathcal {F}_p)_{p}$ inside the local stable manifolds to reduce the problem to the dynamics of a family $(F_p)_{p}$ of skew-products (step 1) with one-dimensional fibers, and such that the restriction on each fiber is $C^r$ . From $(F_p)_{p}$ , we construct a family of skew-products $(G_{p_0})_{p_0}$ acting on s-jets (step 2). Then we extend the latter family into a larger one, $(G_{q_0})_{q_0}$ , to satisfy the transversality assumption (T) (step 3). Finally, we look at the intersection between the unstable set and a family of curves all close to a stable manifold. In each curve, this intersection is equal to the limit set of a perturbation of the skew-product. We then successively apply Theorem C and the Fubini theorem to conclude to a set of jets of positive measure at a.e. parameter, which gives the parablender property (step 4).

Step 1: Dynamically defined family of skew-products. We first need to define local stable and unstable manifolds. Let us fix a small $\varepsilon>0$ and an arbitrary parameter in $\overline {\mathcal {P}}$ , taken arbitrarily close to 0 for simplicity. We recall that $\mathcal {K}_0$ is a hyperbolic basic set for $\mathcal {F}_0$ . Up to a change of metric on the stable (respectively, unstable) bundles of $\mathcal {K}_0$ , we suppose that $D\mathcal {F}_0$ strictly contracts (respectively, expands) the stable (respectively, unstable) bundle by a factor $\lambda <1$ (respectively, $1/\lambda $ ) uniformly over $z \in \mathcal {K}_0$ .

It has been shown by Qian and Zhang (see §4 of [QZ]) that the limit inverse $\overleftrightarrow {\mathcal {K}_0}$ can be endowed with a map $[ \cdot ]$ defined on a subset of $\overleftrightarrow {\mathcal {K}_0} \times \overleftrightarrow {\mathcal {K}_0}$ with values in $\overleftrightarrow {\mathcal {K}_0}$ so that for all sufficiently closed orbits $x, y \in \overleftrightarrow {\mathcal {K}_0}$ , the orbit $z=[x,y]$ is well defined and the 0-coordinate projection $\pi (z) \in \mathcal {K}_0$ of z is the intersection of the local unstable manifold of x and the local stable manifold of $y_0$ . This map endows $\overleftrightarrow {\mathcal {K}_0}$ with a Smale space structure (see Ruelle [Ru, Ch. 7], for the definition of a Smale space). This implies that $\overleftrightarrow {\mathcal {K}_0} $ admits Markov partitions of arbitrarily small diameter (see again Ruelle [Ru, Ch. 7], for this result and the definition and properties of a Markov partition for a Smale space). We pick such a partition of $\overleftrightarrow {\mathcal {K}_0}$ by a finite number of compact rectangles $\overline {\mathcal {R}}_1, \ldots , \overline {\mathcal {R}}_M$ of diameter small compared to $\varepsilon $ . Up to reducing $\varepsilon $ , we can suppose that $\mathcal {F}_0$ is a local diffeomorphism when restricted to the 0-coordinate projection $\mathcal {R}_i:= \pi (\overline {\mathcal {R}}_i) \subset \mathcal {K}_0$ of any rectangle $\overline {\mathcal {R}}_i$ .

The topological entropy of $\mathcal {F}_0 | \mathcal {K}_0$ is larger than $\delta _{d,s} \cdot \lvert \log m(D \mathcal {F}_0) \rvert $ by $(\star )$ . Then for large $\ell $ we can pick $N_{\varepsilon ,\ell }$ orbits $x_a=(x^i_a)_{0 \le i \le \ell -1}$ of length $\ell $ (with $a \in \mathcal {A}:=\{1,2, \ldots , N_{\varepsilon ,\ell }\}$ ) under $\mathcal {F}_0$ which are $(\ell ,\varepsilon )$ -separated (in the sense of Bowen) with

$$ \begin{align*} ({\diamond})\quad\frac{\text{log} N_{\varepsilon,\ell}}{\ell}> \delta_{d,s} \cdot \lvert \log m(D{\mathcal{F}}_0)\rvert. \end{align*} $$

In particular, the cardinality $N_{\varepsilon ,\ell }$ of $\mathcal {A}$ is at least 2. We can extend each of these orbits $x_a$ of length $\ell $ into an infinite orbit in $\overleftrightarrow {\mathcal {K}_{0}} $ , still denoted $x_a$ . Up to slightly perturbing $x_a$ , we can suppose that for every $0 \le k \le \ell -1$ , the orbit $\overleftrightarrow{{\mathcal {F}}_{0}}^{k} (x_a)$ is in the interior of some rectangle $\overline {\mathcal {R}}_{i}$ which we denote by $\overline {\mathcal {R}}_{a,k}$ . In particular, $x_{a}^{k} = {\mathcal {F}}_{0}^{k}(x^{0}_{a})$ is in $\mathcal {R}_{a,k} = \pi ( \overline {\mathcal {R}}_{a,k} )$ . Since $M$ is independent of $\ell$ , up to modifying $N_{\varepsilon ,\ell }$ by a multiplicative constant (independent of $\ell $ large), we can suppose that all orbits $x_a$ ( $a \in \mathcal {A}$ ) begin in the same $\mathcal {R}_i$ and end in the same ${\mathcal {R}}_j$ , with $(\diamond )$ still true. For any $a \in {\mathcal {A}}$ , let us define

$$ \begin{align*} R_a := \mathcal{R}_{a,0} \cap \mathcal{F}^{-1}_0 ( \mathcal{R}_{a,1} ) \cap \cdots \cap \mathcal{F}^{-(\ell-1)}_0 ( \mathcal{R}_{a,\ell-1}). \end{align*} $$

which is not empty by assumption. Since the $\ell $ -orbits $(x^i_a)_{0 \le i \le \ell -1}$ (with $a \in \mathcal {A}$ ) under $\mathcal {F}_0$ are $(\ell ,\varepsilon )$ -separated and since the diameter of each rectangle of the partition is small compared to $\varepsilon $ , the sets $R_a$ ( $a \in \mathcal {A}$ ) are pairwise disjoint compact subsets of $\mathcal {R}_i$ having their images by $\mathcal {F}^\ell _0$ included in $\mathcal {R}_j$ . Up to adding a constant (independent of $\ell $ large) to $\ell $ , using the properties of Markov partitions, we can suppose, moreover, that $\mathcal {R}_i$ and $\mathcal {R}_j$ are equal. We denote this set by R in the following and thus the sets $R_a$ are non-empty pairwise disjoint compact subsets of R with images under $\mathcal {F}^\ell _0$ included in R. This does not change the number $N_{\varepsilon ,\ell }$ of such sets $R_a$ and so inequality $(\diamond )$ is still true. For every non-empty finite word $\beta = \beta _0 \cdots \beta _p \in \mathcal {A}^{p+1}$ , we now set

$$ \begin{align*} R_\beta := R_{\beta_0 } \cap \mathcal{F}^{-\ell}_0 (R_{\beta_1} ) \cap \cdots \cap \mathcal{F}^{-\ell p}_0 (R_{\beta_p}), \end{align*} $$

which is a non-empty compact set by the properties of Markov partitions, and the sets $R_\beta $ among $\beta \in \mathcal {A}^{p+1}$ are pairwise disjoint subsets of R for a fixed value of p. Finally, for $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ , we define

$$ \begin{align*}R_{\mathfrak{a}} := \bigcap_{n>0} R _{\mathfrak{a}_{| n} } \end{align*} $$

which is also a non-empty compact subset of R, included in $\mathcal {K}_0$ , and the sets $R_{\mathfrak {a}}$ ( $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ ) are pairwise disjoint. Finally, since the partition admits a continuation in a neighborhood of 0 in $\mathcal {P}$ , the sets $R_a$ , $R_\beta $ , and $R_{\mathfrak {a}}$ admit continuations $R_{p,a}$ , $R_{p,\beta }$ , and $R_{p,\mathfrak {a}}$ with the same dynamical properties when p varies in a neighborhood of 0 in the parameter space. Up to covering $\overline {\mathcal {P}}$ by finitely many such neighborhoods and extending a finite number of times $(\mathcal {F}_p)_p$ in steps 2,3, and 4, we can suppose that $R_{p,a}$ , $R_{p,\beta }$ , and $R_{p,\mathfrak {a}}$ vary continuously in a neighborhood of $\overline {\mathcal {P}}$ , which we can suppose equal to $\mathcal {P}'$ up to reducing it, and inequality $(\diamond )$ is satisfied in $\overline {\mathcal {P}}$ .

Let us take a small $\varepsilon '>0$ . For any infinite forward sequence $\mathfrak {a} \in \overrightarrow { \mathcal A}$ and any $p \in \mathcal {P}'$ , we notice that all the points in $R_{p, \mathfrak {a}}$ are asymptotic (using hyperbolicity) and thus belong to a same stable manifold W of $\mathcal {K}_p$ (which is one-dimensional). We define $W^{\mathfrak {a}}_p$ as the $\varepsilon '$ -neighborhood in W of the maximal arc of W bounded by points of $R_{p, \mathfrak {a}}$ . We parameterize $W^{\mathfrak {a}}_p$ with $X:=[-1,1]$ via a $C^r$ -map $s_p^{\mathfrak {a}}$ such that $(s_p^{\mathfrak {a}})_p$ Hölder depends on the $C^{r-1}$ -topology on $\mathfrak {a}$ by Remark A.4.1.

Since $\mathcal {F}_p$ is a local diffeomorphism and since $W^{\mathfrak {a}}_p$ is injectively immersed, up to decreasing $\varepsilon '$ , the restriction of $\mathcal {F}_p^\ell $ to each $W_p^{\mathfrak {a}}$ is a diffeomorphism satisfying

$$ \begin{align*} \text{for all } p \in \mathcal{P}', \text{ for all } \mathfrak{a} \in \overrightarrow{ \mathcal A}, \quad \mathcal{F}^\ell_p (W_p^{\mathfrak{a}}) \subset \mathring{W}_p^{\sigma (\mathfrak{a})} :=s_p^{\sigma(\mathfrak{a})} ((-1,1)) \end{align*} $$

by hyperbolicity. We define the $C^r$ -diffeomorphism $f_{p,\mathfrak {a}}:= ( s_p^{\sigma (\mathfrak {a})} )^{-1} \circ \mathcal {F}^\ell _p \circ s_p^{\mathfrak {a}} $ on a small neighborhood $X'$ of X independent of $(p,\mathfrak {a})$ . Its image is such that $f_{p,\mathfrak {a}}(X') \Subset X$ . The differential $D\mathcal {F}^\ell _p$ strictly contracts (respectively, expands) $\mathcal {E}^s_{p,z}$ (respectively, $\mathcal {E}^u_{p,z}$ ) by a factor $\lambda ^\ell <\lambda <1$ (respectively, $1/\lambda $ ) uniformly over $p \in \overline {\mathcal {P}}$ and $z \in \mathcal {K}_p$ . Up to modifying $\mathcal {F}_p^\ell $ outside a neighborhood of $\mathcal {K}_p$ (this operation does not modify the local stable/unstable sets of $\mathcal {K}_p$ in a neighborhood of $\mathcal {K}_p$ nor jets of points inside it), we can suppose that $\mathcal {F}_p^\ell $ contracts strictly each $W_p^{\mathfrak {a}}$ by $\lambda $ and so each map $f_{p,\mathfrak {a}}$ contracts by less than $\lambda $ . Moreover, the map $(p,x) \mapsto f_{p,\mathfrak {a}}(x)$ is $C^r$ . It extends on $\mathcal {P}' \times X'$ , up to reducing $\mathcal {P}'$ and $X'$ . Easy computations show that the map $\mathfrak {a} \mapsto ((p,x) \mapsto f_{p,\mathfrak {a}} (x) )$ is Hölder for the $C^{r-1}$ -topology and continuous for the $C^r$ -topology. We set

$$ \begin{align*} F_{ p} : (\mathfrak{a},x) \in \overrightarrow{ \mathcal A} \times X \mapsto (\sigma (\mathfrak{a}), f_{p,\mathfrak{a}}(x)) \in \overrightarrow{ \mathcal A} \times X , \end{align*} $$

and $(F_p)_p$ is a family of fiberwise $\lambda $ -contracting skew-products (whose restriction on each fiber is $C^r$ ) satisfying the preliminary assumptions of Theorem C.

Lemma 6.0.1. The similarity dimension $\Delta (p)$ is greater than $\delta _{d,s}$ for any $p \in \overline {\mathcal {P}}$ .

Proof. The fact that the similarity dimension does not depend on the fiber is a consequence of Proposition 5.1.3. Moreover, for a fixed fiber $\mathfrak {a}$ , there are $N_{\varepsilon ,\ell }$ contractions $f_{p,\mathfrak {a}}$ which contract by at most $m( D \mathcal {F}_p)^\ell $ . Thus the result follows by the definition of the similarity dimension and inequality $(\diamond )$ .

Using backward sequences instead of forward sequences, we can define similarly families $(W^\alpha _p)_p$ of local unstable manifolds of $\mathcal {K}_p$ parameterized by maps $s^\alpha _p$ so that the map $ \mathcal {F}_p^\ell $ restricted to some subset of each $\mathring {W}_p^{\sigma (\alpha )} := s^{\sigma (\alpha )}_p((-1,1))$ is a diffeomorphism onto $W_p^{\alpha }$ which expands strictly by $1/\lambda $ . Moreover, the local unstable manifold $W_p^{\alpha }$ intersects the local stable manifold $W_p^{\mathfrak {a}}$ at a unique point.

Remark 6.0.2. For any $C^r$ -family $(\mathcal {F}_q)_{q \in \mathcal {Q}}$ of endomorphisms which extends $(\mathcal {F}_p)_{p \in \mathcal {P}}$ (by this we mean that $\mathcal {Q}$ is a neighborhood of $\mathcal {P}$ ), by hyperbolic continuation, we can extend $(\mathcal {K}_p)_{p \in \mathcal {P}}$ into the continuation $(\mathcal {K}_q)_{q \in \mathcal {Q}}$ of a hyperbolic basic set of stable dimension 1, up to reducing $\mathcal {Q}$ . Thus we can also extend the local stable and unstable manifolds (using Theorem F in the Appendix) and so the family $(F_p)_p$ into a family $(F_q)_q$ of fiberwise $\lambda $ -contracting skew-products, whose restriction on each fiber is $C^r$ .

In the following, we prove Theorem B in the case $\ell =1$ . The proof is the same when $\ell>1$ , with a variant in the proof of Proposition 6.0.3 described in Remark 6.0.7.

Step 2: Maps acting on jets. From $(F_p)_p$ , we define a family of skew-products $(G_{p_0})_{p_0}$ acting on s-jets at any $p_0$ . We begin by performing this step for $d=1$ for the sake of simplicity of notation, and then treat the general case $d \ge 2$ .

For any $p_0 \in \mathcal {P}'$ and $\mathfrak {a} \in \overrightarrow {\mathcal A}$ , we define from $(f_{p, \mathfrak {a}})_p$ the following map $g_{p_0, \mathfrak {a}}$ acting on s-jets:

$$ \begin{align*}g_{p_0, \mathfrak{a}} : (x_p, \partial_p x_p , \ldots, \partial^s_p x_p)_{ | p = p_0} \mapsto (f_{p, \mathfrak{a}} (x_p), \partial_p (f_{p, \mathfrak{a}} (x_p)) , \ldots, \partial^s_p (f_{p, \mathfrak{a}} ( x_p)))_{ | p = p_0} , \end{align*} $$

when defined. Since $(f_{p, \mathfrak {a}})_p$ is a $C^r$ -family of maps with $s \le r-2$ , the family $(g_{p_0, \mathfrak {a}})_{p_0}$ is itself a $C^2$ -family of maps. It depends continuously on $\mathfrak {a}$ .

Notice that

$$ \begin{align*} \partial_p (f_{p, \mathfrak{a}} (x_p))_{ | p = p_0} = Df_{p_0} (x_{p_0} ) \cdot \partial_p (x_p)_{ | p = p_0} + \partial_p ( f_p( x_{p_0} ) )_{ | p = p_0}. \end{align*} $$

We notice that $\partial _p (f_{p, \mathfrak {a}} (x_p))_{ | p = p_0} $ has possibly a non-zero derivative relative to $x_{p_0}$ (independent of $\partial _p (x_p)_{ | p = p_0} $ ), a derivative $Df_{p_0} (x_{p_0} )$ relative to $\partial _p (x_p)_{ | p = p_0} $ , and derivatives equal to zero relative to each $ \partial ^j_p (x_p)_{ | p = p_0} $ for $2 \le j \le s$ .

When $1 < k \le s$ , the term $\partial ^k_p (f_{p, \mathfrak {a}} (x_p))_{ | p = p_0} $ has possibly a non-zero derivative relative to $ \partial ^j_p (x_p)_{ | p = p_0} $ for $0 \le j \le k-1$ (independent of $\partial ^k_p (x_p)_{ | p = p_0} $ ), a derivative $Df_{p_0} (x_{p_0} )$ relative to $\partial ^k_p (x_p)_{ | p = p_0} $ , and derivatives equal to zero relative to each $ \partial ^j_p (x_p)_{ | p = p_0} $ for $k< j \le s$ . To see this, it is enough to write the map $p \mapsto f_{p, \mathfrak {a}} (x_p)$ as the composition of $p \mapsto (p,x_p)$ and $(p,x) \mapsto f_{p, \mathfrak {a}} (x)$ and to apply the multidimensional Faà di Bruno formula (see, for example, [CS, Theorem 2.1]).

We recall that by assumption we have $0<\lvert Df_{p_0} (x_{p_0})\rvert <\lambda <1$ . Then for any $p_0 \in \mathcal {P}'$ and $\mathfrak {a} \in \overrightarrow {\mathcal A}$ , the map $g_{p_0, \mathfrak {a}}$ has inferior unipotent differentials with eigenvalues uniformly bounded between 0 and 1 in modulus.

We recall that $\delta _{1,s} = s+1$ . We define a set of s-jets Y, identified with a subset of $\mathbb {R}^{s+1}$ as follows (the term of degree i corresponding to the ith coordinate). We set

$$ \begin{align*} Y:= X \times [-R_1,R_1] \times \cdots \times [-R_s, R_s]. \end{align*} $$

We can choose $R_1$ large compared to the diameter of X and $R_{i+1}$ large compared to $R_i$ for $1 \le i \le s-1$ . Since its differentials are unipotent with non-zero contracting diagonal coefficients, an immediate induction shows that $g_{p_0, \mathfrak {a}}$ is a $C^2$ -diffeomorphism from a small neighborhood $Y'$ of Y onto $g_{p_0, \mathfrak {a}}(Y') \Subset Y$ . Up to rescaling, we can suppose that $Y:=[-1,1]^{s+1}$ . The map $\mathfrak {a} \mapsto ((p,x) \mapsto f_{p,\mathfrak {a}} (x) )$ is Hölder for the $C^{r-1}$ -topology and continuous for the $C^r$ -topology, and we have $s \le r-2$ . This implies that the map $\mathfrak {a} \mapsto ((p_0,y) \mapsto g_{p_0,\mathfrak {a}} (y))$ is Hölder for the $C^1$ -topology and continuous for the $C^2$ -topology. To summarize, we just prove that $(G_{ p_0})_{p_0}$ , with

$$ \begin{align*} G_{ p_0} : (\mathfrak{a},y) \in \overrightarrow{ \mathcal A} \times Y \mapsto (\sigma (\mathfrak{a}), g_{p_0,\mathfrak{a}}(y)) \in \overrightarrow{ \mathcal A} \times Y , \end{align*} $$

is a family of skew-products satisfying the preliminary assumptions of Theorem C, (U) and $\Delta (p_0)>s+1=\delta _{1,s}$ for any $p_0 \in \overline {\mathcal {P}}$ .

As already said, we get the same result with $d \ge 2$ but with painful notation. Indeed, it is enough to remark that the action of $(f_{p, \mathfrak {a}})_p$ on jets of multiorder $(k_1,\ldots , k_d)$ with $\sum \nolimits _i k_i \le s$ only depends on jets of the same multiorder, with a linear coefficient $Df_{p_0} (x_{p_0} )$ , and on jets of multiorder $(k^{\prime }_1,\ldots , k^{\prime }_d)$ with $\sum \nolimits _i k^{\prime }_i < \sum \nolimits _i k_i$ . In particular, its differentials still satisfy assumption (U).

Step 3: Extending the family. We now extend $(G_{p_0})_{p_0}$ to satisfy the transversality assumption (T) inside a larger family. But we have to ensure that this extension comes from an extension of $(F_p)_p$ , itself coming from an extension of $(\mathcal {F}_p)_{p}$ .

Proposition 6.0.3. There exists a family of skew-products $(G_{q_0})_{q_0 \in \mathcal {Q}}$ :

$$ \begin{align*} G_{q_0} : (\mathfrak{a},y) \in \overrightarrow{ \mathcal A} \times Y \mapsto (\sigma (\mathfrak{a}), g_{q_0,\mathfrak{a}}(y)) \in \overrightarrow{ \mathcal A} \times Y , \end{align*} $$

with $\mathcal {Q}:=\mathcal {P} \times (-1,1)^{m}$ for some $m>0$ , satisfying the preliminary assumptions of Theorem C, (U), (T), $\Delta (q_0)> \delta _{d,s}$ for $q_0 \in \overline {\mathcal {Q}}$ , and

$$ \begin{align*} G_{(p_0,0)} = G_{p_0}\quad \text{and}\quad g_{(p_0 ,0), \mathfrak{a}} = g_{p_0 , \mathfrak{a}}\quad \text{for every } p_0 \in \overline{\mathcal{P}} \text{ and } \mathfrak{a } \in \overrightarrow{\mathcal{A}}. \end{align*} $$

Moreover, there exists a $C^r$ -family $(\mathcal {F}_q)_{q \in \mathcal {Q}}$ of local diffeomorphisms extending $(\mathcal {F}_p)_{p \in \mathcal {P}}$ such that if $ (F_q)_q=(f_{q,\mathfrak {a}})_q$ is its associated family of skew-products, we have that $f_{(p_0,0) , \mathfrak {a} } = f_{p_0 , \mathfrak {a}}$ and $g_{q_0,\mathfrak {a}}$ is the map acting on the s-jets at $p_0$ derived from $(f_{q,\mathfrak {a}})_q$ (here the jets at $p_0$ are taken varying p for fixed $q^{\prime }_0$ where $q_0:=(p_0,q^{\prime }_0)$ ). The family $(\mathcal {F}_q)_{q \in \mathcal {Q}}$ is of the form $\mathcal {F}_q = \mathcal {F}_{(p,q')} = \mathcal {F}_p + \Sigma _{p,q'} $ where $(\Sigma _{p,q'})_{(p,q')}$ is a $C^r$ -family such that $\Sigma _{p,0} = 0$ .

We now finish the proof of Theorem B. We give the proof of Proposition 6.0.3 immediately thereafter since it is technical.

Step 4: Conclusion. Let us pick the family $(G_{q_0})_{q_0}$ of skew-products satisfying the preliminary assumptions of Theorem C, (U), (T), and $\Delta (q_0)>\delta _{d,s}$ for every $q_0 \in \overline {\mathcal {Q}}$ given by Proposition 6.0.3. We will pick well-chosen $\vartheta $ - $\mathrm {U}$ -perturbations and apply the second part of Theorem C. We will conclude by using the Fubini theorem.

We pick the continuation $(k_q)_q$ of a point $k_q \in \mathcal {K}_q$ at the intersection of local stable and unstable manifolds $W_q^{\mathfrak {a}}$ and $W_q^\alpha $ for arbitrary $\mathfrak {a} \in \overrightarrow {\mathcal {A}}$ and $\alpha \in \overleftarrow {\mathcal {A}}$ . Up to working locally in the parameter space $\mathcal {Q}$ , we can pick a $C^r$ -family $(\Gamma _{t,q})_{t,q}$ of segments of same direction $u_\Gamma $ (these segments all remain parallel when varying t and q), which do not intersect $W^s(\mathcal {K}_q)$ and which intersect $(W_q^\alpha )_{t,q}$ in a curve $(z_{t,q})_{t,q} \ C^0$ -close to $(k_q)_{t,q}$ . Moreover, we choose $(\Gamma _{t,q})_{t,q}$ so that the $v_\Gamma $ -coordinate $\mathcal {X}(q,t)$ in the basis $(u_\Gamma ,v_\Gamma )$ of $\Gamma _{t,q}$ , for a fixed direction $v_\Gamma $ transverse to $\Gamma _{t,q}$ , is of the form

$$ \begin{align*} \mathcal{X}(q,t):= \sum_{ \lvert i \rvert \le s} t_i p^i\quad \text{for } t=(t_1, \ldots, t_{\delta_{d,s} } ) \in \mathcal{T}:= (-1,1)^{\delta_{d,s} } \end{align*} $$

which does not depend on $q'$ but only on p for $q=(p,q')$ . In particular, for any $q_0 \in \mathcal {Q}$ , the map $t \in \mathcal {T} \mapsto \mathrm {J }_{q_0}^s \mathcal {X}(q,t)$ sends $\mathcal {T}$ diffeomorphically to a non-empty set of s-jets in p (and thus a set of s-jets in p of positive $\delta _{d,s}$ -dimensional measure).

By the parametric inclination lemma (Lemma A.4.2 in the Appendix), up to taking an inverse iterate, for any $t \in \mathcal {T}$ , we can suppose that $(\Gamma _{t,q})_{q}$ is (uniformly in $t \in \mathcal {T}$ ) close to the continuation $(W^{\mathfrak {a}}_q)_q$ of a local stable manifold for some $\mathfrak {a} \in \overrightarrow { \mathcal A}$ . We can parameterize it by the segment X with a $C^r$ -family of charts $C^r$ -close to $(s^{\mathfrak {a}}_q)_q$ . Still iterating backwards, for every $a \in \mathcal {A}$ , there is a family of submanifolds $(\Gamma ^a_{t,q})_{q}$ close to $(W^{a\mathfrak {a}}_q)_q$ (uniformly in $t \in \mathcal {T}$ ) such that $\Gamma ^a_{t,q}$ is sent by $\mathcal {F}_q$ into the interior of $\Gamma _{t,q}$ . We can parameterize $(\Gamma ^a_{t,q})_{q}$ by X with a family of charts close to $(s^{a\mathfrak {a}}_q)_q$ , and in these charts the restriction of $\mathcal {F}_q$ from $\Gamma ^a_{t,q}$ into $\Gamma _{t,q}$ defines a family of maps $(\tilde {f}_{t,q,a \mathfrak {a} })_{q} \ C^r$ -close to $(f_{q,a \mathfrak {a} })_{q}$ . Iterating backwards by induction, we can define, for every $\alpha \in \mathcal {A}^*$ , families of curves $(\Gamma ^\alpha _{t,q})_{q}$ which are (uniformly in $\alpha $ ) $C^r$ -close to $(W^{\alpha \mathfrak {a}}_q)_q$ by Lemma A.4.2 and such that each $\Gamma ^{a \alpha }_{t,q}$ is sent into the interior of $\Gamma ^\alpha _{t,q}$ . This defines families of maps $(\tilde {f}_{t,q,\alpha \mathfrak {a}} )_{q} C^r$ -close to $(f_{q, \alpha \mathfrak {a} })_{q}$ (uniformly in $t \in \mathcal {T}$ and $\alpha \in \mathcal {A}^*$ ). We set $\tilde {f}_{t,q,\mathfrak {b} } := f_{q, \mathfrak {b} }$ when $\mathfrak {b}$ is not of the form $\alpha \mathfrak {a}$ with $\alpha $ a non-empty word. When looking at the action of the families of maps $(\tilde {f}_{t,q,\mathfrak {a} })_{q}$ on s-jets (with respect to p) for fixed values of t and $\mathfrak {a}$ , one obtains a family of maps $(\tilde {g}_{t,q_0, \mathfrak {a}} )_{q_0}$ from Y into itself $C^2$ -close to $(g_{q_0, \mathfrak {a} })_{q_0}$ (uniformly in t and $\mathfrak {a}$ ). This defines a family (indexed by t) of $\vartheta $ - $\mathrm {U}$ -perturbations of the family $(G_{q_0})_{q_0}$ of skew-products, with arbitrarily small $\vartheta $ .

By the second part of Theorem C, we have

$$ \begin{align*} \mathrm{Leb}_{\delta_{d,s} }( \tilde{K}_{t,q_0,\mathfrak{a}} )>0\quad \text{for } \mathrm{Leb}_{d+m}\text{-a.e. } q_0 \in \mathcal{Q} \text{ and } \mathrm{Leb}_{\delta_{d, s}}\text{-a.e. } t \in \mathcal{T} , \end{align*} $$

where the limit set $ \tilde {K}_{t,q_0,\mathfrak {a}}$ is formed by jets at $p_0$ taken while varying p for fixed $q^{\prime }_0$ where $q_0 := (p_0,q^{\prime }_0)$ . By the Fubini theorem, for a.e. $q^{\prime }_0$ we have

$$ \begin{align*} \mathrm{Leb}_{\delta_{d,s} }( \tilde{K}_{t,(p_0,q^{\prime}_0),\mathfrak{a}} )>0\quad \text{for } \mathrm{Leb}_d\text{-a.e. } p_0 \in \mathcal{P} \text{ and } \text{ for } \mathrm{Leb}_{\delta_{d,s}}\text{-a.e. } t \in \mathcal{T}. \end{align*} $$

We fix such a $q^{\prime }_0$ . We consider $(\tilde {\mathcal {F}}_{p})_p := (\mathcal {F}_{(p,q^{\prime }_0)})_p$ and $\tilde {\Gamma }_{t,p}:= \Gamma _{t,(p,q^{\prime }_0)}$ . We notice that $\tilde {K}_{t,(p_0,q^{\prime }_0),\mathfrak {a}}$ is (in the charts) the set of the s-jets at $p_0$ of the intersection points between the local unstable set of $\tilde {\mathcal {K}}_{p}$ and $\tilde {\Gamma }_{t,p}$ . Moreover, the set of s-jets at any $p_0$ of the $v_\Gamma $ -coordinate of $\tilde {\Gamma }_{t,p}$ in the basis $(u_\Gamma ,v_\Gamma )$ when varying t in $\mathcal {T}$ has positive $\delta _{d,s}$ -dimensional Lebesgue measure. Since we have a positive set of one-dimensional s-jets in the direction of $u_\Gamma $ for a.e. $t \in \mathcal {T}$ , we just have to use the Fubini theorem to conclude to a set of bidimensional s-jets of positive measure for a.e. $p_0 \in \mathcal {P}$ .

The same proof works for every family $(\mathcal {G}_p)_p$ which is $C^r$ -close to $(\mathcal {F}_p)_p$ with the extension $(\mathcal {G}_q)_{q \in \mathcal {Q}}$ given by $\mathcal {G}_q = \mathcal {G}_p + \Sigma _{p,q'} $ where $(\Sigma _{p,q'})_{(p,q')}$ is the $C^r$ -family given by Proposition 6.0.3. Indeed, the preliminary conditions of Theorem C, (U), and $\Delta (q_0)> \delta _{d,s}$ are open conditions, and the extension in Proposition 6.0.3 in order to get (T) works for nearby families with the same additive perturbation since having a positive relative speed is an open property. Thus we can apply Theorem C to the family of skew-products derived from $(\mathcal {G}_q)_{q }$ . This achieves the proof that $(\mathcal {K}_{p})_p$ is an almost $C^{r,s}$ -parablender and thus shows Theorem B.□

Remark 6.0.4. When the family $(\mathcal {F}_p)_p$ is of the form $(\mathcal {F})_p$ with $\mathcal {F}$ independent of p and when the order s of the jets is equal to 0, the extension $(\mathcal {F}_q)_q$ satisfying the conclusions of Theorem B can be taken of the form $(\mathcal {F}+\Sigma _{q'})_{(p,q')}$ with $(\Sigma _{q'})_{q'}$ independent of p. This will help to prove Theorem A from Theorem B.

Proof of Proposition 6.0.3

The proof is divided into two steps. The goal and main difficulty of the extension concerns satisfying (T): we want to extend the family $(\mathcal {F}_p)_{p \in \mathcal {P}}$ by adding finitely many new parameters, which will give a positive relative speed to pairs of limit points with different combinatorics, inside each fiber. We set

$$ \begin{align*}\Omega := \{ (\mathfrak{a},\alpha, \beta) \in \overrightarrow{\mathcal{A}} \times \overleftarrow{\mathcal{A}} \times \overleftarrow{\mathcal{A}}: \alpha_{-1} \neq \beta_{-1} \}, \end{align*} $$

and by introducing new parameters we will give a positive relative speed to the points coded by $\alpha $ and $\beta $ inside the fiber coded by $\mathfrak {a}$ for any $(\mathfrak {a},\alpha , \beta ) \in \Omega $ , and this will give the transversality assumption (T) inside the extended family.

We begin by choosing a covering of $ \Omega $ by products of small cylinders. The reason why we work with a covering by small products of cylinders is that this will allow us to control precisely the relative movement of the two limit points. We will then extend iteratively the original family $(\mathcal {F}_p)_{p \in \mathcal {P}}$ by adding new parameters for each set of the covering.

We first pick an arbitrary number $h>0$ . The extended family $(\mathcal {F}_q)_{q \in \mathcal {Q}}$ will be taken so that the families $(\mathcal {F}_{(p,q')})_{p \in \mathcal {P}}$ are (uniformly in $q'$ ) h- $C^r$ -close to $(\mathcal {F}_p)_{p \in \mathcal {P}}$ . We pick an integer M large enough so that $ \lambda ^M/(1-\lambda )$ is small.

Step 1: Working locally. Let us pick any $(\mathfrak {a},\alpha , \beta ) \in \Omega $ . If $\mathfrak {a}$ is periodic of minimal period $\mathfrak {p} \in \mathcal {A}^*$ , since $\alpha _{-1} \neq \beta _{-1}$ , we can choose $\beta $ so that $\beta _{-1}$ is different from the last letter of $\ \mathfrak {p}$ .

Fact 1. The sequences $\sigma ^k(\mathfrak {a})$ where $0 \le k \le M$ are all distinct from $\beta _{-1} \mathfrak {a}$ .

Fact 2. The sequences $\beta _{| k} \mathfrak {a}$ where $2 \le k \le M$ are all distinct from $\beta _{-1} \mathfrak {a}$ .

Proof of Facts 1 and 2

If $\mathfrak {a}$ is not periodic, this is immediate. If $\mathfrak {a}$ is periodic, it is due to the fact that $\beta _{-1}$ is different from the last letter of its minimal period $\mathfrak {p}$ .

From now, we distinguish two disjoint cases which cover all possible tuples in $ \Omega $ :

Case 1: The sequences $\alpha _{| k} \mathfrak {a}$ where $1 \le k \le M$ are all distinct from $\beta _{-1} \mathfrak {a}$ . In this case, we consider the two cylinders $[\rho _\alpha ]$ and $[\rho _\beta ]$ of $ \overleftarrow {\mathcal {A}}$ of length M defined by $\rho _\alpha := \alpha _{|M}$ and $\rho _\beta := \beta _{|M}$ . We pick a small cylinder $[\rho _{\mathfrak {a}}]$ of $\overrightarrow {\mathcal {A}}$ such that $\ \mathfrak {a} \in [\rho _{\mathfrak {a}}]$ and

• • the cylinders $[\alpha _{|k} \rho _{\mathfrak {a}}]$ where $1 \le k \le M$ are all disjoint from $[\beta _{-1} \rho _{\mathfrak {a}}]$ ;

• • the cylinders $[ \beta _{|k} \rho _{\mathfrak {a}}]$ where $2 \le k \le M$ are all disjoint from $[\beta _{-1} \rho _{\mathfrak {a}}]$ .

Case 2: $\mathfrak {a}$ is periodic of minimal period $\mathfrak {p}$ and there exists $f>0$ such that the last letters of $\alpha $ are of the form $\beta _{-1} \cdot \mathfrak {p}^f$ . We consider two small cylinders $[\rho _\alpha ]$ and $[\rho _\beta ]$ of $ \overleftarrow {\mathcal {A}}$ containing $\alpha $ and $\beta $ , and a small cylinder $[\rho _{\mathfrak {a}}]$ of $\overrightarrow {\mathcal {A}}$ such that $\ \mathfrak {a} \in [\rho _{\mathfrak {a}}]$ and

• • the cylinders $\sigma ^k( [\rho _{\mathfrak {a}}])$ where $0 \le k \le M$ are all disjoint from $[\beta _{-1} \rho _{\mathfrak {a}}]$ ;

• • the cylinders $[ \beta _{|k} \rho _{\mathfrak {a}}]$ where $2 \le k \le M$ are all disjoint from $[\beta _{-1} \rho _{\mathfrak {a}}]$ .

We take these cylinders so that their lengths are large compared to M.

We notice that the compact set $ \Omega $ is covered by the union of the (open) products of cylinders $[\rho _{\mathfrak {a}}] \times [\rho _\alpha ] \times [\rho _\beta ]$ associated to any $(\mathfrak {a},\alpha , \beta ) \in \Omega $ (independent of h). We can then find a finite covering of $ \Omega $ by such products.

Step 2: Extension. We now construct an extension of $(\mathcal {F}_p)_p$ obtained by successively extending the family a finite number of times, adding at each step $\delta _{d,s}$ parameters corresponding to a product of cylinders $[\rho ] \times [\rho '] \times [\rho ''] \subset \Omega $ in the finite covering defined in step 1. Here $[\rho ]$ is a cylinder in $\overrightarrow {\mathcal {A}}$ and $[\rho ']$ and $[\rho '']$ are cylinders in $ \overleftarrow {\mathcal {A}}$ such that $\rho ^{\prime }_{-1} \neq \rho ^{\prime \prime }_{-1}$ . The $\delta _{d,s}$ parameters are intended to move the s-jet of the limit point associated to $\alpha $ relative to that corresponding to $\beta $ , inside the fiber encoded by $\mathfrak {a}$ , where $(\mathfrak {a},\alpha , \beta )$ is any tuple in $[\rho ] \times [\rho '] \times [\rho '']$ .

The union of the points of $R_{p,\mathfrak {a}}$ on the local stable manifolds $W^{\mathfrak {a}}_p$ for $\mathfrak {a} \in [ \rho ^{\prime \prime }_{-1} \rho ]$ is disjoint from the union of the points of $R_{p,\mathfrak {a}}$ on the local stable manifolds $W^{\mathfrak {a}}_p$ for $\mathfrak {a} \notin [ \rho ^{\prime \prime }_{-1} \rho ]$ at every parameter $p \in \overline {\mathcal {P}}$ . We then pick a $C^r$ -family $(h_p)_p$ of bump functions $h_p$ equal to h in a neighborhood of the first ones and equal to 0 in a neighborhood of the second ones for every $p \in \overline {\mathcal {P}}$ . We also pick a $C^r$ -family of maps $v^s_p$ such that $v^s_p(z)$ is close to the stable direction of $\mathcal {K}_p$ at $\mathcal {F}_p(z)$ for any point $z \in \mathcal {K}_p$ .

We now extend the family by setting for every $q:=(p,q')$ with $p \in \mathcal {P}'$ and $q'=(q_i)_i \in \mathbb {R}^{\delta _{d,s}}$ small:

(24) $$ \begin{align} \mathcal{F}_q(z) = \mathcal{F}_p(z)+ h_p(z) \cdot \bigg( \sum_{ i } q_{i} \cdot p^i \bigg) \cdot v^s_p(z)\quad \text{for all } z \in \mathbb{R}^2 , \end{align} $$

where we sum over $i=(i_1, \ldots , i_d )$ such that $\sum \nolimits _k i_k \le s$ with $p^i=p_1^{i_1} \cdots p_d^{i_d}$ .

This defines a $C^r$ -family of endomorphisms $(\mathcal {F}_q )_q$ . For small values of $q'$ , say for q in some open neighborhood of $\mathcal {P} \times \{0\}$ , these are still local diffeomorphisms. Moreover, the family of hyperbolic basic sets admits a continuation as a family $(\mathcal {K}_q)_q$ (see the Appendix).

By Remark 6.0.2, we can define the associated family of skew-products $(F_{q})_q$ which extends $(F_p)_p$ and then the associated map $(G_{q_0})_{q_0}$ acting on the s-jets derived from $(F_{q})_q$ , defined as in step 2 of the proof of Theorem B (let us recall that the jets are taken while varying only p for fixed $q'$ if we set $q:=(p,q')$ ). In particular, $(G_{q_0})_{q_0}$ is a family of skew-products satisfying the preliminary assumptions of Theorem C and also (U). By Proposition 5.1.3, the map $q \mapsto \Delta (q)$ is continuous. Thus, up to restricting the parameter space, we have $\Delta (q_0)>\delta _{d,s}$ for any $q_0$ .

Here is the counterpart of Lemma 5.1.4 in the $C^s$ -case.

Lemma 6.0.5. The map $q \mapsto \pi _{q,\mathfrak {a}}(\alpha )$ is $C^s$ for every $\mathfrak {a} \in \overrightarrow { \mathcal A}$ and $\alpha \in \overleftarrow { \mathcal A}$ . Moreover, for every family $\mathbb {F}$ of $\vartheta $ - $\mathrm {U}$ -perturbations with small $\vartheta>0$ , the map $(t,q) \mapsto \tilde {\pi }_{t,q,\mathfrak {a}}(\alpha )$ is $C^s$ and $q \mapsto \tilde {\pi }_{t,q,\mathfrak {a}}(\alpha )$ is $C^s$ -close to $q \mapsto \pi _{q,\mathfrak {a}}(\alpha )$ uniformly in $t \in \mathcal {T}$ .

Proof. We already saw in the proof of Lemma 5.1.4 that the map $q \mapsto \pi _{q,\mathfrak {a}}(\alpha )$ is the uniform limit (when $n \rightarrow + \infty $ ) of the maps $q \mapsto \psi _{q,\mathfrak {a}}^{\alpha _{| n}} (0)$ . On the other hand, we saw in the second step of the proof of Theorem B that the contraction $\psi _{q,\mathfrak {a}}^{\alpha _{| n}}$ induces a map on s-jets which has inferior unipotent differentials with eigenvalues exponentially small in n. Thus by Lemma 5.1.2 this induced map acting on s-jets contracts by a factor independent of $(q,\mathfrak {a})$ and exponentially small in n. Thus the sequences of the s first derivatives of the maps $q \mapsto \psi _{q,\mathfrak {a}}^{\alpha _{| n}} (0)$ all converge uniformly. Then the map $q \mapsto \pi _{q,\mathfrak {a}}(\alpha )$ is $C^s$ . The same argument works for small $\vartheta $ - $\mathrm {U}$ -perturbations.

We now prove that $(G_{q_0})_{q_0}$ satisfies property (T) restricted to any $(\mathfrak {a},\alpha , \beta ) \in [\rho ] \times [\rho '] \times [\rho '']$ . More precisely we show below the following technical lemma.

Lemma 6.0.6. Up to reducing $\mathcal {Q}$ , for every $\mathfrak {a} \in [\rho ]$ , $\alpha \in [\rho ']$ , $\beta \in [\rho '']$ , $p_0 \in \mathcal {P}$ , and $r>0$ , with $q:=(p,q')$ , the set of $q^{\prime }_0$ such that $\mathrm {J}_{p_0}^s \pi _{q,\mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^s \pi _{q,\mathfrak {a}}(\beta )$ are r-close is of Lebesgue measure dominated by $r^{\delta _{d,s}}$ , with a constant independent of $\mathfrak {a}$ , $\alpha $ , $\beta $ , and $p_0$ . Moreover, for every family of $\vartheta $ -perturbations of $(G_{q_0})_{q_0}$ with $\vartheta $ small enough, $t \in \mathcal {T}$ , $\mathfrak {a} \in [\rho ]$ , $\alpha \in [\rho ']$ , $\beta \in [\rho '']$ , $p_0 \in \mathcal {P}$ , and $r>0$ , the set of $q^{\prime }_0$ such that $\mathrm {J}_{p_0}^s \tilde {\pi }_{t,q,\mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^s \tilde {\pi }_{t,q,\mathfrak {a}}(\beta )$ are r-close is of Lebesgue measure dominated by $r^{\delta _{d,s}}$ , with the same constant.

We recall that the s-jets at $p_0$ are taken while varying p around $p_0$ for a fixed value of $q'$ equal to $q^{\prime }_0 $ . In particular, by the Fubini theorem, the set of $(p_0,q^{\prime }_0)$ such that $\mathrm {J}_{p_0}^s \pi _{q,\mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^s \pi _{q,\mathfrak {a}}(\beta )$ are r-close is also of measure dominated by $r^{\delta _{d,s}}$ and the same holds for $\vartheta $ -perturbations. The proof of Lemma 6.0.6 is given below. We first finish the proof of Proposition 6.0.3.

We extend iteratively the initial family $(\mathcal {F}_p)_p$ a finite number of times by the same method. At each step, we extend by adding $\delta _{d,s}$ new parameters corresponding to a new product of cylinders $[\rho ] \times [\rho '] \times [\rho '']$ in the finite covering of $ \Omega $ defined in step 1. The adaptation of Lemma 6.0.6 is straightforward. This proves property (T) and so $(G_{q_0})_{q_0 \in \mathcal {Q}}$ is a family of skew-products, with $\mathcal {Q}:=\mathcal {P} \times (-1,1)^{m}$ for some $m>0$ (up to rescaling). This concludes the proof.

Proof of Lemma 6.0.6

We first prove the result for case 1 in step 1 of the proof of Proposition 6.0.3. Let $\Pi _{q,\mathfrak {a}}(\alpha )$ and $\Pi _{q,\mathfrak {a}}(\beta )$ be the points of the phase space $\mathbb {R}^2$ equal to $\pi _{q,\mathfrak {a}}(\alpha )$ and $\pi _{q,\mathfrak {a}}(\beta )$ in the parameterization of $W^{\mathfrak {a}}_q$ . Both belong to $W^{\mathfrak {a}}_q$ .

We pick local coordinates in a neighborhood of $\Pi _{0,\mathfrak {a}}(\alpha )$ centered at $\Pi _{0,\mathfrak {a}}(\alpha ) $ with a basis given by $\mathbb {R} \cdot e_0^s+\mathbb {R} \cdot e_0^u$ , where we denote by $\mathbb {R} \cdot e_0^s$ and $\mathbb {R} \cdot e_0^u$ the stable and unstable directions of $\overleftrightarrow {\mathcal {K}_0}$ at $\Pi _{0,\mathfrak {a}}(\alpha )$ . We write

$$ \begin{align*} \mathrm{J}_{p_0}^s \Pi_{q,\mathfrak{a}}(\alpha) =: \mathcal{J}_{p_0}^{\kern2pt s} \Pi_{q,\mathfrak{a}}(\alpha) \cdot e_0^s + \mathcal{J}_{p_0}^{\kern2pt u} \Pi_{q,\mathfrak{a}}(\alpha) \cdot e_0^u. \end{align*} $$

Take care that the s in the left-hand term is the order of the jet and the s in the right hand term just means ‘stable’. We can proceed similarly for $\Pi _{0,\mathfrak {a}}(\beta )$ .

Case 1. Easy case: $0$ -jets. We first perform the proof for 0-jets to show the general idea. We fix the parameter $p_0=0$ . We begin by studying $\Pi _{q,\mathfrak {a}}(\alpha )$ , more precisely the variations of the 0-jet of $\Pi _{(0,q'),\mathfrak {a}}(\alpha )$ , that is, $\Pi _{(0,q'),\mathfrak {a}}(\alpha )$ itself, when moving $q'$ . In order to do this, we set

$$ \begin{align*} \Pi_0(q'):=\Pi_{(0,q'),\mathfrak{a}}(\alpha) \end{align*} $$

and denote by $\Pi _k(q')$ its preimage by $\mathcal {F}^{\kern2pt k}_{(0,q')}$ on $W^{\alpha _{ | k } \mathfrak {a}}_{(0,q')}$ (this is the point equal to $\pi _{(0,q'), \alpha _{ | k } \mathfrak {a}}(\sigma ^k ( \alpha ))$ in the parameterization of the stable manifold). We pick local coordinates centered at each $P_k:=\Pi _k(0)$ with a basis given by the corresponding preimages $\mathbb {R} \cdot e_k^s$ and $\mathbb {R} \cdot e_k^u$ of $\mathbb {R} \cdot e_0^s$ and $\mathbb {R} \cdot e_0^u$ by $\mathcal {F}^{\kern2pt k}_0$ for $k>0$ . These are the stable and unstable directions of $\overleftrightarrow {\mathcal {K}_0}$ at $P_k$ . In the decomposition $\mathbb {R} \cdot e_k^s+\mathbb {R} \cdot e_k^u$ , we write

$$ \begin{align*} \Pi_k(q') =: \Pi^s_k(q') \cdot e_k^s + \Pi^u_k(q') \cdot e_k^u. \end{align*} $$

In the coordinates given by $P_{k+1} + \mathbb {R} \cdot e_{k+1}^s + \mathbb {R} \cdot e_{k+1}^u$ and $P_k + \mathbb {R} \cdot e_k^s+\mathbb {R} \cdot e_{k}^u$ with $P_{k+1}=P_k=(0,0)$ , the map $\mathcal {F}_0$ restricted to a neighborhood of $P_{k+1} = (0,0)$ sends $P_{k+1}=(0,0)$ to $P_k=(0,0)$ and is $C^1$ -close to its differential which is diagonal. In particular, the $(1,1)$ -coefficient is a real number $\lambda _k$ such that $\lvert \lambda _k\rvert < \lambda <1$ .

By hyperbolic continuation, there exists $B>0$ independent of k and M such that

$$ \begin{align*} C_k := \bigg\lvert \frac{d \Pi^s_k }{dq'} (0) \bigg\rvert < B. \end{align*} $$

We recall that $\Pi _{k+1} (q')$ is sent onto $\Pi _k(q')$ by $\mathcal {F}_{(0,q')}$ . Moreover, $\mathcal {F}_{(0,q')} (\Pi _{k+1} (q'))$ is the sum of $\mathcal {F}_{0} (\Pi _{k+1} (q'))$ and a term $\Sigma _{k}(q') = \Sigma ^s_k(q') \cdot e_k^s+ \Sigma ^u_k(q') \cdot e_k^u$ coming from equation (24). In particular, we notice that

$$ \begin{align*} \bigg\lvert \frac{d \Sigma^s_k}{dq'} (0) \bigg\rvert < 2h. \end{align*} $$

We have

$$ \begin{align*} \frac{d\Pi^s_{k}}{dq'} (0)&= \lambda_k \cdot \frac{d \Pi^s_{k+1}}{d q'} (0) + \frac{d \Sigma^s_k}{d q'} (0) , \\[5pt] C_k &< \lambda_k \cdot C_{k+1} + 2h. \end{align*} $$

We also notice that $\Sigma _k(q')$ is equal to zero when $0 \le k \le M-1$ by the first item of case 1. This gives, for every $M'>M$ ,

$$ \begin{align*} C_0 < B \cdot \lambda^{M'} + 2h \cdot (\lambda^{M'-1} + \cdots +\lambda^M). \end{align*} $$

By taking $M'$ large, we get

$$ \begin{align*} C_0 \le 2 h \frac{ \lambda^{M}} {1- \lambda} , \end{align*} $$

which is small compared to h by assumption. We finally get that the derivative of $ \Pi _{(0,q'),\mathfrak {a}}(\alpha ) = \Pi _0(q')$ at $q'=0$ in the stable direction $\mathbb {R} \cdot e_0^s$ is small compared to h:

$$ \begin{align*}\frac{d}{dq'} \Pi^s_{(0,q'),\mathfrak{a}}(\alpha) \text{ is small compared to } h. \end{align*} $$

The same holds true when replacing $\Pi _{(0,q'),\mathfrak {a}}(\alpha )$ by the preimage $\Pi ^{\prime }_1(q')$ of $\Pi _{(0,q'),\mathfrak {a}}(\beta )$ by $\mathcal {F}_{(0,q')}$ on $W^{\beta _{ -1 } \mathfrak {a}}_{(0,q')}$ , using the second item of case 1. By equation (24) and since $ e_0^s$ is close to $ v_0^s(\Pi ^{\prime }_1(0))$ , the derivative $({d}/{dq'}) \Pi ^s_{(0,q'),\mathfrak {a}}(\beta )$ is then close to h. Thus

$$ \begin{align*} \frac{d}{dq'} ( \Pi^s_{(0,q'),\mathfrak{a}}(\alpha) - \Pi^s_{(0,q'),\mathfrak{a}}(\beta) ) \text{ is bounded away from }(0,0). \end{align*} $$

In particular, up to reducing $\mathcal {Q}$ , the set of $q^{\prime }_0$ such that $\mathrm {J}_{0}^0 \Pi _{q, \mathfrak {a}}(\alpha )$ and $\mathrm {J}_{0}^0 \Pi _{q,\mathfrak {a}}(\beta )$ are r-close is of Lebesgue measure dominated by r with a constant which is independent of $\mathfrak {a}$ , $\alpha $ , $\beta $ , and $p_0$ (since the above relative speed is locally uniformly bounded from below while varying $\mathfrak {a}$ , $\alpha $ , $\beta $ , and $p_0$ ). We can proceed the same way for any $p_0$ , and the domination constant is independent. Integrating using the Fubini theorem, the set $(p_0,q^{\prime }_0)$ such that $\mathrm {J}_{p_0}^0 \Pi _{q, \mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^0 \Pi _{q,\mathfrak {a}}(\beta )$ are r-close is of Lebesgue measure dominated by r, and thus it is also the case for $\mathrm {J}_{p_0}^0 \pi _{q, \mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^0 \pi _{q,\mathfrak {a}}(\beta )$ . The statement about $\vartheta $ -perturbations follows easily with the same arguments when $\vartheta $ is small.

Case 2. General case: s-jets. We now turn to the general but more difficult case of s-jets. This time, we also vary the parameter p. We fix $p_0=0$ . We investigate the differentials of $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\alpha )$ and $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\beta )$ when deriving relative to $q'$ .

We keep the same coordinates for each k: we keep expressing the point $\Pi _k(p,q')$ (depending also on p this time) in the coordinates $P_k+\mathbb {R} \cdot e_k^s+\mathbb {R} \cdot e_k^u$ (independent of $(p,q')$ small). Let $\mathcal {J}_{0}^s \Pi _k(q')$ and $\mathcal {J}_{0}^u \Pi _k(q')$ be the components in this basis of the s-jet of $\Pi _k(p,q')$ at $p_0=0$ for $q'$ fixed. Note that by hyperbolic continuation, there exists $B'>0$ independent of k and M such that

$$ \begin{align*} C^{\prime}_k := \bigg\lvert \frac{d \mathcal{J}_{0}^{\kern2pt s} \Pi_k}{dq'} (0) \bigg\rvert < B'. \end{align*} $$

For $q'$ fixed the family $(\Pi _k(p,q'))_p$ is the image of $(\Pi _{k+1}(p,q'))_p$ by the family of maps $(\mathcal {F}_{(p,q')})_p$ . The family $(\mathcal {F}_{(p,q')} (\Pi _{k+1} (p,q')))_p$ is the sum of $(\mathcal {F}_{p} (\Pi _{k+1} (p,q')))_p$ and of the family $(\Sigma _{p,k})_p$ of perturbations coming from equation (24). In particular,

$$ \begin{align*} \bigg\lvert \frac{d \mathcal{J}_{0}^{\kern2pt s} \Sigma_{k,p}}{dq'} (0) \bigg\rvert \preceq 2 \delta_{d,s} h. \end{align*} $$

Again $\Sigma _{k,p}(q')$ is equal to zero when $0 \le k \le M-1$ by the second item of case 1. However, this time the action of $(\mathcal {F}_p)_p$ on s-jets is more complicated than on 0-jets. For each k, we write the Taylor expansion of $\mathcal {F}_p$ in $P_{k+1}$ of order s. Then we replace each of its coefficients by its s-jet in p, and the variables by $ \mathcal {J}_{0}^s \Pi _{k+1}(q')$ and $\mathcal {J}_{0}^u \Pi _{k+1}(q')$ , and we expand this expression. This shows that

$$ \begin{align*} \mathcal{J}_{0}^{\kern2pt s} \Pi_{k}(q') = \mathcal{M}_k \cdot \mathcal{J}_{0}^{\kern2pt s} \Pi_{k+1}(q')+\mathcal{M}_k' \cdot \mathcal{J}_{0}^{\kern2pt u} \Pi_{k+1}(q')+ \mathcal{J}_{0}^{\kern2pt s} \Sigma_{k,p}(q'). \end{align*} $$

The term $\mathcal {J}_{0}^s \Sigma _{k,p}(q')$ has its derivative at $q'=0$ bounded by $2 \delta _{d,s} h$ , and its coefficients in $p^i$ depends only on $q_i$ . The matrix $\mathcal {M}_k$ is lower triangular with all its diagonal coefficients equal to $\lambda _k$ with $\lvert \lambda _k\rvert <\lambda <1$ . On the other hand, the matrix $\mathcal {M}_k'$ is lower triangular with all its diagonal coefficients equal to $0$ . Iterating, using the first item of step 1 and then taking the derivative at $q'=0$ , we see that

$$ \begin{align*} \frac{d}{dq'} \mathcal{J}_{0}^{\kern2pt s} \Pi_{(p ,q'),\mathfrak{a}}(\alpha) \end{align*} $$

is lower triangular with all its diagonal coefficients bounded by

$$ \begin{align*} 2 \delta_{d,s} h \frac{ \lambda^{M}} {1- \lambda} , \end{align*} $$

which is small compared to h by assumption on M. Thus for every $i = (i_1, \ldots , i_d)$ such that $\sum \nolimits _k i_k \le s$ , the coordinate in $p^i$ of the stable component of the s-jet $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\alpha )$ has small derivative compared to h while moving the parameter $q_i$ around 0.

The same holds for the preimage $\Pi ^{\prime }_1(p,q')$ of $\Pi _{q,\mathfrak {a}}(\beta )$ by $\mathcal {F}_{q}$ . By equation (24) and since $\mathbb {R} \cdot e_0^s$ is close to $\mathbb {R} \cdot v_0^s(\Pi ^{\prime }_1(0,0))$ , for every $i = (i_1, \ldots , i_d)$ such that $\sum \nolimits _k i_k \le s$ , the coordinate in $p^i$ of the stable component of the s-jet $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\beta )$ has a derivative close to h (up to a non-zero independent multiplicative constant) while moving the parameter $q_i$ . Thus the coordinate in $p^i$ of $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\alpha )-\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\beta )$ has a non-zero derivative while moving $q_i$ around 0. The same still holds for other values of $p_0$ . Then, up to reducing $\mathcal {Q}$ , the set of $(p_0,q^{\prime }_0)$ such that $\mathrm {J}_{p_0}^s \Pi _{q, \mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^s \Pi _{q,\mathfrak {a}}(\beta )$ are r-close is of Lebesgue measure dominated by $r^{\delta _{d,s}}$ , and thus it is also the case for $\mathrm {J}_{p_0}^s \pi _{q, \mathfrak {a}}(\alpha )$ and $\mathrm {J}_{p_0}^s \pi _{q,\mathfrak {a}}(\beta )$ . This gives the result, using the Fubini theorem. The statement about $\vartheta $ -perturbations follows easily with similar arguments. This ends the proof for case 1.

For case 2, the proof is simpler. First take the original sequences $\alpha $ , $\beta $ , and $\mathfrak {a}$ of case 2 around which the cylinders $[\rho _\alpha ]$ , $[\rho _\beta ]$ , and $[\rho _{\mathfrak {a}}]$ are centered. The coordinate in $p^i$ of the stable component of the s-jet $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\beta )$ still has a non-zero derivative while moving the parameter $q_i$ (this only needs the second item of case 2 and not the first item of case 1 not present in case 2). On the other hand, $\Pi _{q,\mathfrak {a}}(\alpha )$ is the image of $\Pi _{q,\mathfrak {a}}(\beta )$ by some iterate of $\mathcal {F}_q$ restricted on the M first images of $W^{\mathfrak {a}}_q$ since $\mathfrak {a}$ is periodic of period $\mathfrak {p}$ and the last letters of $\alpha $ are of the form $\beta _{-1} \cdot \mathfrak {p}^f$ for some $f>0$ . These local stable manifolds do not depend on $q'$ by the first item of case 2. Moreover, the action of $(\mathcal {F}_q)_q$ restricted to these stable manifolds on stable components of jets is lower triangular with diagonal coefficients between 0 and 1. Thus the coordinate in $p^i$ of $\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\alpha )-\mathrm {J}_{0}^s \Pi _{q,\mathfrak {a}}(\beta )$ has again a non-zero derivative while moving $q_i$ . Since the lengths of the cylinders are large, this remains true for any $(\alpha , \beta , \mathfrak {a}) $ in $[\rho _\alpha ] \times [\rho _\beta ] \times [\rho _{\mathfrak {a}}]$ . We conclude as in case 1, which ends the proof.

Remark 6.0.7. When $\ell =1$ , to give a relative movement inside the $\mathfrak {a}$ -fiber to the points encoded by $\alpha $ and $\beta $ , we considered the preimages of these two points by $\mathcal {F}_p$ (respectively on $W^{\alpha _{-1} \mathfrak {a}}_p$ and $W^{\beta _{-1} \mathfrak {a}}_p$ ). The second one was not periodic and distinct from the first one. We then perturbed $\mathcal {F}_p$ in a neighborhood of this second preimage.

In the case where $\ell>1$ , we shall look at the $\ell $ respective successive preimages of these two points by $\mathcal {F}_p$ (respectively, on $W^{\alpha _{-1} \mathfrak {a}}_p$ and $W^{\beta _{-1} \mathfrak {a}}_p$ and their $\ell -1$ successive images by $\mathcal {F}_p$ ) and take the first pair of preimages which are distinct. One of them is not periodic, and we perform the same perturbation as before in a neighborhood of this point. Then the proof is the same with the same distinction in two cases whether the M successive preimages of the other preimage intersect this neighborhood or not.