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 [Reference Simon, Solomyak and UrbańskiSiSoU] 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.

4.1 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 [Reference Simon, Solomyak and UrbańskiSiSoU].

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 [Reference MattilaMa].

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 }$ .

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.

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.

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 $\tilde {K}_{t,p,\mathfrak {a}} :=\tilde {\pi }_{t,p,\mathfrak {a}}( \overleftarrow { \mathcal A})$ .

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 [Reference BowenBo1, 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.

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

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).