SRB measures for $C^{\infty }$ surface diffeomorphisms

Abstract

A $C^{\infty }$ smooth surface diffeomorphism admits an SRB measure if and only if the set $\{ x, \ \limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}\|>0\}$ has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for $C^{r}$ surface diffeomorphisms with $+\infty >r>1$.

1 Introduction

One fundamental problem in dynamics consists in understanding the statistical behaviour of the system. Given a topological system $(X,f)$ we are more precisely interested in the asymptotic distribution of the empirical measures $\left (\frac{1}{n}\sum _{k=0}^{n-1}\delta _{f^{k}x}\right )_{n}$ for typical points $x$ with respect to a reference measure. In the setting of differentiable dynamical systems the natural reference measure to consider is the Lebesgue measure on the manifold.

The basin of a $f$-invariant measure $\mu$ is the set $\mathcal {B}(\mu )$ of points whose empirical measures are converging to $\mu$ in the weak-∗ topology. By Birkhoff’s ergodic theorem the basin of an ergodic measure $\mu$ has full $\mu$-measure. An invariant measure is said to be physical when its basin has positive Lebesgue measure. We may wonder when such measures exist and then study their basins.

In the works of Y. Sinai, D. Ruelle and R. Bowen [10, 38, 42] these questions have been successfully solved for uniformly hyperbolic systems. An SRB measure of a $C^{1+}$ system is an invariant probability measure with at least one positive Lyapunov exponent almost everywhere, which has absolutely continuous conditional measures on unstable manifolds [45]. Physical measures may neither be SRB measures nor sinks (as in the famous figure-eight attractor), however hyperbolic ergodic SRB measures are physical measures [30]. For uniformly hyperbolic systems, there is a finite number of such measures and their basins cover a full Lebesgue subset of the manifold. Beyond the uniformly hyperbolic case such a picture is known for large classes of partially hyperbolic systems [1, 2, 9, 35]. Corresponding results have been established for unimodal maps with negative Schwartzian derivative [25]. SRB measures have been also deeply investigated for parameter families such as the quadratic family and Hénon maps [4–6, 24]. In his celebrated ICM’s talk, M. Viana conjectured that a $C^{1+}$ diffeomorphism admits an SRB measure, whenever the set of points with non-zero Lyapunov exponents has full Lebesgue measure. In recent works some weaker versions of the conjecture (with some additional assumptions of recurrence and Lyapunov regularity) have been proved [7, 18, 19]. Finally we mention that J. Buzzi, S. Crovisier, O. Sarig have also recently shown the existence of an SRB measure for $C^{\infty}$ surface diffeomorphims when the set of points with a positive Lyapunov exponent has positive Lebesgue measure [16] (Corollary 2).

In this paper we define a general entropic approach to build SRB measures. We strongly believe that this approach may be used to recover the existence of SRB measures for weakly mostly expanding partially hyperbolic systems [1] and to give another proof of Ben Ovadia’s criterion for $C^{1+}$ diffeomorphisms in any dimension [7].

We state now the main results of our paper. Let $(M,\|\cdot \|)$ be a compact Riemannian surface and let $\mathop{\mathrm{Leb}}$ be a volume form on $M$, called Lebesgue measure. We consider a $C^{\infty}$ surface diffeomorphism $f:M\circlearrowleft$. The maximal Lyapunov exponent at $x\in M$ is given by $\chi (x)=\limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}\|$. When $\mu$ is a $f$-invariant probability measure, we let $\chi (\mu )=\int \chi (x)\, d\mu (x)$. For two Borel subsets $A$ and $B$ of $M$ we write $A\stackrel{o}{\subset} B$ (resp. $A\stackrel{o}{=} B$) when we have $\mathop{\mathrm{Leb}}(A\setminus B)=0$ (resp. $\mathop{\mathrm{Leb}}(A\Delta B)=0$). For $c\in \mathbb{R}$ and $\Gamma \subset \mathbb{R}$ we also let $\{\chi >c\}:=\{x\in M, \ \chi (x)>c\}$, $\{\chi =c\}:=\{x\in M, \ \chi (x)=c\}$ and $\{\chi \in \Gamma \}:=\{x\in M, \ \chi (x)\in \Gamma \}$.

Theorem 1

Let $f:M\circlearrowleft$ be a $C^{\infty}$ surface diffeomorphism. There are countably many ergodic SRB measures $(\mu _{i})_{i\in I}$, such that we have with $\Lambda =\{\chi (\mu _{i}), \ i\in I\}\subset \mathbb{R}_{>0}$:

• $\{\chi >0\}\stackrel{o}{=} \{\chi \in \Lambda \}$,

• $\{\chi =\lambda \}\stackrel{o}{\subset}\bigcup _{i, \chi (\mu _{i})= \lambda}\mathcal {B}(\mu _{i})$ for all $\lambda \in \Lambda$.

Corollary 1

Let $f:M\circlearrowleft$ be a $C^{\infty}$ surface diffeomorphism. Then

$$ \{\chi >0\}\stackrel{o}{\subset}\bigcup _{\mu \textit{ SRB ergodic}} \mathcal {B}(\mu ). $$

Corollary 2

Buzzi-Crovisier-Sarig [16]

Let $f:M\circlearrowleft$ be a $C^{\infty}$ surface diffeomorphism.

If $\mathop{\mathrm{Leb}}(\chi >0)>0$, then there exists an SRB measure.

In fact we establish a $C^{r}$, $1< r< +\infty$, stronger version, which implies straightforwardly Theorem 1:

Main Theorem 1

Let $f:M\circlearrowleft$ be a $C^{r}$, $\mathbb{R}\ni r>1$, surface diffeomorphism. Let $R(f):=\lim _{n}\frac{1}{n}\log ^{+} \sup _{x\in M}\|d_{x}f^{n}\|$. There are countably many ergodic SRB measures $(\mu _{i})_{i\in I}$ with $\Lambda :=\{\chi (\mu _{i}), \ i\in I\}\subset ]\frac{R(f)}{r},+ \infty [$, such that we have:

• $\left \{\chi >\frac{R(f)}{r}\right \}\stackrel{o}{=} \{\chi \in \Lambda \}$,

• $\{\chi =\lambda \}\stackrel{o}{\subset}\bigcup _{i, \chi (\mu _{i})= \lambda}\mathcal {B}(\mu _{i})$ for all $\lambda \in \Lambda$.

In others terms, Lebesgue almost every point $x$ with $\chi (x)>\frac{R(f)}{r}$ lies in the basin of an ergodic SRB measure $\mu$ with $\chi (x)=\chi (\mu )$.

When $f$ is a $C^{1+}$ topologically transitive surface diffeomorphism, there is at most one SRB measure, i.e. $\sharp I\leq 1$ [23]. If moreover the system is topologically mixing, then the SRB measure when it exists is Bernoulli [30]. By the spectral decomposition of $C^{r}$ surface diffeomorphisms for $1< r\leq +\infty$ [14] there are at most finitely many ergodic SRB measures with entropy and thus maximal exponent larger than a given constant $b>\frac{R(f)}{r}$. Therefore, in the Main Theorem, the set $\Lambda =\{\chi (\mu _{i}), \ i\in I\}$ is either finite or a sequence decreasing to $\frac{R(f)}{r}$. When $r$ is finite, there may also exist ergodic SRB measures $\mu$ with $\chi (\mu )\leq \frac{R(f)}{r}$.

We prove in a forthcoming paper [11] that the above statement is sharp by building for any finite $r>1$ a $C^{r}$ surface diffeomorphism $f:M\circlearrowleft$ with a periodic saddle hyperbolic point $p$ such that $\chi (x)=\frac{\chi (\delta _{p})}{r}>0$ for all $x\in U$ for some set $U\subset \mathcal {B}(\mu _{p})$ with $\mathop{\mathrm{Leb}}(U)>0$, where $\mu _{p}$ denotes the periodic measure associated to $p$ (see [13] for such an example of interval maps).

In higher dimensions we let $\Sigma ^{k}\chi (x):=\limsup _{n}\frac{1}{n}\|\Lambda ^{k} d_{x}f^{n} \|$ where $\Lambda ^{k} df$ denotes the action induced by $f$ on the $k$th exterior power of $TM$ for $k=1,\ldots , d$ with $d$ being the dimension of $M$. By convention we also let $\Sigma ^{0}\chi =0$. For any $C^{1}$ diffeomorphism $(M,f)$ we have $\mathop{\mathrm{Leb}}(\Sigma ^{d}\chi >0)=0$ (see [3]). The product of a figure-eight attractor with a surface Anosov diffeomorphism does not admit any SRB measure whereas $\chi$ is positive on a set of positive Lebesgue measure. However we conjecture:

Conjecture 1

Let $f:M\circlearrowleft$ be a $C^{\infty}$ diffeomorphism on a compact manifold (of any dimension).

If $\mathop{\mathrm{Leb}}\left (\Sigma ^{k}\chi >\Sigma ^{k-1}\chi \geq 0 \right )>0$, then there exists an ergodic measure with at least $k$ positive Lyapunov exponents, such that its entropy is larger than or equal to the sum of its $k$ smallest positive Lyapunov exponents.

In the present two-dimensional case the semi-algebraic tools used to bound the distortion and the local volume growth of $C^{\infty}$ curves are elementary. This is a challenging problem to adapt this technology in higher dimensions.

When the empirical measures from $x\in M$ are not converging, the point $x$ is said to have historic behaviour [39]. A set $U$ is contracting when the diameter of $f^{n}U$ goes to zero when $n\in \mathbb{N}$ goes to infinity. In a contracting set the empirical measures of all points have the same limit set, however they may not converge. P. Berger and S. Biebler have shown that $C^{\infty}$ densely inside the Newhouse domains [8] there are contracting domains with historic behaviour. In intermediate smoothness, such domains have been previously built in [27]. As a consequence of the Main Theorem, Lebesgue almost every point $x$ with historic behaviour satisfies $\chi (x)\leq 0$ for $C^{\infty}$ surface diffeomorphisms. We also show the following statement.

Theorem 2

Let $f$ be a $C^{\infty}$ diffeomorphism on a compact manifold (of any dimension). Then Lebesgue a.e. point $x$ in a contracting set satisfies $\chi (x)\leq 0$.

We explain now in few lines the main ideas to build an SRB measure under the assumptions of the Main Theorem. The geometric approach for uniformly hyperbolic systems (see e.g. [17]) consists in considering a weak limit of $\left (\frac{1}{n}\sum _{k=0}^{n-1}f_{*}^{k}\mathop{\mathrm{Leb}}_{D_{u}} \right )_{n}$, where $D_{u}$ is a local unstable disc and $\mathop{\mathrm{Leb}}_{D_{u}}$ denotes the normalized Lebesgue measure on $D_{u}$ induced by its inherited Riemannian structure as a submanifold of $M$. Here we take a smooth $C^{r}$ embedded curve $D$ such that

$$ \chi (x,v_{x}):=\limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}(v_{x})\|>b> \frac{R(f)}{r} $$

for $(x,v_{x})$ in the unit tangent space $T^{1}D$ of $D$ with $x$ in a subset $B$ of $D$ of positive $\mathop{\mathrm{Leb}}_{D}$-measure. For $x$ in $B$ we define a subset $E(x)$ of positive integers, called the geometric set, such that the following properties hold for any $n\in E(x)$:

• the geometry of $f^{n}D$ around $f^{n}x$ is bounded meaning that for some uniform $\epsilon >0$, the connected component $D_{n}^{\epsilon}(x)$ of $f^{n}D$ intersected with the ball at $f^{n}x$ of radius $\epsilon >0$ is a curve with bounded $s$-derivative for $s\leq r$,

• the distortion of $df^{-n}$ on the tangent space of $D_{n}^{\epsilon}(x)$ is controlled,

• for some $\tau >0$ we have $\frac{\|d_{x}f^{l}(v_{x})\|}{\|d_{x}f^{k}(v_{x})\|}\geq e^{(l-k)\tau}$ for any $l>k\in E(x)$.

We show that $E(x)$ has positive upper asymptotic density for $x$ in a subset $A$ of $B$ of positive $\mathop{\mathrm{Leb}}_{D}$-measure. Let $F:\mathbb{P}TM\circlearrowleft$ be the map induced by $f$ on the projective tangent bundle $\mathbb{P}TM$. We build an SRB measure by considering a weak limit $\mu$ of a sequence of the form $\left (\frac{1}{\sharp F_{n}}\sum _{k\in F_{n}}F_{*}^{k}\mu _{n} \right )_{n}$ such that:

• $(F_{n})_{n}$ is a Følner sequence, so that the weak limit $\mu$ will be invariant by $F$,

• for all $n$, the measure $\mu _{n}$ is the probability measure induced by $\mathop{\mathrm{Leb}}_{D}$ on $A_{n}\subset A$, the $\mathop{\mathrm{Leb}}_{D}$-measure of $A_{n}$ being not exponentially small,

• the sets $(F_{n})_{n}$ are in some sense filled with the geometric set $E(x)$ for $x\in A_{n}$. Then the measure $\mu$ on $\mathbb{P}TM$ will be supported on the unstable Oseledec’s bundle.

Finally we check with some Følner Gibbs property that the limit empirical measure $\mu$ projects to an SRB measure on $M$ by using the Ledrappier-Young entropic characterization.

The paper is organized as follows. In Sect. 2 we recall for general sequences of integers the notion of asymptotical density and we build for any sequence $E$ with positive upper density a Følner set $F$ filled with $E$. Then we use a Borel-Cantelli argument to define our sets $(A_{n})_{n}$ and the Følner sequence $(F_{n})_{n}$. In Sect. 3, we study the maximal Lyapunov exponent and the entropy of the generalized empirical measure $\mu$ assuming some Gibbs property. We introduce the geometric set in Sect. 4 by using the Reparametrization Lemma of [12]. We build then SRB measures in Sect. 5 by using the abstract formalism of Sect. 2 and 3. Then we prove the covering property of the basins in Sect. 6 by the standard argument of absolute continuity of Pesin stable foliation. The last section is devoted to the proof of Theorem 2.

Comment: In a first version of this work, by following [12] (incorrectly) the author claimed that, at $b$-hyperbolic times $n$ of the sequence $\left (\|d_{x}f^{k}(v_{x})\|\right )_{k}$ for some $b>0$, the geometry of $f^{n}D$ at $f^{n}x$ was bounded. J. Buzzi, S. Crovisier and O. Sarig gave then in [16] another proof of Corollary 2 by using their analysis of the entropic continuity of Lyapunov exponents from [15]. But our claim on the geometry at hyperbolic times is wrong in general and we manage to show it only when $\chi (x)>\frac{R(f)}{2}$. Here we correct our proof based on the Reparametrization Lemma of [12] by showing directly that the set of times with bounded geometry has positive upper asymptotic density on a set of positive $\mathop{\mathrm{Leb}}_{D}$-measure.

2 Some asymptotic properties of integers

2.1 Asymptotic density

We first introduce some notations. The set of positive integers is denoted by $\mathbb{N}^{*}$. In the following we let $\mathcal {P}_{\mathbb{N}}$ and $\mathcal {P}_{n}$ be respectively the power sets of ℕ and $\{0, 1,2,\ldots ,n-1\}$, $n\in \mathbb{N}$. For $a\leq b\in \mathbb{N}$ we write $[\!\![a,b ]\!\!]$ (resp. $[\!\![a,b [\!\![$, $]\!\!]a,b]\!\!]$) the interval of integers $k$ with $a\leq k\leq b$ (resp $a\leq k< b$, $a< k\leq b$). The connected components of $E\in \mathcal {P}_{\mathbb{N}}$ are the maximal intervals of integers contained in $E$. An interval of integers $[\!\![a, b[\!\![$ is said to be $E$-irreducible when we have $a, b\in E$ and $[\!\![a,b [\!\![\cap E=\{a\}$. The boundary $\partial E$ of $E$ is the symmetric difference of $E$ and $E+1$ with $E+1:=\{k+1, \ k\in E\}$. In particular $\partial \,[\!\![a, b[\!\![= \{a,b\}$. Observe that $\partial E$ completely determines $E$. For $M\in \mathbb{N}^{*}$, we denote by $E_{M}$ the union of the intervals $[\!\![a,b [\!\![$ with $a,b\in E$ and $0< b-a\leq M$. We let $\mathfrak {N}$ be the set of increasing sequences of natural integers, which may be identified with the subset of $\mathcal {P}_{\mathbb{N}}$ given by infinite subsets of ℕ. For $\mathfrak {n}\in \mathfrak {N}$ we define the generalized power set of $\mathfrak {n}$ as $\mathcal {Q}_{\mathfrak {n}}:=\prod _{n\in \mathfrak {n}} \mathcal {P}_{n}$.

We recall now the classical notion of upper and lower asymptotic densities. For $n\in \mathbb{N}^{*}$ and $F\in \mathcal {P}_{\mathbb{N}}$ we let $d_{n}(F)$ be the frequency of $F$ in $[\!\![0,n [\!\![$:

$$ d_{n}(F_{n})=\frac{\sharp F\cap [\!\![0 ,n[\!\![}{n}. $$

The upper and lower asymptotic densities $\overline{d}(E)$ and $\underline{d}(E)$ of $E\in \mathcal {P}_{\mathbb{N}}$ are respectively defined by

$$ \overline{d}(E):=\limsup _{n\in \mathbb{N}}d_{n}(E ) \text{ and} $$

$$ \underline{d}(E):=\liminf _{n\in \mathbb{N}}d_{n}(E ). $$

We just write $d(E)$ for the limit, when the frequencies $d_{n}(E)$ are converging. For any $\mathfrak {n}\in \mathfrak {N}$ we let similarly $\overline{d}^{\mathfrak {n}}(E):=\limsup _{n\in \mathfrak {n}}d_{n}(E )$ and $\underline{d}^{\mathfrak {n}}(E):=\liminf _{n\in \mathfrak {n}}d_{n}(E )$. The concept of upper and lower asymptotic densities of $E\in \mathcal {P}_{\mathbb{N}}$ may be extended to generalized power sets as follows. For $\mathfrak {n}\in \mathfrak {N}$ and $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}\in \mathcal {Q}_{\mathfrak {n}}$ we let

$$ \overline{d}^{\mathfrak {n}}(\mathcal {F}):=\limsup _{n\in \mathfrak {n}}d_{n}(F_{n}) \text{ and } $$

$$ \underline{d}^{\mathfrak {n}}(\mathcal {F}):=\liminf _{n\in \mathfrak {n}}d_{n}(F_{n}). $$

Again we just write $d^{\mathfrak {n}}(E)$ and $d^{\mathfrak {n}}(\mathcal {F})$ when the corresponding frequencies are converging.

2.2 Følner sequence and density along subsequences

We say that $E\in \mathcal {P}_{\mathbb{N}}$ is Følner along $\mathfrak {n}\in \mathfrak {N}$ when its boundary $\partial E$ has zero upper asymptotic density with respect to $\mathfrak {n}$, i.e. $d^{\mathfrak {n}}(\partial E)=0$. More generally $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}\in \mathcal {Q}_{\mathfrak {n}}$ with $\mathfrak {n}\in \mathfrak {N}$ is Følner when we have $d^{\mathfrak {n}}(\partial \mathcal {F})=0$ with $\partial \mathcal {F}=(\partial F_{n} )_{n\in \mathfrak {n}}$. In general this property seems to be weaker than the usual Følner property, which requires $\lim _{n\in \mathfrak {n}}\frac{\sharp \partial F_{n}}{\sharp F_{n}}=0$. But in the following we will work with sequences ℱ with $\underline{d}^{\mathfrak {n}}(\mathcal {F})>0$. In this case our definition coincides with the standard one.

Let $E,F\in \mathcal {P}_{\mathbb{N}}$ and $\mathfrak {n}\in \mathfrak {N}$. We say that $F$ is $\mathfrak {n}$-filled with $E$ or $E$ is dense in $F$ along $\mathfrak {n}$ when we have

$$ d^{\mathfrak {n}}(F\setminus E_{M})\xrightarrow{M\rightarrow +\infty}0. $$

Observe that $\left (\overline{d}(E_{M})\right )_{M}$ is converging nondecreasingly to some $a\geq \overline{d}(E)$ when $M$ goes to infinity. The limit $a$ is in general strictly less than 1. For example if $E:=\bigcup _{n} [\!\![2^{2n},2^{2n+1}]\!\!]$ one easily computes $\overline{d}(E_{M})=\overline{d}(E)=2/3$ for all $M$. In this case, the set $E$ is moreover a Følner set.

Also $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}\in \mathcal {Q}_{\mathfrak {n}}$ is said to be filled with $E$ when we have with $\mathcal {F}\setminus E_{M}:=(F_{n}\setminus E_{M})_{n\in \mathfrak {n}}$:

$$ d^{\mathfrak {n}}(\mathcal {F}\setminus E_{M}) \xrightarrow{M\rightarrow +\infty}0. $$

2.3 Følner set $F$ filled with a given $E$ with $\overline{d}(E)>0$

Given a set $E$ with positive upper asymptotic density we build a Følner set $F$ filled with $E$ by using a diagonal argument. More precisely we will build $F$ by filling the holes in $E$ of larger and larger size when going to infinity.

Lemma 1

For any $E\in \mathcal {P}_{\mathbb{N}}$ with $\overline{d}(E)>0$ there is a subsequence $\mathfrak {m}\in \mathfrak {N}$ and $F\in \mathcal {P}_{\mathbb{N}}$ with $\partial F\subset E$ such that

• $d^{\mathfrak{m}}(F)\geq d^{\mathfrak {m}}(E\cap F )= \overline{d}(E)$;

• $F$ is Følner along $\mathfrak {m}$;

• $E$ is dense in $F$ along $\mathfrak {m}$.

Proof

By using a Cantor diagonal argument there is a subsequence $\mathfrak {N}\ni \mathfrak {p}\subset E$ with $d^{\mathfrak{p}}(E)=\overline{d}(E)$ such that the limits $\Delta _{k}:=d^{\mathfrak{p}}(E_{k})=\lim _{p\in \mathfrak {p}}d_{p}(E_{k})$ exist for all $k\in \mathbb{N}^{*}$. The sequence $(\Delta _{k})_{k}$ is nondecreasing and bounded from above by 1. We let $\Delta _{\infty}=\lim _{k\rightarrow +\infty}\Delta _{k}$. For $k\in \mathbb{N}^{*}$, we take $\mathfrak {m}_{k}\in \mathfrak {p}$, such that

$$\begin{aligned} \forall \mathfrak {m}_{k}\leq p\in \mathfrak {p}, \ \left | d_{p}(E_{k})- \Delta _{k}\right |< \frac{1}{2^{k}}. \end{aligned}$$

(2.1)

One can ensure that the sequence $(\mathfrak {m}_{k})_{k\in \mathbb{N}^{*}}$ is increasing. We put

$$ \mathfrak {m}=(\mathfrak {m}_{k})_{k} \text{ and }F=\bigcup _{k} [\!\![\mathfrak {m}_{k}, \mathfrak {m}_{k+1}[\!\![\cap E_{k}. $$

Clearly we have $\partial F\subset \bigcup _{k} (\partial [\!\![\mathfrak {m}_{k}, \mathfrak {m}_{k+1}[\!\![)\cup (\partial E_{k})\subset E$.

Any two integers $l< l'$ lying both in $E\setminus E_{k}$ satisfy $l'-l> k$, therefore $\overline{d}(E\setminus E_{k})\leq 1/k$ for all $k\in \mathbb{N}^{*}$. Since $F$ contains $[\!\![\mathfrak {m}_{k},+\infty [\!\![\cap E_{k}$ for all $k$, we have then:

$$\begin{aligned} \forall k\in \mathbb{N}^{*}, \underline{d}^{\mathfrak {m}}(E\cap F)& \geq \underline{d}^{\mathfrak {m}}(E\cap E_{k}), \\ &\geq d^{\mathfrak {p}}(E)-\overline{d}(E\setminus E_{k}), \\ &\geq \overline{d}(E)-1/k. \end{aligned}$$

By letting $k$ go to infinity we get $\underline{d}^{\mathfrak {m}}(E\cap F)\geq \overline{d}(E)\geq \overline{d}^{\mathfrak {m}}(E\cap F)$, thus $d^{\mathfrak {m}}(E\cap F)= \overline{d}(E)$.

Let us prove now the Følner property of the set $F$. For $\mathfrak {m}_{k}< K\in \partial F\subset E$ either $K$ or $K-1$ does not belong to $F$. But $K-1\geq \mathfrak {m}_{k}$ and $F\supset [\!\![\mathfrak {m}_{k}, +\infty [\!\![\cap E_{k}$. Consequently

• either $K$ does not belong to $E_{k}$, then $[\!\![K, K+k [\!\![\subset \mathbb{N}\setminus E\subset \mathbb{N}\setminus \partial F$,

• or $K-1$ does not belong to $E_{k}$, then $[\!\![K-k, K [\!\![\subset \mathbb{N}\setminus E\subset \mathbb{N}\setminus \partial F$.

Therefore $\overline{d}^{\mathfrak {m}}(\partial F)\leq \frac{2}{k}$. As it holds for all $k$, the set $F$ is Følner along $\mathfrak {m}$.

Finally we have $[\!\![0,\mathfrak {m}_{k}[\!\![\cap F\subset [\!\![0, \mathfrak {m}_{k}[\!\![\cap E_{k}$, therefore for any $M< k$ we get:

By taking the limits when $k$ goes to infinity we get

$$\begin{aligned} \overline{d}^{\mathfrak {m}}(F\setminus E_{M})\leq \Delta _{\infty}- \Delta _{M}+\frac{1}{2^{M}}, \end{aligned}$$

therefore

$$ \overline{d}^{\mathfrak {m}}(F\setminus E_{M}) \xrightarrow{M\rightarrow +\infty}0. $$

By using once more a Cantor diagonal argument, we may finally assume the limits $d^{\mathfrak {m}}(F)$ and $d^{\mathfrak {m}}(F\setminus E_{M})$, $M\in \mathbb{N}^{*}$, exist. □

2.4 Borel-Cantelli argument

Let $(X, \mathcal {A}, \lambda )$ be a measure space with $\lambda$ being a finite measure. A map $E:X\rightarrow \mathcal {P}_{\mathbb{N}}$ is said to be measurable, when for all $n\in \mathbb{N}$ the set $\{x, \ n\in E(x)\}$ belongs to $\mathcal {A}$ (equivalently writing $E$ as an increasing sequence $\mathfrak {n}= (\mathfrak {n}_{i})_{i\in \mathbb{N}}$ the integers valued functions $\mathfrak {n}_{i}$ are measurable). For such measurable maps $E$ and $\mathfrak {n}$, the upper asymptotic density $\overline{d}^{\mathfrak {n}}(E)$ defines a measurable function.

Lemma 2

Assume $E$ is a measurable sequence of integers such that $\overline{d}(E(x))>\beta >0$ for $x$ in a measurable set $A$ of a positive $\lambda$-measure. Then there exist $\mathfrak {n}\in \mathfrak {N}$, measurable subsets $(A_{n})_{n\in \mathfrak {n}}$ of $X$ and $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}\in \mathcal {Q}_{\mathfrak {n}}$ with $\partial F_{n}\subset E(x)$ for all $x\in A_{n}$, $n\in \mathfrak {n}$ such that:

• $\lambda (A_{n})\geq \frac{e^{-n\delta _{n}}}{n^{2}}$ for all $n\in \mathfrak {n}$ with $\delta _{n}\xrightarrow{\mathfrak {n} \ni n\rightarrow +\infty}0$;

• ℱ is a Følner sequence;

• $E$ is dense in ℱ uniformly on $A_{n}$, i.e.

  $$ \limsup _{n\in \mathfrak {n}}\sup _{x\in A_{n}}d_{n}\left ( F_{n} \setminus E_{M}(x)\right ) \xrightarrow{M\rightarrow +\infty}0. $$
• $$\begin{aligned} \underline{d}^{\mathfrak {n}}(\mathcal {F})\geq \liminf _{n\in \mathfrak {n}} \inf _{x\in A_{n}}d_{n}\left (E(x)\cap F_{n}\right )& \geq \beta . \end{aligned}$$

Proof

The sequences $\mathfrak {m}=(\mathfrak {m}_{k})_{k\in \mathbb{N}}$ and $F$ built in the previous lemma define measurable sequences on $A$. By taking a smaller subset $A$ we may assume that $\mathfrak {m}_{k}(x)$ is bounded on $A$ for all $k$ and that the following sequences of functions are converging uniformly in $x\in A$ by using Egorov’s theorem:

• (1) $d_{\mathfrak {m}_{k}(x)}(\partial F(x))\xrightarrow{k}0$,

• (2) $d^{\mathfrak {m}(x)}(F(x)\setminus E_{M}(x)) \xrightarrow{M\rightarrow +\infty}0$,

• (3) $d_{\mathfrak {m}_{k}(x)}(F(x)\setminus E_{M}(x)) \xrightarrow{k\rightarrow +\infty} d^{\mathfrak {m}}\left (F(x) \setminus E_{M}(x)\right )$ for any $M\in \mathbb{N}^{*}$,

• (4) $d_{\mathfrak {m}_{k}(x)}(E(x)\cap F(x))\xrightarrow{k}d^{\mathfrak {m}(x)}(E(x) \cap F(x))\geq \beta$.

For a uniformly converging sequence of real bounded functions $f=\lim _{n}f_{n}$ we have $\sup f=\lim _{n}\sup f_{n}$ and $\inf f=\lim _{n} \inf f_{n}$. Applying this fact to the above third item we get:

$$\begin{aligned} \sup _{x\in A}d_{\mathfrak {m}_{k}(x)}(F(x)\setminus E_{M}(x)& ) \xrightarrow{k\rightarrow +\infty} \sup _{x\in A} d^{\mathfrak {m}(x)} \left (F(x)\setminus E_{M}(x)\right ). \end{aligned}$$

Together with the second item, we get:

$$ \lim _{k} \sup _{x\in A}d_{\mathfrak {m}_{k}(x)}(F(x)\setminus E_{M}(x)) \xrightarrow{M}0. $$

For the last item we obtain by taking the infimum:

$$ \lim _{k}\inf _{x\in A} d_{\mathfrak {m}_{k}(x)}(E(x)\cap F(x))=\inf _{x \in A} d^{\mathfrak {m}(x)}(E(x)\cap F(x))\geq \beta . $$

By Borel-Cantelli Lemma, the subset $A_{n}:=\{x\in A, \ n\in \mathfrak {m}(x)\}$ has $\lambda$-measure larger than $1/n^{2}$ for infinitely many $n\in \mathbb{N}$. We let $\mathfrak {n}$ be this infinite subset of integers. We observe firstly that:

$$\begin{aligned} \alpha _{n}:=\sup _{x\in A_{n}}d_{n}(\partial F(x)) \xrightarrow{n\in \mathfrak {n}}0. \end{aligned}$$

(2.2)

For any $n$ we let $k_{n}$ be the largest integer $k$ satisfying $\sup _{x\in A}\mathfrak {m}_{k}(x)< n$. As the functions $(\mathfrak {m}_{k})_{k}$ are bounded on $A$, we get $k_{n}\xrightarrow{n\rightarrow +\infty}+\infty$. Then, by item (1), for any given $\epsilon >0$ we may find $k'_{\epsilon}\in \mathbb{N}$ such that $\sup _{x\in A}d_{\mathfrak {m}_{k}(x)}(\partial F(x))<\epsilon$ for any $k\geq k'_{\epsilon}$. Therefore for any $n$ so large that $k_{n}>k'_{\epsilon}$ and for any $x\in A_{n}\subset A$ we have $n=\mathfrak {m}_{l}(x)$ for some $l=l(x)>k_{n}$, therefore $d_{n}(\partial F(x))<\epsilon$. This shows (2.2). By a similar argument, we can show that:

$$\begin{aligned} \limsup _{n\in \mathfrak {n} } \sup _{x\in A_{n}}d_{n}(F(x)\setminus E_{M}(x)) \leq \lim _{k} \sup _{x\in A}d_{\mathfrak {m}_{k}(x)}(F(x)\setminus E_{M}(x)) \xrightarrow{M}0 \end{aligned}$$

and

$$\begin{aligned} \liminf _{n\in \mathfrak {n}} \inf _{x\in A_{n}}d_{n}\left (E(x)\cap F(x) \right )\geq \lim _{k} \inf _{x\in A}d_{\mathfrak {m}_{k}(x)}(E(x)\cap F(x)) \geq \beta . \end{aligned}$$

As $\partial F(x)\cap [\!\![0,n[\!\![$ determines $F(x)\cap [\!\![0,n[\!\![$, there are at most $\sum _{k=1}^{[n\alpha _{n}]}{n \choose k}$ choices for $F(x)\cap [\!\![0,n[\!\![$, $x\in A_{n}$, and thus it may be fixed by dividing the measure of $A_{n}$ by $\sum _{k=1}^{[n\alpha _{n}]}{n \choose k}=e^{n\delta _{n}}$ for some $\delta _{n}\xrightarrow{n}0$. We let $F_{n}$ be the common value of $F(x)\cap [\!\![0,n[\!\![$ for $x$ in this new set $A_{n}$ for any $n\in \mathfrak {n}$. Then the conclusions of the lemma hold with $(A_{n})_{n\in \mathfrak {n}}$ and $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}$. □

3 Empirical measures associated to Følner sequences

Let $(X,T)$ be a topological system, i.e. $X$ is a compact metrizable space and $T:X\circlearrowleft$ is continuous. We denote by $\mathcal {M}(X)$ the set of Borel probability measures on $X$ endowed with the weak-∗ topology and by $\mathcal {M}(X,T)$ the compact subset of invariant measures. We will write $\delta _{x}$ for the Dirac measure at $x\in X$. We let $T_{*}$ be the induced (continuous) action on $\mathcal {M}(X)$, where for $\mu \in \mathcal {M}(X)$ the measure $T_{*}\mu$ is defined as $T_{*}\mu (A)=\mu (T^{-1}A)$ for any Borel set $A$. For $\mu \in \mathcal {M} (X)$ and a finite subset $F$ of ℕ, we let $\mu ^{F}$ be the empirical measure $\mu ^{F}:=\frac{1}{\sharp F }\sum _{k\in F}T_{*}^{k}\mu$.

3.1 Invariant measures

The following lemma is standard, but we give a proof for the sake of completeness. We fix $\mathfrak {n}\in \mathfrak {N}$ and $\mathcal {F}=(F_{n})_{n\in \mathfrak {N}}\in \mathcal {Q}_{\mathfrak {n}}$.

Lemma 3

Assume ℱ is a Følner sequence and $\underline{d}^{\mathfrak {n}}(\mathcal {F})>0$. Let $(\mu _{n})_{n\in \mathfrak {n}}$ be a family in $\mathcal {M}(X)$ indexed by $\mathfrak {n}$. Then any limit of $\left (\mu _{n}^{F_{n}}\right )_{n\in \mathfrak {n}}$ is a $T$-invariant Borel probability measure.

Proof

Let $\mathfrak {n}'$ be a subsequence of $\mathfrak {n}$ such that $\left (\mu _{n}^{F_{n}}\right )_{n\in \mathfrak {n}'}$ is converging to some $\mu '$. It is enough to check that $\left |\int \phi \, d\mu _{n}^{F_{n}}-\int \phi \circ T\, d\mu _{n}^{ F_{n}}\right |$ goes to zero when $\mathfrak {n}'\ni n\rightarrow +\infty$ for any $\phi :X\rightarrow \mathbb{R}$ continuous.

This follows from the following inequalities:

$$\begin{aligned} \int \phi \, d\mu _{n}^{F_{n}}-\int \phi \circ T\, d\mu _{n}^{ F_{n}}&= \frac{1}{\sharp F_{n}} \int \left (\sum _{ \substack{k\in F_{n} \\ k-1\notin F_{n}}}\phi \circ T^{k}- \sum _{ \substack{k\notin F_{n} \\ k-1\in F_{n}}} \phi \circ T^{k}\right ) \, d \mu _{n}, \end{aligned}$$

$$\begin{aligned} \left |\int \phi \, d\mu _{n}^{F_{n}}-\int \phi \circ T\, d\mu _{n}^{F_{n}} \right | & \leq \sup _{x\in X}|\phi (x)| \frac{\sharp \partial F_{n}}{\sharp F_{n}}, \\ \limsup _{n\in \mathfrak {n}} \left |\int \phi \, d\mu _{n}^{F_{n}}- \int \phi \circ T\, d\mu _{n}^{F_{n}}\right | &\leq \sup _{x\in X}| \phi (x)| \limsup _{n\in \mathfrak {n}} \frac{\sharp \partial F_{n}}{\sharp F_{n}}, \\ &\leq \sup _{x\in X}|\phi (x)| \frac{d^{\mathfrak {n}}(\partial \mathcal {F}) }{ \underline{d}^{\mathfrak {n}}(\mathcal {F})}= 0. \end{aligned}$$

 □

3.2 Positive exponent of empirical measures for superadditive cocycles

We fix a general continuous superadditive cocycle $\Phi =(\phi _{n})_{n\in \mathbb{N}}$ with respect to $(X,T)$, i.e. $\phi _{0}=0$, $\phi _{n}:X\rightarrow \mathbb{R}$ is a continuous function for all $n$ and $\phi _{n+m}\geq \phi _{n}+\phi _{m}\circ T^{n}$ for all $m$, $n$. By the subadditive ergodic theorem [26], the limit $\phi _{*}(x)=\lim _{n}\frac{\phi _{n}(x)}{n}$ exists for $x$ in a set of full measure with respect to any invariant measure $\mu$. In the proof of the main theorem we will only consider additive cocycles, but we think it could be interesting to consider general superadditive cocycles in other contexts.

Let $E:Y\rightarrow \mathcal {P}_{\mathbb{N}}$ be a measurable sequence of integers defined on a Borel subset $Y$ of $X$. The set valued map $E$ is said to be$a$-large with respect to $\Phi$ for some $a\in \mathbb{R}$ when for any $x\in Y$ we have $\phi _{q-p}(T^{p}x)\geq (q-p)a$ for all integers $q> p$ in $E(x)$. For a finite subset of integers $J$ we let $\phi _{J}(x)$ be the sum of $\phi _{q-p}(T^{p}x)$ where $[\!\![p, q[\!\![$ runs over all connected components of $J$. By superadditivity of $\Phi$ we always have $\phi _{J}(x)\geq \phi _{1}(x)\sharp J$.

Lemma 4

Let $(A_{n})_{n\in \mathfrak {n}}$ and $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}$ as in Lemma 2and let $(\mu _{n})_{n\in \mathfrak {n}}$ be a family in $\mathcal {M}(X)$ indexed by $\mathfrak {n}$ with $\mu _{n}(A_{n})=1$ for all $n\in \mathfrak {n}$. Assume $E$ is $a$-large with $a\in \mathbb{R}$. Then for any weak-∗ limit $\mu$ of $\mu _{n}^{F_{n}}=\frac{1}{\sharp F_{n}}\sum _{k\in F_{n}}T_{*}^{k} \mu _{n}$, $n\in \mathfrak {n}$, we have

$$ \phi _{*}(x)\geq a\textit{ for } \mu \textit{ a.e. }x. $$

Proof

Without loss of generality we can take $a=0$ by considering the cocycle $(\phi _{n}-na)_{n}$, which is again superadditive. By taking a subsequence we can also assume that $\left (\mu _{n}^{F_{n}}\right )_{n\in \mathfrak {n}}$ is converging to $\mu$. Fix $\alpha <0$ and $M< N\in \mathbb{N}^{*}$. For $x\in A_{n}$, $n\in \mathfrak {n}$, we let

$$ F_{n}^{\alpha}(x):=\left \{k\in F_{n}, \ \frac{\phi _{N}(T^{k}x)}{N} \leq \alpha \right \}. $$

For $l\in \mathbb{N}$ and $k\in [\!\![0,N[\!\![$ with $k+lN\in F_{n}$, the interval of integers $J_{k,l}=[\!\![k+lN,k+(l+1)N[\!\![$ may be written as

$$ J_{k,l}=I_{k,l}^{1}\coprod I_{k,l}^{2}\coprod I_{k,l}^{3}\coprod I_{k,l}^{4} $$

where

• $I_{k,l}^{1}$ is the union of disjoint $E(x)$-irreducible intervals of length less than $M$ contained in $J_{k,l}$. As $E(x)$ is 0-large, we have $\phi _{I_{k,l}^{1}}(x)\geq 0$ by superadditivity of $\Phi$,

• $I_{k,l}^{2}$ consists of a subinterval of an $E(x)$-irreducible interval of length less than $M$, which does not lie entirely in $J_{k,l}$ but may contain the right extreme of $J_{k,l}$. In particular $\sharp I_{k,l}^{2}\leq M$, therefore $\phi _{I_{k,l}^{2}}(x)\geq -M\sup _{y\in X}|\phi _{1}(y)|$,

• $I_{k,l}^{3}\subset F_{n}\setminus E_{M}(x)$,

• $I_{k,l}^{4} \subset \mathbb{N}\setminus F_{n}$.

Note that the subsets $(I_{k,l}^{j})_{j}$ depend on $x$, but we do not make this dependence explicit to simplify the notations. If $I_{k,l}^{4}$ is non empty then $J_{k,l}$ contains an element of $\partial F_{n}$, therefore for a fixed $k$ we get from $\sharp \left ( \bigcup _{l,\ I_{k,l}^{4}\neq \emptyset }J_{k,l} \right )\leq N\sharp \partial F_{n}$:

$$\begin{aligned} \sum _{ \substack{l, \ k+lN\in F_{n}^{\alpha }\\ I_{k,l}^{4}\neq \emptyset}} \phi _{J_{k,l}}(x)\geq -N\sharp \partial F_{n}\sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

(3.1)

Then, if $I_{k,l}^{4}= \emptyset$ we have by superadditivity of $\Phi$:

$$\begin{aligned} \phi _{J_{k,l}}(x)&\geq \phi _{I_{k,l}^{1}}(x) +\sum _{j=2,3}\phi _{I_{k,l}^{j}}(x), \\ &\geq -(\sharp I_{k,l}^{3}+M)\sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

As $F_{n}$ is a subset of $[\!\![0,n[\!\![$, the cardinality of $\{l, \ k+lN\in F_{n}\}$ is less than or equal to $\lceil \frac{n}{N}\rceil$. Therefore by summing this last inequality over $l$ we obtain

$$\begin{aligned} \sum _{ \substack{l, \ k+lN\in F_{n}^{\alpha }\\ I_{k,l}^{4}= \emptyset}} \phi _{J_{k,l}}(x)\geq -\left (\sharp (F_{n}\setminus E_{M}(x))+M \lceil \frac{n}{N}\rceil \right )\sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

(3.2)

By combining (3.1) and (3.2) we have

$$\begin{aligned} \sum _{\substack{l, \ k+lN\in F_{n}^{\alpha }}} \phi _{J_{k,l}}(x)& \geq -\left (N\sharp \partial F_{n}+\sharp (F_{n}\setminus E_{M}(x))+M \lceil \frac{n}{N}\rceil \right )\sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

After summing over $k\in [\!\![0,N[\!\![$ and dividing by $N\sharp F_{n}$ we obtain

$$\begin{aligned} \int _{ \{\frac{\phi _{N}}{N}\leq \alpha \}}\frac{ \phi _{N}}{N}\, d \delta _{x}^{F_{n}}&=\frac{1}{N\sharp F_{n}} \sum _{ \substack{k,l, \ k+lN\in F_{n}^{\alpha }}} \phi _{J_{k,l}}(x), \\ &\geq -\frac{1}{\sharp F_{n}}\left (N\sharp \partial F_{n}+\sharp (F_{n} \setminus E_{M}(x))+M\lceil \frac{n}{N}\rceil \right )\sup _{y\in X}| \phi _{1}(y)|. \end{aligned}$$

We integrate then with respect to $\mu _{n}$ (recall that $A_{n}$ has full $\mu _{n}$-measure):

$$\begin{aligned} &\int _{\{\frac{\phi _{N}}{N}\leq \alpha \}} \frac{ \phi _{N}}{N} \, d \mu _{n}^{F_{n}}\\ &\quad \geq -\frac{1}{\sharp F_{n}}\left (N\sharp \partial F_{n}+\sup _{x\in A_{n}}\sharp (F_{n}\setminus E_{M}(x))+M \lceil \frac{n}{N}\rceil \right )\sup _{y\in X}|\phi _{1}(y)|, \\ &\quad \geq -\frac{1}{d_{n}( F_{n})}\left (Nd_{n}( \partial F_{n})+\sup _{x \in A_{n}}d_{n} (F_{n}\setminus E_{M}(x))+\frac{M}{n}\lceil \frac{n}{N}\rceil \right )\sup _{y\in X}|\phi _{1}(y)|, \end{aligned}$$

Since the sequence $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}$ is Følner and $\underline{d}^{\mathfrak{n}}(\mathcal {F})\geq \beta$, we get by taking the limsup when $n\in \mathfrak {n}$ goes to infinity:

$$\begin{aligned} \limsup _{n\in \mathfrak {n}}\int _{\{\frac{\phi _{N}}{N}\leq \alpha \}} \frac{ \phi _{N}}{N} \, d\mu _{n}^{F_{n}}&\geq -\frac{1}{\beta} \limsup _{n\in \mathfrak {n}}\left ( \sup _{x\in A_{n}}d_{n}(F_{n} \setminus E_{M}(x))+\frac{M}{N} \right )\sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

Then, the set $\{\frac{\phi _{N}}{N}\leq \alpha \}$ being closed, we get:

$$\begin{aligned} \int _{\{\frac{\phi _{N}}{N}\leq \alpha \}} \frac{ \phi _{N}}{N}\, d \mu &\geq \limsup _{n\in \mathfrak {n}}\int _{\{\frac{\phi _{N}}{N} \leq \alpha \}} \frac{ \phi _{N}}{N} \, d\mu _{n}^{F_{n}}. \end{aligned}$$

On the other hand we have:

$$\begin{aligned} \alpha \mu \left (\left \{\frac{\phi _{N}}{N}\leq \alpha \right \} \right )\geq \int _{\{\frac{\phi _{N}}{N}\leq \alpha \}} \frac{ \phi _{N}}{N} \, d\mu , \end{aligned}$$

therefore we get for all $M$:

$$\begin{aligned} \lim _{N\rightarrow +\infty} \mu \left (\left \{\frac{\phi _{N}}{N} \leq \alpha \right \}\right )\leq \frac{1}{-\alpha \beta}\limsup _{n \in \mathfrak {n}} \sup _{x\in A_{n}}d_{n} (F_{n}\setminus E_{M}(x)) \sup _{y\in X}|\phi _{1}(y)|. \end{aligned}$$

By taking the limit in $M$ we finally have $\lim _{N\rightarrow +\infty} \mu \left (\left \{\frac{\phi _{N}}{N} \leq \alpha \right \}\right )=0$ by the third item of Lemma 2, therefore $\phi _{*}\geq \alpha$ almost everywhere. As it holds for any $\alpha <0$, we conclude that $\phi _{*}=\lim _{n}\frac{\phi _{n}}{n}\geq 0$ almost everywhere. □

3.3 Entropy of empirical measures

Following Misiurewicz’s proof of the variational principle, we estimate the entropy of empirical measures from below. For a finite partition $P$ of $X$ and a finite subset $F$ of ℕ, we let $P^{F}$ be the iterated partition $P^{F}=\bigvee _{k\in F}f^{-k}P$. When $F=[\!\![0,n[\!\![$, $n\in \mathbb{N}$, we just let $P^{F}=P^{n}$. We denote by $P(x)$ the element of $P$ containing $x\in X$.

For a Borel probability measure $\mu$ on $X$, the static entropy $H_{\mu}(P)$ of $\mu$ with respect to a (finite measurable) partition $P$ is defined as follows:

$$\begin{aligned} H_{\mu}(P)&=-\sum _{A\in P}\mu (A)\log \mu (A), \\ &=-\int \log \mu \left (P(x)\right )\, d\mu (x). \end{aligned}$$

When $\mu$ is $T$-invariant, we recall that the measure theoretical entropy of $\mu$ with respect to $P$ is then

$$ h_{\mu}(P)=\lim _{n}\frac{1}{n}H_{\mu}(P^{n}) $$

and the entropy $h(\mu )$ of $\mu$ is

$$ h(\mu )=\sup _{P}h_{\mu}(P). $$

We will use the two following standard properties of the static entropy [20]:

• for a fixed partition $P$, the map $\mu \mapsto H_{\mu}(P)$ is concave on $\mathcal {M}(X)$,

• for two partitions $P$ and $Q$, the joined partition $P\vee Q$ satisfies

  $$\begin{aligned} H_{\mu}(P\vee Q)&\leq H_{\mu}(P)+H_{\mu}(Q). \end{aligned}$$

  (3.3)

Lemma 5

Let $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}$ be a Følner sequence with $\underline{d}^{\mathfrak {n}}(\mathcal {F})>0$. For any measurable finite partition $P$ and $m\in \mathbb{N}^{*}$, there exist a sequence $(\epsilon _{n})_{n\in \mathfrak {n}}$ converging to 0 such that

$$ \forall n\in \mathfrak {n}, \ \frac{1}{m}H_{\mu _{n}^{F_{n}}}(P^{m}) \geq \frac{1}{\sharp F_{n}}H_{\mu _{n}}(P^{F_{n}})-\epsilon _{n}. $$

Proof

When $F_{n}$ is an interval of integers, we have [32]:

$$ \frac{1}{m}H_{\mu _{n}^{F_{n}}}(P^{m})\geq \frac{1}{\sharp F_{n}}H_{ \mu _{n}}(P^{F_{n}})-\frac{3m\log \sharp P}{\sharp F_{n}}. $$

(3.4)

Consider a general set $F_{n}\in \mathcal {P}_{n}$. We decompose $F_{n}$ into connected components $F_{n}=\coprod _{k=1,\ldots , K } F_{n}^{k}$. Observe $K\leq \sharp \partial F_{n}$. Then we get:

$$\begin{aligned} \frac{1}{m}H_{\mu _{n}^{F_{n}}}(P^{m})&\geq \sum _{k=1}^{K} \frac{\sharp F_{n}^{k}}{m\sharp F_{n}}H_{\mu _{n}^{F^{k}_{n}}}(P^{m}), \text{by concavity of $\mu \mapsto H_{\mu}(P^{m})$,} \\ &\geq \frac{1}{\sharp F_{n}}\sum _{k=1}^{K}H_{\mu _{n}}(P^{F_{n}^{k}})- \frac{3mK\log \sharp P}{\sharp F_{n}}, \text{ by applying (3.4) to each $F_{n}^{k}$}, \\ &\geq \frac{1}{\sharp F_{n}}H_{\mu _{n}}(P^{F_{n}})-3m\log \sharp P \frac{\sharp \partial F_{n}}{\sharp F_{n}}, \text{ according to (3.3)}. \end{aligned}$$

This concludes the proof with $\epsilon _{n}=3m\frac{\sharp \partial F_{n}}{\sharp F_{n}} \log \sharp P$, because ℱ is a Følner sequence with $\underline{d}^{\mathfrak {n}}(\mathcal {F})>0$. □

With the notations of Lemma 2 we let $\mu _{n}$ be the probability measure induced by $\lambda$ on $A_{n}$, i.e. $\mu _{n}=\frac{\lambda (A_{n}\cap \cdot )}{\lambda (A_{n})}$. In the following we consider an additive cocycle $\Psi =(\psi _{n})_{n}$ associated to a continuous function $\psi :X\rightarrow \mathbb{R}$, i.e. $\psi _{0}=0$ and $\psi _{n}=\sum _{k=0}^{n-1}\psi \circ T^{k}$ for a positive integer $n$. Then for any finite subset of integers $J$ we have $\psi _{J}=\sum _{k\in J }\psi \circ T^{k}$. The measure $\lambda$ is said to satisfy the Følner Gibbs property with respect to the additive cocycle $\Psi =(\psi _{n})_{n}$ and the Følner sequence $\mathcal {F}=(F_{n})_{n}$ when:

$$\begin{aligned} &\text{There exists $\epsilon >0$ such that } \\ & \text{we have for any partition $P$ with diameter less than $\epsilon $:} \\ & \exists N \ \forall x\in A_{n} \textrm{ with } N< n\in \mathfrak{n}, \ \ \ \frac{1}{\lambda \left (P^{F_{n}}(x)\cap A_{n}\right )}\geq e^{ \psi _{F_{n}}(x)}. \end{aligned}$$

(H)

Proposition 3

Under the above hypothesis (H), any weak-∗ limit $\mu$ of $(\mu _{n}^{F_{n}})_{n\in \mathfrak {n}}$ satisfies

$$ h(\mu )\geq \psi (\mu ). $$

Proof

Assume again without loss of generality that $(\mu _{n}^{F_{n}})_{n\in \mathfrak {n}}$ is converging to $\mu$. Take a partition $P$ with $\mu (\partial P)=0$ and with diameter less than $\epsilon$. Then for $n\gg N\gg m$ we obtain:

$$\begin{aligned} \frac{1}{m}H_{\mu}(P^{m})&\geq \limsup _{n\in \mathfrak {n}} \ \frac{1}{\sharp F_{n}}H_{\mu _{n}}(P^{F_{n}}), \text{ by Lemma 5}, \\ &\geq \limsup _{n\in \mathfrak {n}} \frac{1}{\sharp F_{n}}\int \left (- \log \lambda \left ( P^{F_{n}}(x)\cap A_{n} \right )+\log \lambda (A_{n}) \right ) \, d\mu _{n}(x). \end{aligned}$$

Note that by the first item of Lemma 2 we have $\log \lambda (A_{n})\geq -n\delta _{n} -2\log n$, then

$$ \frac{\log \lambda (A_{n})}{\sharp F_{n}}\geq \frac{-n\delta _{n}-2\log n}{d_{n}(F_{n}) n} \xrightarrow{\mathfrak {n}\ni n\rightarrow +\infty}0 \text{ because } \delta _{n}\rightarrow 0 \text{ and }\underline{d}^{\mathfrak{n}}( \mathcal {F})>0, \text{ thus} $$

$$\begin{aligned} \frac{1}{m}H_{\mu}(P^{m}) &\geq \limsup _{n\in \mathfrak {n}} - \frac{1}{\sharp F_{n}}\int \log \lambda \left ( P^{F_{n}}(x)\cap A_{n} \right ) \, d\mu _{n}(x). \end{aligned}$$

It follows from our definitions of $\psi _{F_{n}}$ and $\delta _{x}^{F_{n}}$ that:

$$ \forall x, \ \frac{\psi _{F_{n}}(x)}{\sharp F_{n}}=\int \psi \, d \delta _{x}^{F_{n}}. $$

By Hypothesis (H) we get therefore:

$$\begin{aligned} \frac{1}{m}H_{\mu}(P^{m})&\geq \limsup _{n\in \mathfrak {n}} \int \frac{\psi _{F_{n}}(x)}{\sharp F_{n}} \, d\mu _{n}(x), \\ & \geq \limsup _{n\in \mathfrak {n}} \int \psi \, d \mu _{n}^{F_{n}}, \\ & \geq \psi (\mu ). \end{aligned}$$

Letting $m$ go to infinity, we conclude that $h(\mu )\geq h_{\mu}(P)\geq \psi (\mu )$. □

4 Geometric times

Let $(M,\|\cdot \|)$ be a $C^{r}$, $r>1$, smooth compact Riemannian manifold, not necessarily a surface for the moment. We denote by $\mathrm {d}$ the distance induced by the Riemannian structure on $M$. We also consider a distance $\hat{\mathrm{d}}$ on the projective tangent bundle $\mathbb{P} TM$ (compatible with the standard topology on $\mathbb{P} TM$), such that $\hat{\mathrm{d}}(\hat{x}, \hat{y})\geq \mathrm{d}(\pi \hat{x}, \pi \hat{y})$ for all $\hat{x}, \hat{y} \in \mathbb{P} TM$ with $\pi :\mathbb{P} TM\rightarrow M$ being the natural projection. For a $C^{r}$ map $f:M\rightarrow M$ or a $C^{r}$ curve $\sigma :[0,1]\rightarrow M$ we may define the norm $\|d^{s}f\|_{\infty}$ and $\|d^{s}\sigma \|_{\infty}$ for $1\leq s\leq r$ as the supremum norm of the $s$-derivative of the induced maps through the charts of a given atlas or through the exponential map exp. In the following, to simplify the presentation we lead the computations as $M$ was an Euclidean space or a flat torus. For a $C^{1}$ embedded curve $\sigma :I\rightarrow M$, $I$ being a compact interval of ℝ, we let $\sigma _{*}=\sigma (I)$. The length of $\sigma _{*}$ for the induced Riemannian metric is denoted by $\ell (\sigma _{*})$. For $x\in \sigma _{*}$ we also let $v_{x}\in \mathbb{P}TM$ be the line tangent to $\sigma _{*}$ at $x$ and we write $\hat{x}=(x,v_{x})$.

We denote by $F$ the projective action $F:\mathbb{P}TM\circlearrowleft$ induced by $f$, i.e. $F(x,v)=\left (f(x), \frac{d_{x}f(v)}{\|d_{x}f(v)\|}\right )$, and we consider the additive derivative cocycle $\Phi =(\phi _{k})_{k}$ for $F$ on $\mathbb{P}TM$ given by $\phi (x,v)=\phi _{1}(x,v)=\log \|d_{x}f(v)\|$, where we have identified the line $v$ of $T_{x} M$ with one of its unit generating vectors.

4.1 Bounded curve

Following [12] a $C^{r}$ smooth curve $\gamma :[-1,1]\rightarrow M$ is said to be bounded when

$$ \max _{s=2,\ldots , r}\|d^{s}\gamma \|_{\infty}\leq \frac{1}{6}\|d \gamma \|_{\infty}. $$

We first recall some basic properties of bounded curves (see Lemma 7 in [12]). A bounded curve has bounded distortion meaning that

$$ \forall t,s\in [-1,1], \ \frac{\|d\gamma (t)\|}{\|d\gamma (s)\|}\leq 3/2. $$

(4.1)

Indeed, we have for all $t,s\in [-1,1]$,

$$\begin{aligned} \|d\gamma (t)-d\gamma (s)\|&\leq 2 \|d^{2}\gamma \|_{\infty}, \\ &\leq \frac{1}{3}\|d\gamma (t)\|, \\ \text{ therefore }\frac{2}{3}\|d\gamma (t)\|&\leq \|d\gamma (s)\|. \end{aligned}$$

The projective component of $\gamma$ oscillates also slowly. If we identify $M$ with $\mathbb{R}^{2}$,¹ we have

$$\begin{aligned} \|d\gamma (t)\|\cdot \sin \measuredangle (d\gamma (t), d\gamma (s))& \leq \|d\gamma (t)-d\gamma (s)\|\leq \frac{1}{3}\|d\gamma (t)\|, \\ \measuredangle (d\gamma (t), d\gamma (s))&\leq \pi /6. \end{aligned}$$

(4.2)

When moreover $\|d\gamma \|_{\infty}\leq \epsilon$ we say that $\gamma$ is strongly $\epsilon$-bounded. In particular such a map satisfies $\|\gamma \|_{r}:=\max _{1\leq s\leq r}\|d^{s}\gamma \|_{\infty}\leq \epsilon$, which is the standard $C^{r}$ upper bound required for the reparametrizations in the usual Yomdin’s theory. But this last condition does not allow to control the distortion along the curve in general.

If $\gamma$ is bounded then so is $\gamma _{a,b}=\gamma (a\cdot + b ):[-1,1]\rightarrow M$ for any $a$, $b$ with $|a|\leq \frac{2}{3}$ and $|a|+|b|\leq 1$:

$$\begin{aligned} \forall s\geq 2, \ \|d^{s}\gamma _{a,b}\|_{\infty}&\leq \frac{1}{6} |a|^{s} \|d\gamma \|_{\infty}, \\ &\leq \frac{1}{6} |a|^{s}\frac{3}{2}\|d\gamma (0)\|, \\ &\leq \frac{1}{6}|a|^{s-1}\|d\gamma (0)\|, \\ &\leq \frac{1}{6}\|d\gamma _{a,b}\|_{\infty}. \end{aligned}$$

As $\|d\gamma _{a,b}\|_{\infty}\leq |a|\|d\gamma \|_{\infty}$, if $\gamma$ is moreover strongly $\epsilon$-bounded, then $\gamma _{a,b}$ is strongly $|a|\epsilon$-bounded.

Lemma 6

Let $\gamma :[-1,1]\rightarrow M$ be a $C^{r}$ bounded curve with $\|d\gamma \|_{\infty}\geq \epsilon$. Then there is a family of affine maps $\iota _{j}:[-1,1]\circlearrowleft$, $j\in L:=\underline{L}\cup \overline{L}$ such that:

• each $\gamma \circ \iota _{j}$ is strongly $\epsilon$-bounded and $\|d(\gamma \circ \iota _{j})(0)\|\geq \frac{\epsilon}{6}$,

• $[-1,1]$ is the union of $\bigcup _{j\in \underline{L}}\iota _{j}([-1,1])$ and $\bigcup _{j\in \overline{L}}\iota _{j}([-\frac{1}{3},\frac{1}{3}])$,

• $\sharp \underline{L}\leq 2$ and $\sharp \overline{L}\leq 6\left ( \frac{\|d\gamma \|_{\infty}}{\epsilon}+1\right )$,

• for any $x\in \gamma _{*}$, we have $\sharp \{j\in L, \ (\gamma \circ \iota _{j})_{*}\cap B(x,\epsilon ) \neq \emptyset \}\leq 100$.

Sketch of proof

For the first three items it is enough to consider affine reparametrizations of $[-1,1]$ with rate $\frac{2\epsilon}{3\|d\gamma \|_{\infty}}$. As the bounded map $\gamma$ stays in a cone of opening angle $\pi /6$, its intersection with $B(x,\epsilon )$ is a curve of length less than $2\epsilon$. The last item follows then easily. □

Fix a $C^{r}$ smooth diffeomorphism $f:M\circlearrowleft$. A curve $\gamma :[-1,1]\rightarrow M$ is said$n$-bounded (resp. strongly $(n,\epsilon )$-bounded) when $f^{k}\circ \gamma$ is bounded (resp. strongly $\epsilon$-bounded) for $k=0,\ldots , n$. A strongly $\epsilon$-bounded curve $\gamma$ is contained in the dynamical ball $B_{n}(x,\epsilon ):=\{y\in M, \ \forall k=0,\ldots , n-1, \ \mathrm{d}(f^{k}x,f^{k}y )<\epsilon \}$ with $x=\gamma (0)$.

Fix a $C^{r}$ curve $\sigma :I\rightarrow M$. For $x\in \sigma _{*}$, a positive integer $n$ is called an $(\alpha , \epsilon )$-geometric time of $x$ when there exists an affine map $\theta _{n}:[-1,1]\rightarrow I$ such that $\gamma _{n}:=\sigma \circ \theta _{n}$ is strongly $(n,\epsilon )$-bounded, $\gamma _{n}(0)=x$ and $\|d(f^{n}\circ \gamma _{n})(0)\|\geq \frac{3}{2}\alpha \epsilon$. One can easily check that the curvature of $f^{n}\circ \sigma$ at $f^{n}x$ is bounded from above by $\frac{1}{\alpha \epsilon}$, when $n$ is a $(\alpha , \epsilon )$-geometric time of $x$. It follows from the discussion just before Lemma 6 that, if $n$ is a $(\alpha , \epsilon )$-geometric time of $x$, then it is also a $(\alpha , \epsilon ')$-geometric time for $\epsilon '<\frac{2\epsilon}{3}$.

We let $D_{n}(x)$ and $H_{n}(x)$ be the images of $f^{n}\circ \gamma _{n}$ and $\gamma _{n}$ respectively with $\gamma _{n}$ as above of maximal length. Observe that for all $y=\gamma _{n}(t)$, $t\in [-1,1]$, we have for any $0\leq l< n$:

$$ e^{\phi _{n-l}(F^{l}\hat{y})}= \frac{\|d(f^{n}\circ \gamma _{n})(t)\|}{\|d(f^{l}\circ \gamma _{n})(t)\|}. $$

The bounded distortion property of bounded curves (4.1) then implies:

$$ \forall y,z\in H_{n}(x) \, \forall 0\leq l< n,\ \ \ \frac{e^{\phi _{n-l}(F^{l}\hat{y})}}{e^{\phi _{n-l}(F^{l}\hat{z})}} \leq \frac{9}{4}.$$

(4.3)

4.2 Reparametrization lemma

We consider a $C^{r}$ smooth diffeomorphism $g:M\circlearrowleft$ and a $C^{r}$ smooth curve $\sigma :I\rightarrow M$ with $r>1$. To simplify the exposition we deal with $r\in \mathbb{N}$. The general case follows from standard arguments, see e.g. [12]. We state a global reparametrization lemma to describe the dynamics on $\sigma _{*}$. We will apply this lemma to $g=f^{p}$ for large $p$ with $f$ being the $C^{r}$ smooth system under study. We denote by $G$ the map induced by $g$ on $\mathbb{P} TM$.

We will encode the dynamics of $g$ on $\sigma _{*}$ with a tree, in a similar way the symbolic dynamic associated to monotone branches encodes the dynamic of a continuous piecewise monotone interval map. A weighted directed rooted tree $\mathcal {T}$ is a directed rooted tree (by making all its edges point away from the root) whose edges are labelled. Here the weights on the edges are pairs of integers. Moreover the nodes of our tree will be coloured, either in blue or in red. The level of a node is the number of edges along the unique path between it and the root node.

We let $\mathcal {T}_{n}$ (resp. $\underline{\mathcal {T}_{n}}$, $\overline{\mathcal {T}_{n}}$) be the set of nodes (resp. blue, red nodes) of level $n$. For all $k\leq n-1$ and for all $\mathbf {i}^{n}\in \mathcal {T}_{n}$, we also let $\mathbf {i}^{n}_{k}$ be the node of level $k$ leading to $\mathbf {i}^{n}$. For $\mathbf {i}^{n}\in \mathcal {T}_{n}$, we let $k(\mathbf {i}^{n})=(k_{1}(\mathbf {i}^{n}),k'_{1}(\mathbf {i}^{n}), k_{2}( \mathbf {i}^{n}) \cdots ,k'_{n}(\mathbf {i}^{n}) )$ be the $2n$-tuple of integers given by the sequence of labels along the path from the root $\mathbf {i}^{0}$ to $\mathbf {i}^{n}$, where $\left (k_{l}(\mathbf {i}^{n}),k'_{l}(\mathbf {i}^{n})\right )$ denotes the label of the edge joining $\mathbf {i}^{n}_{l-1}$ and $\mathbf {i}^{n}_{l}$.

For $x\in \sigma _{*}$, we recall that $\hat{x}=(x,v_{x})\in \mathbb{P} TM$ denotes the line tangent to $\sigma$ at $x$. Then we let $k(x)\geq k'(\hat{x})$ be the following integers:

$$ k(x):=\left [\log \|d_{x}g \|\right ], $$

$$ k'( \hat{x}):=\left [\log \|d_{x}g(v_{x})\|\right ]. $$

Moreover for all $n\in \mathbb{N}^{*}$ we define:

$$ k^{n}( x)=(k( x),k'(\hat{x}), k(gx),\ldots k'(G^{n-2}\hat{x}), k(g^{n-1}x), k'(G^{n-1}\hat{x}) ). $$

For a $2n$-tuple of integers $\mathbf {k}^{n}=(k_{1},k'_{1}, \ldots k_{n},k'_{n})$ we consider then

$$ \mathcal {H}(\mathbf {k}^{n}):=\left \{x\in \sigma _{*}, \, k^{n}( x)= \mathbf {k}^{n}\right \}. $$

We restate the Reparametrization Lemma (RL for short) proved in [12] in a global version. Let $\exp _{x}$ be the exponential map at $x$ and let $R_{inj}$ be the radius of injectivity of $(M, \|\cdot \|)$. For $\frac{R_{inj}}{2}>\epsilon >0$ we let $g^{x}_{2\epsilon}= g\circ \exp _{x}(2\epsilon \cdot ): \{w\in T_{x}M, \ \|w\|\leq 1\}\rightarrow M$. Then $\|d^{s} g^{x}_{2\epsilon}\|_{\infty }\leq (2\epsilon )^{s} \sup _{ \stackrel{w\in T_{x}M,}{ \|w\|\leq 2\epsilon}}\|d^{s}(g\circ \exp _{x})(w) \|$. In particular there is $\epsilon _{0}=\epsilon _{0}(g)<\frac{R_{inj}}{2}$ depending only on $M$ and $\|d^{k}g\|_{\infty}$, $k=1, \ldots , r$, such that $\|d^{s}g_{2\epsilon}^{x}\|_{\infty}\leq 3\epsilon \|d_{x}g\|$ for all $s=1,\ldots ,r$, all $x\in M$ and all $\epsilon \leq \epsilon _{0}(g)$.

Reparametrization Lemma 1

Let $\epsilon _{0}(g)\geq \epsilon >0$ and let $\sigma :[-1,1]\rightarrow M$ be a strongly $\epsilon$-bounded curve.

Then there is $\mathcal {T}$, a bicoloured weighted directed rooted tree, and $\left (\theta _{\mathbf{i}^{n}}\right )_{\mathbf{i}^{n} \in \mathcal {T}_{n}}$, $n\in \mathbb{N}$, families of affine reparametrizations of $[-1,1]$, such that for some universal constant $C_{r}$ depending only on $r$:

• (1) $\forall \mathbf {i}^{n}\in \mathcal {T}_{n}$, the curve $\sigma \circ \theta _{\mathbf {i}^{n}}$ is strongly $(n,\epsilon )$-bounded.

• (2) $\forall \mathbf {i}^{n}\in \mathcal {T}_{n}$, the affine map $\theta _{\mathbf{i}^{n}}$ may be written as $\theta _{\mathbf{i}^{n}_{n-1}}\circ \varphi _{\mathbf {i}^{n}}$ with $\varphi _{\mathbf {i}^{n}}$ being an affine contraction with rate smaller than $1/100$. Moreover, when $\mathbf{i}^{n}_{n-1}$ belongs to $\overline{\mathcal {T}_{n-1}}$, we have also $\theta _{\mathbf{i}^{n}}([-1,1])\subset \theta _{\mathbf{i}^{n}_{n-1}}([-1/3,1/3])$.

• (3) $\forall \mathbf {i}^{n}\in \overline{\mathcal {T}_{n}}$, we have $\left \|d\left (g^{n}\circ \sigma \circ \theta _{\mathbf{i}^{n}} \right )(0)\right \|\geq \epsilon /6$.

• (4) $\forall \mathbf {k}^{n}\in (\mathbb{Z}\times \mathbb{Z})^{n}$, the set $\sigma ^{-1}\mathcal {H}(\mathbf {k}^{n})$ is contained in the union of

  $$ \displaystyle{\bigcup _{ \stackrel{\mathbf {i}^{n}\in \overline{\mathcal {T}_{n}}}{ k(\mathbf {i}^{n})=\mathbf {k}^{n}}} \theta _{\mathbf{i}^{n}}([-1/3,1/3])} \textit{ and } \displaystyle{ \bigcup _{ \stackrel{\mathbf {i}^{n}\in \underline{\mathcal {T}_{n}}}{ k(\mathbf {i}^{n})=\mathbf {k}^{n}}} \theta _{\mathbf{i}^{n}}([-1,1])}. $$

  Moreover any term of these unions have a non-empty intersection with $\sigma ^{-1} \mathcal {H}(\mathbf {k}^{n})$.

• (5) $\forall \mathbf {i}^{n-1}\in \mathcal {T}_{n-1}$ and $(k_{n},k'_{n})\in \mathbb{Z}\times \mathbb{Z}$ we have

  $$ \sharp \left \{ \mathbf {i}^{n}\in \overline{\mathcal {T}_{n}}, \ \mathbf {i}^{n}_{n-1}= \mathbf {i}^{n-1} \textit{ and } (k_{n}(\mathbf {i}^{n}),k'_{n}( \mathbf {i}^{n})) =(k_{n},k'_{n})\right \}\leq C_{r}e^{\max \left (k'_{n}, \frac{k_{n}-k'_{n}}{r-1}\right )}, $$
  $$ \sharp \left \{ \mathbf {i}^{n}\in \underline{\mathcal {T}_{n}}, \ \mathbf {i}^{n}_{n-1}= \mathbf {i}^{n-1} \textit{ and } (k_{n}(\mathbf {i}^{n}),k'_{n}( \mathbf {i}^{n})) =(k_{n},k'_{n})\right \}\leq C_{r} e^{ \frac{k_{n}-k'_{n}}{r-1}}. $$

The images of the curve $\sigma$ together with the tree are represented in Figure 1.

Fig. 1

Picture by J. Paik. Color figure online.

Proof

We argue by induction on $n$. For $n=0$ we let $\mathcal {T}_{0}=\underline{\mathcal {T}_{0}}=\{\mathbf {i}^{0}\}$ and we just take $\theta _{\mathbf {i}^{0}}$ equal to the identity map on $[-1,1]$. Assume the tree and the associated reparametrizations have been built till the level $n$.

Fix $\mathbf {i}^{n}\in \mathcal {T}_{n}$ and let

$$\begin{aligned} \hat{\theta}_{\mathbf {i}^{n}}:=&\left \{ \textstyle\begin{array}{l@{\quad}l} \theta _{\mathbf {i}^{n}}(\frac{1}{3}\cdot ) & \text{if } \mathbf {i}^{n} \in \overline{\mathcal {T}_{n}}, \\ \theta _{\mathbf {i}^{n}} & \text{if } \mathbf {i}^{n}\in \underline{\mathcal {T}_{n}}. \end{array}\displaystyle \right . \end{aligned}$$

We will define the children $\mathbf {i}^{n+1}$ of $\mathbf {i}^{n}$, i.e. the nodes $\mathbf {i}^{n +1}\in \mathcal {T}_{n+1}$ with $\mathbf {i}_{n}^{n+1}=\mathbf {i}^{n}$. The label on the edge joining $\mathbf {i}^{n}$ to $\mathbf {i}^{n+1}$ is a pair $(k_{n+1}, k'_{n+1})$ such that the $2(n+1)$-tuple $\mathbf{k}^{n+1}=(k_{1}(\mathbf {i}^{n}), \ldots , k'_{n}(\mathbf {i}^{n}),k_{n+1},k'_{n+1})$ satisfies $\mathcal {H}(\mathbf {k}^{n+1})\cap \left (\sigma \circ \hat{\theta}_{ \mathbf {i}^{n}}\right )_{*}\neq \emptyset$. We fix such a pair $(k_{n+1}, k'_{n+1})$ and the associated sequence $\mathbf{k}^{n+1}$. We let $\eta ,\psi :[-1,1]\rightarrow M$ be the curves defined as:

$$\begin{aligned} \eta :=&\sigma \circ \hat{\theta}_{\mathbf {i}^{n}}, \\ \psi :=&g^{n}\circ \eta . \end{aligned}$$

We will make use of the two following well-known multivariate formulas for the derivatives of a product and a composition of $C^{r}$ functions on $\mathbb{R}^{d}$. For positive integers $m$, $p$, $q$ we let $M_{p,q}(\mathbb{R})$ be the set of real valued $p\times q$ matrices and we denote $A\cdot B\in M_{p,m}(\mathbb{R})$ the product of two matrices $A\in M_{p,q}(\mathbb{R})$ and $B\in M_{q,m}(\mathbb{R})$. We have with the standard multi-index notations:

• General Leibniz rule: Let $u:\mathbb{R}^{d}\rightarrow M_{p,q}(\mathbb{R})$ and $v:\mathbb{R}^{d}\rightarrow M_{q,m}(\mathbb{R})$ be $C^{r}$ maps, then for any $\alpha =(\alpha _{1}, \ldots ,\alpha _{d} )\in \mathbb{N}^{d}$ with $|\alpha |:=\sum _{i}\alpha _{i}\leq r$, we have

  $$\begin{aligned} \partial ^{\alpha }(u\cdot v)=\sum _{\beta : \beta \leq \alpha} \binom{\alpha}{\beta}(\partial ^{\beta }u)\cdot (\partial ^{\alpha - \beta}v). \end{aligned}$$

  (4.4)

• Faà di Bruno’s formula (see e.g. [22]): Let $u:\mathbb{R}^{d}\rightarrow \mathbb{R}$ and $v=(v_{1},\ldots ,v_{d}):\mathbb{R}^{e}\rightarrow \mathbb{R}^{d}$ be $C^{r}$ maps, then for any $\alpha \in \mathbb{N}^{e}$ with $|\alpha |\leq r$, we have

  $$\begin{aligned} \partial ^{\alpha }(u\circ v)=\sum _{\beta \in \mathbb{N}^{d}, \ | \beta |\leq |\alpha |}(\partial ^{\beta }u)\circ v \times P_{\beta} \left (\left (\partial ^{\gamma}v_{i}\right )_{ \gamma , i}\right ), \end{aligned}$$

  (4.5)

  where $P_{\beta}\left (\left (\partial ^{\gamma}v_{i}\right )_{ \gamma , i} \right )$ is a universal polynomial in $\partial ^{\gamma}v_{i}$ for $i=1,\ldots , d$ and $\gamma \in \mathbb{N}^{e}$ with $|\gamma |\leq |\alpha |$.

First step: Taylor polynomial approximation. One computes for an affine map $\theta :[-1,1]\circlearrowleft$ with contraction rate $b$ made precise later and with $y= \psi (t)\in g^{n}\mathcal {H}(\mathbf {k}^{n+1})$, $t\in \theta ([-1,1])$:

$$\begin{aligned} \|d^{r}(g\circ \psi \circ \theta )\|_{\infty }&\leq b^{r} \left \|d^{r} \left (g_{2\epsilon}^{y} \circ \psi _{2\epsilon}^{y}\right )\right \|_{ \infty }, \text{with } \psi _{2\epsilon}^{y}:=(2\epsilon )^{-1}\exp _{y}^{-1}\circ \psi , \\ &\leq b^{r}\left \|d^{r-1}\left ( d_{\psi _{2\epsilon}^{y}}g_{2 \epsilon}^{y}\circ d\psi _{2\epsilon}^{y} \right ) \right \|_{\infty}. \end{aligned}$$

From Leibniz rule (4.4) we get for any $\alpha \in \mathbb{N}^{2}$ with $|\alpha |=r-1$:

$$\begin{aligned} \left \|\partial ^{\alpha}\left ( d_{\psi _{2\epsilon}^{y}}g_{2 \epsilon}^{y}\circ d\psi _{2\epsilon}^{y} \right )\right \|_{\infty}& \leq \sum _{\beta \leq \alpha} \binom{\alpha}{\beta}\|\partial ^{ \beta }(d_{\psi _{2\epsilon}^{y}}g_{2\epsilon}^{y})\|_{\infty }\| \partial ^{\alpha -\beta}(d\psi _{2\epsilon}^{y})\|_{\infty}, \\ & \leq \max _{s=0,\ldots ,r-1}\left \|d^{s}\left (d_{\psi _{2\epsilon}^{y}}g_{2 \epsilon}^{y}\right )\right \| _{\infty}\|\psi _{2\epsilon}^{y} \|_{r} \sum _{\beta \leq \alpha} \binom{\alpha}{\beta}. \end{aligned}$$

By the multi-binomial formula we have $\sum _{\beta \leq \alpha} \binom{\alpha}{\beta}=2^{r}$, so that we obtain finally

$$\begin{aligned} \|d^{r}(g\circ \psi \circ \theta )\|_{\infty }&\leq b^{r} 2^{r} \max _{s=0, \ldots ,r-1}\left \|d^{s}\left (d_{\psi _{2\epsilon}^{y}}g_{2\epsilon}^{y} \right )\right \|_{\infty}\|\psi _{2\epsilon}^{y} \|_{r}. \end{aligned}$$

By assumption on $\epsilon$, we have $\|d^{s} g_{2\epsilon}^{y}\|_{\infty}\leq 3\epsilon \|d_{y} g\|$ for any $r\geq s\geq 1$. Moreover $\|\psi _{2\epsilon}^{y} \|_{r}\leq (2\epsilon )^{-1}\|d\psi \|_{ \infty}\leq 1$ as $\psi$ is strongly $\epsilon$-bounded. Therefore by Faà di Bruno’s formula (4.5), we get for some² constants $C_{r}>0$ depending only on $r$:

$$\begin{aligned} \max _{s=0,\ldots ,r-1}\|d^{s}\left (d_{\psi _{2\epsilon}^{y}}g_{2 \epsilon}^{y}\right )\|_{\infty} &\leq \epsilon C_{r}\|d_{y} g\|, \end{aligned}$$

then,

$$\begin{aligned} \|d^{r}(g\circ \psi \circ \theta )\|_{\infty }&\leq \epsilon C_{r}b^{r} \|d_{y} g\|\|\psi _{2\epsilon}^{y} \|_{r}, \\ &\leq C_{r}b^{r} \|d_{y} g\|\|d\psi \|_{\infty}, \\ &\leq ( C_{r} b^{r-1}\|d_{y}g\|) \|d(\psi \circ \theta )\|_{\infty}, \\ &\leq (C_{r} b^{r-1}e^{k_{n+1}}) \|d(\psi \circ \theta )\|_{\infty}, \textrm{ because }y\text{ belongs to }g^{n}\mathcal {H}(\mathbf {k}^{n+1}), \\ & \leq e^{k'_{n+1}-4}\|d(\psi \circ \theta )\|_{\infty}, \textrm{ by taking } b=\left (C_{r}e^{k_{n+1}-k'_{n+1}+4 }\right )^{-\frac{1}{r-1}}. \end{aligned}$$

The Taylor polynomial $P$ at 0 of degree $r-1$ of $d(g\circ \psi \circ \theta )$ satisfies on $[-1,1]$:

$$\begin{aligned} \|P-d(g\circ \psi \circ \theta )\|_{\infty}&\leq e^{k'_{n+1}-4}\|d( \psi \circ \theta )\|_{\infty}. \end{aligned}$$

We may cover $[-1,1]$ by at most $b^{-1}+1$ such affine maps $\theta$. This term $b^{-1}+1$ is the source of the factor $e^{\frac{k_{n}-k'_{n}}{r-1}}$ in the last item of RL.

Second step: Bezout theorem. Let $a_{n}:=e^{k'_{n+1}}\|d(\psi \circ \theta )\|_{\infty}$. Note that for $s\in [-1,1]$ with $\eta \circ \theta (s)\in \mathcal {H}(\mathbf {k}^{n+1})$ we have

$$\begin{aligned} e^{k'\left (\widehat{\psi \circ \theta (s)}\right )}\|d(\psi \circ \theta )(s)\|\leq \|d(g\circ \psi \circ \theta )(s)\|&\leq e^{1+k' \left (\widehat{\psi \circ \theta (s)}\right )}\|d(\psi \circ \theta )(s) \|, \\ e^{k'_{n+1}} \|d(\psi \circ \theta )(s)\|\leq \|d(g\circ \psi \circ \theta )(s)\|&\leq e^{1+k'_{n+1}}\|d(\psi \circ \theta )(s)\|. \end{aligned}$$

Then, as $\psi \circ \theta$ is bounded, we get by the bounded distortion property (4.1):

$$\begin{aligned} \frac{2}{3}e^{k'_{n+1}}\|d(\psi \circ \theta )\|_{\infty}\leq \|d(g \circ \psi \circ \theta )(s)\|&\leq e^{1+k'_{n+1}}\|d(\psi \circ \theta )\|_{\infty}=a_{n}e. \end{aligned}$$

In particular, $\|d(g\circ \psi \circ \theta )(s)\|\in [a_{n}e^{-1},a_{n}e]$, therefore $\|P(s)\|\in [a_{n}e^{-3},a_{n}e^{3}]$. Moreover if we have now $\|P(s)\|\in [a_{n}e^{-3},a_{n}e^{3}]$ for some $s\in [-1,1]$ we get also $\|d(g\circ \psi \circ \theta )(s)\|\in [a_{n}e^{-4},a_{n}e^{4}]$.

By Bezout theorem the semi-algebraic set $\{ s\in [-1,1],\ \|P(s)\|\in [e^{-3}a_{n}, e^{3}a_{n}]\}$ is the disjoint union of closed intervals $(J_{i})_{i\in I}$ with $\sharp I$ depending only on $r$ (see e.g. Proposition 4.5 in [44]). Let $\theta _{i}$ be the composition of $\theta$ with an affine reparametrization from $[-1,1]$ onto $J_{i}$.

Third step: Landau-Kolmogorov inequality. By the Landau-Kolmogorov inequality on the interval (see Lemma 6 in [12]), we have for some constants $C_{r}\in \mathbb{N}$ and for all $1\leq s\leq r$ with $|J_{i}|$ being the length of $J_{i}$:

$$\begin{aligned} \|d^{s}(g\circ \psi \circ \theta _{i})\|_{\infty }& \leq C_{r}\left ( \|d^{r}(g\circ \psi \circ \theta _{i})\|_{\infty }+\|d(g\circ \psi \circ \theta _{i})\|_{\infty}\right ), \\ &\leq C_{r}\frac{|J_{i}|}{2}\left ( \|d^{r}(g\circ \psi \circ \theta ) \|_{\infty}+ \sup _{t\in J_{i}}\|d(g\circ \psi \circ \theta )(t)\| \right ), \\ &\leq C_{r} a_{n}\frac{|J_{i}|}{2}. \end{aligned}$$

We cut again each $J_{i}$ into $1000C_{r}$ intervals $\tilde{J_{i}}$ of the same length with $(\eta \circ \theta )(\tilde{J}_{i})\cap \mathcal {H}(\mathbf {k}^{n+1}) \neq \emptyset$. Let $\tilde{\theta _{i}}$ be the affine reparametrization from $[-1,1]$ onto $\theta (\tilde{J_{i}})$. We check that $g\circ \psi \circ \tilde{\theta _{i}}$ is bounded:

$$\begin{aligned} \forall s=2,\ldots , r, \ \|d^{s}(g\circ \psi \circ \tilde{\theta _{i}})\|_{\infty }& \leq (1000C_{r})^{-2} \|d^{s}(g \circ \psi \circ \theta _{i})\|_{\infty}, \\ &\leq \frac{1}{6}(1000C_{r})^{-1}\frac{|J_{i}|}{2}a_{n}e^{-4}, \\ &\leq \frac{1}{6}(1000C_{r})^{-1}\frac{|J_{i}|}{2}\min _{s\in J_{i}} \|d(g\circ \psi \circ \theta )(s)\|, \\ &\leq \frac{1}{6}(1000C_{r})^{-1}\frac{|J_{i}|}{2}\min _{s\in \tilde{J}_{i}}\|d(g\circ \psi \circ \theta )(s)\|, \\ &\leq \frac{1}{6} \|d(g\circ \psi \circ \tilde{\theta _{i}})\|_{ \infty}. \end{aligned}$$

Last step: Strongly $\epsilon$-bounded curve. Either $g\circ \psi \circ \tilde{\theta _{i}}$ is strongly $\epsilon$-bounded and $\hat{\theta}_{\mathbf {i}^{n}}\circ \tilde{\theta _{i}}=\theta _{ \mathbf {i}^{n+1}}$ for some $\mathbf {i}^{n+1}\in \underline{\mathcal {T}}_{n+1}$. Or we apply Lemma 6 to $g\circ \psi \circ \tilde{\theta _{i}}$: the new affine parametrizations $\hat{\theta}_{\mathbf {i}^{n}}\circ \tilde{\theta _{i}}\circ \iota _{j}$, $j\in \underline{L}$ (resp. $j\in \overline{L}$) then define $\theta _{\mathbf {i}^{n+1}}$ for a node $\mathbf {i}^{n+1}$ in $\underline{\mathcal {T}}_{n+1}$ (resp. $\overline{\mathcal {T}}_{n+1}$). Note finally that:

$$\begin{aligned} \sharp \overline{L} &\leq 6\left ( \frac{\|d(g\circ \psi \circ \tilde{\theta _{i}})\|_{\infty}}{\epsilon}+1 \right ), \\ & \leq 100 \max (e^{k'_{n+1}}b,1), \text{ as $\psi $ is strongly $\epsilon $-bounded and $\|d\tilde{\theta _{i}}\|_{\infty}\leq b$}, \\ &\leq C_{r} \max \left ( \frac{e^{k'_{n+1}}}{e^{\frac{k_{n+1}-k'_{n+1}}{r-1}}},1\right ), \end{aligned}$$

therefore

$$\begin{aligned} &\sharp \left \{ \mathbf {i}^{n+1}\in \overline{\mathcal {T}_{n+1}} \ \Big| \ \overset{\mathbf {i}^{n+1}_{n}= \mathbf {i}^{n} \text{ and }}{ (k_{n+1}(\mathbf {i}^{n+1}),k'_{n+1}(\mathbf {i}^{n+1})) =(k_{n+1},k'_{n+1})} \right \}\\ &\quad \leq \sum _{\tilde{\theta}_{i}}C_{r} \max \left ( \frac{e^{k'_{n+1}}}{e^{\frac{k_{n+1}-k'_{n+1}}{r-1}}},1\right ), \\ &\quad \leq \sharp \{\tilde{\theta}_{i}\}C_{r} \max \left ( \frac{e^{k'_{n+1}}}{e^{\frac{k_{n+1}-k'_{n+1}}{r-1}}},1\right ), \\ &\quad \leq C_{r}e^{\max \left (k'_{n+1},\frac{k_{n+1}-k'_{n+1}}{r-1} \right )}. \end{aligned}$$

and

$$\begin{aligned} \sharp \left \{ \mathbf {i}^{n+1}\in \underline{\mathcal {T}_{n+1}} \ \Big| \ \overset{\mathbf {i}^{n+1}_{n}= \mathbf {i}^{n} \text{ and }}{ (k_{n+1}(\mathbf {i}^{n+1}),k'_{n+1}(\mathbf {i}^{n+1})) =(k_{n+1},k'_{n+1})} \right \}&\leq \sharp \{\tilde{\theta}_{i}\}, \\ &\leq C_{r} e^{\frac{k_{n+1}-k'_{n+1}}{r-1}}. \end{aligned}$$

 □

As a corollary of the proof of RL we state a local reparametrization lemma, i.e. we only reparametrize the intersection of $\sigma _{*}$ with some given dynamical ball (with respect to the projective action $G$ induced by $g$). For $x\in \sigma _{*}$, $n\in \mathbb{N}$ and $\epsilon >0$ we let

$$\begin{aligned} B^{G}_{\sigma}(x,\epsilon ,n):=\left \{y\in \sigma _{*}, \ \forall k=0, \ldots , n, \ \hat{\mathrm{d}}(G^{k}\hat{x},G^{k}\hat{y})< \epsilon \right \}. \end{aligned}$$

(4.6)

For all $(x,v)\in \mathbb{P}TM$, we also let $w(x,v)=w_{g}(x,v):=\log \|d_{x}g\|-\log \|d_{x}g(v)\|$ and for all $n\in \mathbb{N}$ we let $w^{n}(x,v)=w^{n}_{g}(x,v):=\sum _{k=0}^{n-1}w(G^{k}(x,v))$ with the convention $w^{0}=0$. We consider $\epsilon >0$ as in the Reparametrization Lemma. We assume moreover that

$$\begin{aligned}{} [\hat{\mathrm{d}}((x,v), (y,w) )< \epsilon ]\Rightarrow [ \left | \log \|d_{x}g(v)\|-\log \|d_{y}g(w)\|\right |< 1 \text{ and } \\ \left |\log \|d_{x}g\|-\log \|d_{y}g\|\right |< 1]. \end{aligned}$$

(4.7)

Corollary 3

For any strongly $\epsilon$-bounded curve $\sigma :[-1,1]\rightarrow M$ and for any $x\in \sigma _{*}$, we have for some constant $C_{r}$ depending only on $r$:

$$ \forall n\in \mathbb{N}, \ \ \sharp \left \{\mathbf {i}^{n}\in \mathcal {T}_{n}, \ (\sigma \circ \theta _{\mathbf {i}^{n}})_{*}\cap B^{G}_{ \sigma}(x,\epsilon , n)\neq \emptyset \right \}\leq C_{r}^{n} e^{ \frac{w^{n}(\hat{x})}{r-1}}.$$

(4.8)

Sketch of proof

The Corollary follows from the Reparametrization Lemma together with the two following facts:

• for $y \in B^{G}_{\sigma}(x,\epsilon ,n)$ we have $k^{n}(x)\simeq k^{n}(y)$ up to 1 on each coordinate,

• for any $\mathbf {i}^{n-1}$ there is at most $C_{r}e^{\frac{k(g^{n}x)-k'(G^{n}\hat{x})}{r-1}}$ nodes $\mathbf {i}^{n}\in \overline{\mathcal {T}}_{n}$ with $\mathbf {i}_{n-1}^{n}=\mathbf {i}^{n-1}$ and $\theta _{\mathbf {i}^{n}}([-1,1])\cap \sigma ^{-1} B^{G}_{\sigma}(x, \epsilon , n+1)\neq \emptyset$.

This last point is a consequence of the last item of Lemma 6 applied to the bounded map $g\circ \psi \circ \tilde{\theta _{i}}$ introduced in the third step of the proof of the reparametrization lemma. □

4.3 The geometric set $E$

We apply the Reparametrization Lemma to $g=f^{p}$ for some positive integer $p$. For $x\in \sigma _{*}$ we define the set $E_{p}(x)\subset p\mathbb{N}$ of integers $mp$ such that there is $\mathbf {i}^{m}\in \overline{\mathcal {T}_{m}}$ with $k(\mathbf {i}^{m})=k^{m}(x)$ and $x\in \sigma \circ \theta _{\mathbf {i}^{m}}([-1/3,1/3])$.

Lemma 7

There are $\alpha _{p}>0$ and $\epsilon _{p}>0$ depending only on $r$, $f$ and $p$ such that any $n\in E_{p}(x)$ is a $(\alpha _{p},\epsilon _{p})$-geometric time of $x$ (with respect to $f$).

Proof

Write $x=\sigma \circ \theta _{\mathbf {i}^{m}}(b)$ with $b\in [-1/3,1/3]$ and let $n=mp$. Then for $\epsilon =\epsilon _{0}(f^{p})$, the curve $\gamma _{n}=\sigma \circ \theta _{\mathbf {i}^{m}}(b+\frac{2}{3}\cdot )$ is strongly $(m,\epsilon )$-bounded with respect to $f^{p}$ according to the discussion before Lemma 6. By item (3) of RL we have also $\|d(f^{n}\circ \gamma _{n})(0)\|=\frac{2}{3}\|d(f^{n}\circ \sigma \circ \theta _{\mathbf {i}^{m}} )(b)\|\geq \frac{4}{9}\|d(f^{n}\circ \sigma \circ \theta _{\mathbf {i}^{m}})(0)\|\geq \frac{4}{9} \frac{\epsilon}{6}=\frac{2}{27}\epsilon$. Consequently $m$ is a $(\frac{4}{81},\epsilon )$-geometric time of $x$ with respect to $f^{p}$. For $0< a\leq 1$ we let $\gamma _{n}^{a}=\gamma _{n}(a\cdot )$. Let $0\leq n'\leq n$ and let $m'\in \mathbb{N}$ with $m'p\leq n'<(m'+1)p$. By arguing as in the first step of RL, we have for some constant $C>1$ depending only on $f$, $p$ and $r$:

$$\begin{aligned} \forall s=2,\ldots , r, \ \|d^{s}(f^{n'}\circ \gamma _{n}^{a})\|_{ \infty } \leq & a^{s} \|d^{s}(f^{n'-m'p}\circ (f^{m'p}\circ \gamma _{n})) \|_{\infty}, \\ \leq & Ca^{s} \|d(f^{m'p}\circ \gamma _{n})\|_{\infty}, \end{aligned}$$

then as $f^{m'p}\circ \gamma _{n}$ is bounded:

$$\begin{aligned} \forall s=2,\ldots , r, \ \|d^{s}(f^{n'}\circ \gamma _{n}^{a})\|_{ \infty } \leq & \frac{3}{2}Ca^{s} \|d(f^{m'p}\circ \gamma _{n})(0)\|, \\ \leq & \frac{3}{2}Ca^{s} \max (1,\|df^{-1}\|)^{p} \|d(f^{n'}\circ \gamma _{n})(0)\|, \\ \leq & \frac{3}{2}Ca^{s-1} \max (1,\|df^{-1}\|)^{p} \|d(f^{n'}\circ \gamma _{n}^{a})\|_{\infty}. \end{aligned}$$

We fix $a=\frac{1}{9C\max (1,\|df^{-1}\|)^{p} }$ so that $\gamma _{n}^{a}$ is $n$-bounded with respect to $f$. As $\gamma _{n}$ is strongly $(m,\epsilon )$-bounded with respect to $f^{p}$, the curve $\gamma _{n}^{a}$ is strongly $(n,\epsilon _{p})$-bounded with respect to $f$ with $\epsilon _{p}=a\epsilon \|df\|^{p}_{\infty}$. Finally $\|d(f^{n}\circ \gamma _{n}^{a})(0)\|=a\|d(f^{n}\circ \gamma _{n})(0) \|\geq \frac{2}{27}a\epsilon =\frac{3}{2}\alpha _{p}\epsilon _{p}$ with $\alpha _{p}=\frac{4}{81\|df\|^{p}_{\infty}}$, therefore $n$ is a $(\alpha _{p},\epsilon _{p})$-geometric time of $x$ with respect to $f$. □

Observe now that if $k< m$ we have with $x=\sigma \circ \theta _{\mathbf {i}^{m}}(t)$ and $\theta _{\mathbf {i}^{m}}(t)=\theta _{\mathbf {i}^{m}_{k}}(s)$:

$$\begin{aligned} &e^{\phi _{mp-kp}(F^{kp}\hat{x})}\\ &\quad = \frac{\left \|d\left (f^{mp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(t)\right \|}{\left \|d\left (f^{kp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(t)\right \|}, \\ &\quad \geq \frac{2}{3} \frac{\left \|d\left (f^{mp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(0)\right \|}{\left \|d\left (f^{kp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(t)\right \|}, \text{since $\sigma \circ \theta _{\mathbf {i}^{m}}$ is $m$-bounded}, \\ &\quad \geq \frac{2}{3} \frac{\left \|d\left (f^{mp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(0)\right \|}{\left \|d\left (f^{kp}\circ \sigma \circ \theta _{\mathbf{i}^{m}_{k}}\right )(s)\right \|}100^{m-k}, \text{by item (2) of RL,} \\ &\quad \geq \frac{2}{3\epsilon}\left \|d\left (f^{mp}\circ \sigma \circ \theta _{\mathbf{i}^{m}}\right )(0)\right \|100^{m-k}, \text{as $\sigma \circ \theta _{\mathbf {i}^{n}_{k}}$ is strongly $(k,\epsilon )$-bounded,} \\ &\quad \geq \frac{1}{9}100^{m-k}\geq 10^{m-k}, \text{ by item (3) of RL.} \end{aligned}$$

Therefore $E_{p}$ is $\tau _{p}$-large with $\tau _{p}=\frac{\log 10}{p}$.

The next proposition is the key statement, which will ensure positive density of geometric times on a set of a positive Lebesgue measure of a curve with exponential growth length (see the beginning of Sect. 5.3). In the following, $\mathop{\mathrm{Leb}}_{\sigma _{*}}$ denotes the Lebesgue measure on $\sigma _{*}$ induced by its inherited Riemannian structure as a submanifold of $M$. This is a finite measure with $\mathop{\mathrm{Leb}}_{\sigma _{*}}(M)=\ell \left (\sigma _{*} \right )$.

Proposition 4

Let $f:M\circlearrowleft$ be a $C^{r}$ diffeomorphism and $b>\frac{R(f)}{r}$. For $p$ large enough there exists $\beta _{p}>0$ such that

$$ \limsup _{n}\frac{1}{n}\log \textrm{Leb}_{\sigma _{*}}\left (\left \{x, \ d_{n}(E_{p}(x))< \beta _{p} \textrm{ and } \|d_{x}f^{n}(v_{x})\| \geq e^{nb}\right \}\right )< 0. $$

Proof

Let

$$ \mathcal {E}_{n}:=\left \{x\in A, \ d_{n}(E_{p}(x))< \beta _{p} \textrm{ and } \|d_{x}f^{n}(v_{x})\|\geq e^{nb}\right \}. $$

It is enough to consider $n=mp\in p\mathbb{N}$. We apply the Reparametrization Lemma to $g=f^{p}$ with $\epsilon >0$ being the scale. Let $\mathcal {T}$ be the corresponding tree and $(\theta _{\mathbf {i}^{m}})_{\mathbf {i}^{m}\in \mathcal {T}_{m}}$, $m\in \mathbb{N}$, its associated affine reparametrizations. Let $A_{f}:=\log \|df\|_{\infty}+\log \|df^{-1}\|_{\infty}+1$. We will show the following three claims later on:

• (i) for a node $\mathbf {i}^{m}\in \mathcal {T}_{m}$ with $(\sigma \circ \theta _{\mathbf {i}^{m}})_{*}\cap \mathcal {E}_{n}\neq \emptyset$ the length of $(\sigma \circ \theta _{\mathbf {i}^{m}})_{*}$ is less than $3\epsilon e^{-nb}$,

• (ii) the number of sequences $\mathbf {k}^{m}$ with $\mathcal {H}(\mathbf {k}^{m})\cap \mathcal {E}_{n}\neq \emptyset$ is bounded from above by $(2pA_{f}+1)^{2m}$,

• (iii) for a fixed sequence $\mathbf {k}^{m}$ the number of nodes $\mathbf {i}^{m}\in \mathcal {T}_{m}$ with $k(\mathbf {i}^{m})=\mathbf {k}^{m}$ and $(\sigma \circ \theta _{\mathbf {i}^{m}})_{*}\cap \mathcal {H}(\mathbf {k}^{m}) \cap \mathcal {E}_{n}\neq \emptyset$ is bounded from above by $2^{m}C_{r}^{m} e^{m}\|df^{p}\|_{\infty}^{\frac{m}{r}}\|df\|_{\infty}^{ \beta _{p} p^{2}m}$ for some constant $C_{r}$ depending only on $r$.

Assume these three items already shown and let us conclude the proof of Proposition 4:

$$\begin{aligned} \textrm{Leb}_{\sigma _{*}}(\mathcal {E}_{n})\leq{} &\sum _{\mathbf {k}^{m}, \ \mathcal {H}(\mathbf {k}^{m})\cap \mathcal {E}_{n}\neq \emptyset} \textrm{Leb}_{\sigma _{*}}(\mathcal {H}(\mathbf {k}^{m})\cap \mathcal {E}_{n}), \\ \leq {}&\sum _{\mathbf {k}^{m}, \ \mathcal {H}(\mathbf {k}^{m})\cap \mathcal {E}_{n}\neq \emptyset} \sum _{ \substack{\mathbf {i}^{m}\in \mathcal {T}_{m}, k(\mathbf {i}^{m})=\mathbf {k}^{m},\\ (\sigma \circ \theta _{\mathbf {i}^{m}})_{*}\cap \mathcal {H}(\mathbf {k}^{m})\cap \mathcal {E}_{n}\neq \emptyset}} \ell ((\sigma \circ \theta _{\mathbf {i}^{m}})_{*}), \text{ by (4) of RL,} \\ \leq {}& \left [(2pA_{f}+1)^{2m}\right ]\times \left [2^{m}C_{r}^{m}e^{m} \|df^{p}\|_{\infty}^{\frac{m}{r}} \|df\|_{\infty}^{\beta _{p} p^{2}m} \right ]\\ &{}\times \left [3\epsilon e^{-nb}\right ], \text{ by using (i), (ii) and (iii)}. \end{aligned}$$

Finally we obtain:

$$\begin{aligned} \limsup _{n\in p\mathbb{N}} \frac{1}{n}\log \textrm{Leb}_{\sigma _{*}} \left (\mathcal {E}_{n}\right ) \leq {}&\frac{2}{p}\log (2pA_{f}+1) \\ &{}+ \frac{\log (2C_{r}e)}{p}+\frac{\log \|df^{p}\|_{\infty}}{pr} +p\beta _{p}A_{f}- b. \end{aligned}$$

As $p$ goes to infinity the right member is bounded from above by $\frac{R(f)}{r}-b+A_{f}\limsup _{p}(p\beta _{p})$. As $b$ is larger than $\frac{R(f)}{r}$ one can choose firstly $p\in \mathbb{N}^{*}$ large then $\beta _{p}>0$ small in such a way this right member is negative.

We show now the three items (i), (ii), (iii):

• (i) Let $\mathbf {i}^{m}\in \mathcal {T}_{m}$ with $(\sigma \circ \theta _{\mathbf {i}^{m}})_{*}\cap \mathcal {E}_{n}\neq \emptyset$. For $x=\sigma \circ \theta _{\mathbf {i}^{m}}(t)\in \mathcal {E}_{n}$ we have $\|d_{x}f^{n}(v_{x})\|\geq e^{nb}$. Then by the distortion property (4.1) of the bounded maps $f^{n}\circ \sigma \circ \theta _{\mathbf {i}^{m}}$ and $\sigma \circ \theta _{\mathbf {i}^{m}}$ we get

  $$\begin{aligned} \ell \left ((\sigma \circ \theta _{\mathbf {i}^{m}})_{*}\right )&\leq 2 \|d(\sigma \circ \theta _{\mathbf {i}^{m}})\|_{\infty}, \\ &\leq 3\|d(\sigma \circ \theta _{\mathbf {i}^{m}})(t)\|, \text{ as $\sigma \circ \theta _{\mathbf {i}^{m}}$ is bounded,} \\ &\leq 3 \frac{\|d(f^{n}\circ \sigma \circ \theta _{\mathbf {i}^{m}})(t)\|}{\|d_{x}f^{n}(v_{x})\|} , \\ &\leq 3\epsilon e^{-nb}, \text{as $f^{n}\circ \sigma \circ \theta _{\mathbf {i}^{m}}$ is strongly $\epsilon $-bounded.} \end{aligned}$$

• (ii) As the functions $k$ and $k'$ associated to $g$ takes values in $[-pA_{f},pA_{f}]$ the number of sequences $\mathbf {k}^{m}$ with $\mathbf {k}^{m}=k^{m}(x)$ for some $x\in \sigma _{*}$ is bounded from above by $(2pA_{f}+1)^{2m}$.

• (iii) For a fixed sequence $\mathbf {k}^{m}$ we estimate now the number of nodes $\mathbf {i}^{m}\in \mathcal {T}_{m}$ whose path to the root is labelled with $\mathbf {k}^{m}$ and such that $(\sigma \circ \theta _{\mathbf {i}^{m}})_{*}$ has a non empty intersection with $\mathcal {E}_{n}\cap \mathcal {H}(\mathbf {k}^{m})$. When $x$ belongs to $(\sigma \circ \theta _{\mathbf{i}^{m}})_{*}$ for some $\mathbf{i}^{m}\in \mathcal {T}_{m}$ and satisfies $d_{n}(E_{p}(x))<\beta _{p}$, then we have $\sharp \left \{ 0< k< m, \ \mathbf{i}^{m}_{k} \in \overline{\mathcal {T}_{k}} \right \}\leq n\beta _{p}$. But, by the estimates on the valence of $\mathcal {T}$ given in the last item of RL, the number of $m$-paths from the root labelled with $\mathbf {k}^{m}$ and with at most $n\beta _{p}$ red nodes are less than $2^{m}C_{r}^{m}e^{\sum _{i}\frac{k_{i}-k'_{i}}{r-1}} \|df\|_{\infty}^{ \beta _{p} p^{2}m}$ for some constant $C_{r}$ depending only on $r$ (the factor $2^{m}$ is a rough upper bound for the number of ways to distribute the colours blue and red along the path). Then if $x\in \mathcal {H}(\mathbf {k}^{m})$ satisfies $\|d_{x}f^{n}(v_{x})\|\geq e^{nb}$, we have $e^{\sum _{i}\frac{k_{i}-k'_{i}}{r-1}}\leq e^{m}e^{m \frac{\log \|df^{p}\|_{\infty}-bp}{r-1}}$. But, as $b$ is larger than $\frac{\log \|df^{p}\|_{\infty}}{pr}$ for large $p$, we get for such values of $p$: $\frac{\log \|df^{p}\|_{\infty}-bp}{r-1}\leq \frac{1-\frac{1}{r}}{r-1}\cdot \log \|df^{p}\|_{\infty}= \frac{\log \|df^{p}\|_{\infty}}{r}$.

 □

From now we fix $p$ and the associated quantities satisfying the conclusion of Proposition 4 and we will simply write $E$, $\tau$, $\alpha$, $\epsilon$, $\beta$ for $E_{p}$, $\tau _{p}$, $\alpha _{p}$, $\epsilon _{p}$, $\beta _{p}$. The set $E(x)$ is called the geometric set of $x$.

4.4 Cover of $F$-dynamical balls by bounded curves

As a consequence of Corollary 3, we give now an estimate of the number of strongly $(n,\epsilon ')$-bounded curves reparametrizing the intersection of a given strongly $\epsilon '$-bounded curve with a $F$-dynamical ball of length $n$ and radius $\epsilon '$. This estimate will be used in the proof of the Følner Gibbs property (Proposition 6).

For any $q\in \mathbb{N}^{*}$ we let $\omega _{q}:\mathbb{P}TM\rightarrow \mathbb{R}$ be the map defined for all $(x,v)\in \mathbb{P} TM$ by

$$ \omega _{q}(x,v):=\frac{1}{q}\left (\log \|d_{x} f^{q}\|-\log \|d_{x}f^{q}(v) \|\right ). $$

Note that $\omega _{q}=\frac{w_{f^{q}}}{q}$. We also write $(\omega _{q}^{n})_{n}$ for the additive associated $F$-cocycle, i.e.

$$ \omega _{q}^{n}(x,v)=\sum _{0\leq k< n}\omega _{q}(F^{k}(x,v)). $$

Recall that the dynamical ball $B_{\sigma}^{F}(x,\epsilon ', n)$ has been defined in (4.6).

Lemma 8

For any $q\in \mathbb{N}^{*}$, there exist $\epsilon '_{q}>0$ and $B_{q}>0$ such that for any strongly $\epsilon '_{q}$-bounded curve $\sigma :[-1,1]\rightarrow M$, for any $x\in \sigma _{*}$ and for any $n\in \mathbb{N}^{*}$ there exists a family $(\theta _{i})_{i\in I_{n}}$ of affine maps of $[-1,1]$ such that:

• $B_{\sigma}^{F}(x,\epsilon '_{q}, n)\subset \bigcup _{i\in I_{n}}( \sigma \circ \theta _{i})_{*}$,

• $\sigma \circ \theta _{i}$ is strongly $(n,\epsilon '_{q})$-bounded (with respect to $f$) for any $i\in I_{n}$,

• $\sharp I_{n}\leq B_{q} C_{r}^{\frac{n}{q}}e^{ \frac{\omega _{q}^{n}(\hat{x})}{r-1}}$, with $C_{r}$ being a universal constant depending only on $r$.

Proof

Fix $q$. Let $\epsilon '_{q}=\epsilon /2$ with $\epsilon$ as in Corollary 3 for $g=f^{q}$. There is a finite family $\Theta$ of affine maps of $[-1,1]$ with $\bigcup _{\theta \in \Theta}\theta _{*}=[-1,1]$ such that for any strongly $\epsilon '_{q}$-bounded map $\gamma :[-1,1]\rightarrow M$ and for any $\theta \in \Theta$, the map $\gamma \circ \theta$ is strongly $(q,\epsilon '_{q})$-bounded.

Fix now a strongly $\epsilon '_{q}$-bounded curve $\sigma :[-1,1]\rightarrow M$ and let $x\in \sigma _{*}$. We consider only the map $\theta \in \Theta$ such that $B_{\sigma}^{F}(x,\epsilon '_{q},n)\cap (\sigma \circ \theta )_{*} \neq \emptyset$. For such a map $\theta$ we let $x_{\theta }\in B_{\sigma}^{F}(x,\epsilon '_{q},n)\cap (\sigma \circ \theta )_{*}$. By the choice of $\epsilon '_{q}$ and the inequalities (4.7), we have $|w^{m_{k}}_{f^{q}}(F^{k}\hat{x}_{\theta})-w^{m_{k}}_{f^{q}}(F^{k} \hat{x})|\leq 2$.

Take any $0\leq k< q$ and let $m_{k}=\left [\frac{n-k}{q}\right ]$. By applying Corollary 3 to “$g=f^{q}$”, “$\sigma = f^{k}\circ \sigma \circ \theta$”, “$x=f^{k}(x_{\theta})$” and “$n=[\frac{n-k}{q}]$”, we get a family $\Psi _{\theta , k}$ of affine maps of $[-1,1]$ with $\sharp \Psi _{\theta ,k}\leq C_{r}^{m_{k}} e^{ \frac{w^{m_{k}}_{f^{q}}(F^{k}\hat{x}_{\theta})}{r-1}}\leq C_{r}^{m_{k}} e^{\frac{w^{m_{k}}_{f^{q}}(F^{k}\hat{x})+2}{r-1}}$ such that

$$ \bigcup _{\psi \in \Psi _{\theta ,k}}(\sigma \circ \theta \circ \psi )_{*} \supset f^{-k}B_{ f^{k}\circ \sigma \circ \theta}^{F^{q}}\left (f^{k}(x_{ \theta}),\epsilon ,\left [\frac{n-k}{q}\right ]\right )\supset B_{ \sigma \circ \theta}^{F}(x,\epsilon '_{q},n) $$

such that $f^{mq+k}\circ \sigma \circ \theta \circ \psi$ is strongly $\epsilon$-bounded for $\psi \in \Psi _{\theta ,k}$ and integers $m$ with $0\leq mq+k\leq n$. Then $\Theta _{k}=\{\theta \circ \psi \circ \theta ', \ \psi \in \Psi _{ \theta , k} \text{ and } (\theta , \theta ')\in \Theta ^{2}\}$ satisfies the two first items of the conclusion. Moreover we have:

$$\begin{aligned} \sharp \Theta _{k}& \leq C_{r}^{m_{k}} \sharp \Theta ^{2} e^{ \frac{w^{m_{k}}_{f^{q}}(F^{k}\hat{x})+2}{r-1}}. \end{aligned}$$

But for some constant $A_{q}$ depending only on $q$ we have

$$ \min _{0\leq k< q}e^{w^{m_{k}}_{f^{q}}(F^{k}\hat{x})}\leq \left (\prod _{0 \leq k< q} e^{w^{m_{k}}_{f^{q}}(F^{k}\hat{x})}\right )^{1/q}\leq A_{q} e^{ \omega _{q}^{n}(\hat{x})}. $$

Take $0\leq k= l< q$ achieving the minimum in $\min _{0\leq k< q}e^{w^{m_{k}}_{f^{q}}(F^{k}\hat{x})}$. As $\sharp \Theta$ depends only on $q$, we get for some $B_{q}>0$:

$$ \sharp \Theta _{l}\leq B_{q} C_{r}^{\frac{n}{q}}e^{ \frac{\omega _{q}^{n}(\hat{x})}{r-1}}. $$

This concludes the proof of the lemma by taking $(\theta _{i})_{i\in I_{n}}:=\Theta _{l}$. □

5 Existence of SRB measures

5.1 Entropy formula

By Ruelle’s inequality [40], for any $C^{1}$ system, the entropy of an invariant measure is less than or equal to the integral of the sum of its positive Lyapunov exponents. For $C^{1+}$ systems, the following entropy characterization of SRB measures was obtained by Ledrappier and Young:

Theorem 5

[28] An invariant measure of a $C^{1+}$ diffeomorphism on a compact manifold is an SRB measure if and only if it has a positive Lyapunov exponent almost everywhere and the entropy is equal to the integral of the sum of its positive Lyapunov exponents.

As the entropy is harmonic (i.e. preserves the ergodic decomposition), the ergodic components of an SRB measures are also SRB measures.

5.2 Lyapunov exponents

We consider in this subsection a $C^{1}$ diffeomorphism $f:M\circlearrowleft$. Let $\|\|$ be a Riemannian structure on $M$. The (forward upper) Lyapunov exponent of $(x,v)$ for $x\in M$ and $v\in T_{x}M$ is defined as follows (see [34] for an introduction to Lyapunov exponents):

$$ \chi (x,v):=\limsup _{n\rightarrow +\infty}\frac{1}{n}\log \|d_{x}f^{n}(v) \|. $$

The function $\chi (x,\cdot )$ admits only finitely many values $\chi _{1}(x)>\cdots>\chi _{p(x)}(x)$ on $T_{x}M\setminus \{0\}$ and generates a flag $0\subsetneq V_{p(x)}(x) \subsetneq \cdots \subsetneq V_{1}=T_{x}M$ with $V_{i}(x)=\{ v\in T_{x}M, \ \chi (x,v)\leq \chi _{i}(x)\}$. In particular, $\chi (x,v)=\chi _{i}(x)$ for $v\in V_{i}(x)\setminus V_{i+1}(x)$. The function $p$ as well the functions $\chi _{i}$ and the vector spaces $V_{i}(x)$, $i=1,\ldots,p(x)$ are invariant and depend Borel measurably on $x$. One can show the maximal Lyapunov exponent $\chi$ introduced in the introduction coincides with $\chi _{1}$ (see the Appendix).

A point $x$ is said to be regular when there exists a decomposition

$$ T_{x}M=\bigoplus _{i=1}^{p(x)} H_{i}(x) $$

such that

$$ \forall v\in H_{i}(x)\setminus \{0\}, \ \lim _{n\rightarrow \pm \ \infty }\frac{1}{n}\log \|d_{x}f^{n}(v)\|=\chi _{i}(x) $$

with uniform convergence in $\{v\in H_{i}(x), \ \|v\|=1\}$ and

$$ \lim _{n\rightarrow \pm \ \infty }\frac{1}{|n|}\log \mathop{ \mathrm{Jac}}\left (d_{x}f^{n}\right )=\sum _{i} \dim (H_{i}(x))\chi _{i}(x). $$

In particular we have $V_{i}(x)=\bigoplus _{j=i}^{p(x)} H_{j}(x)$ for all $i$. The set ℛ of regular points is an invariant measurable set of full measure for any invariant measure [29]. The invariant subbundles $H_{i}$ are called the Oseledec’s bundles. We also let $\mathcal {R}^{*}:=\{x\in \mathcal {R}, \ \forall i \ \chi _{i}(x)\neq 0 \}$. For $x\in \mathcal {R}^{*}$ we put $E_{u}(x)= \bigoplus _{i, \ \chi _{i}(x)>0} H_{i}(x)$ and $E_{s}(x)= \bigoplus _{i, \ \chi _{i}(x)<0} H_{i}(x)$.

In the following we only consider surface diffeomorphisms. Therefore we always have $p(x)\leq 2$ and when $p(x)$ is equal to 1, we let $\chi _{2}(x)=\chi _{1}(x)$. When $\nu$ is $f$-invariant we let $\chi _{i}(\nu )=\int \chi _{i}\, d\nu$.

5.3 Building SRB measures

Assume $f$ is a $C^{r}$, $r>1$, surface diffeomorphism and $\limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}\|>b>\frac{R(f)}{r}$ on a set of positive Lebesgue measure as in the Main Theorem. Take $p$ (depending only on $b-\frac{R(f)}{r}>0$) as in Proposition 4, then $\epsilon =\epsilon _{0}(f^{p})$ as in RL. From now $\beta =\beta _{p}$ is also fixed. By using Fubini’s theorem as in [13] there is a $C^{r}$ smooth embedded curve $\sigma :I\rightarrow M$, which can be assumed to be strongly $\epsilon$-bounded, and a subset $A$ of $\sigma _{*}$ with $\mathop{\mathrm{Leb}}_{\sigma _{*}}(A)>0$, such that we have $\limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}(v_{x})\|>b$ for all $x\in A$ (recall $v_{x}$ is the line tangent to $\sigma _{*}$ at $x$, which is identified with an associated unit vector). We can also assume that the countable set of periodic sources has an empty intersection with $\sigma _{*}$.

It follows from Proposition 4 that

$$ \sum _{n} \textrm{Leb}_{\sigma _{*}}\left (\left \{x\in A, \ d_{n}(E(x))< \beta \textrm{ and } \|d_{x}f^{n}(v_{x})\|\geq e^{nb}\right \}\right )< + \infty . $$

Therefore, by taking a smaller subset $A$ (still with positive $\mathop{\mathrm{Leb}}_{\sigma _{*}}$-measure), we may assume that there is $N>0$ such that for any $n>N$ it holds that

$$ \forall x\in A, \ \left [\|d_{x}f^{n}(v_{x})\|\geq e^{nb}\right ] \Rightarrow \left [d_{n}(E(x))\geq \beta \right ]. $$

As we have $\limsup _{n}\frac{1}{n}\log \|d_{x}f^{n}(v_{x})\|>b$ for all $x\in A$, the set of geometric times has positive upper density in $A$:

$$ \forall x\in A, \ \overline{d}(E(x))\geq \beta . $$

We prove now the existence of an SRB measure. This is a first step in the proof of the Main Theorem. For any $q\in \mathbb{N}^{*}$ we let:

$$ \psi _{q}=\phi -\frac{\omega _{q}}{r-1}. $$

We will apply the results of the first sections to the projective action $F:\mathbb{P}TM\circlearrowleft$ induced by $f$, where we consider:

• the additive derivative cocycle $\Phi =(\phi _{k})_{k}$ given by $\phi _{k}(x,v)=\log \|d_{x}f^{k}(v)\|$,

• the measure $\lambda =\lambda _{\sigma}$ on $\mathbb{P}TM$ given by $\mathfrak {s}^{*}\mathop{\mathrm{Leb}}_{\sigma _{*}}$ with $\mathfrak {s}:x\in \sigma _{*}\mapsto (x,v_{x})$,

• the geometric set $E$, which is $\tau$-large with respect to $\Phi$,

• the additive cocycles $\Psi _{q}$ associated to $\psi _{q}-\delta _{q}$ for any $q\in \mathbb{N}^{*}$.

The topological extension $\pi :(\mathbb{P} TM,F)\rightarrow (M,f)$ is principal³ by a straightforward application of Ledrappier-Walters variational principle [31] and Lemma 3.3 in [41]. In fact this holds in any dimension and more generally for any finite dimensional vector bundle morphism instead of $df:TM\circlearrowleft$.

Let $\mathcal {F}=(F_{n})_{n\in \mathfrak {n}}$ and $(A_{n})_{n\in \mathfrak {n}}$ be the sequences associated to $E$ given by Lemma 2. As in Proposition 3 we denote by $\mu _{n}$ the probability measure induced by $\lambda$ on $A_{n}$, i.e. $\mu _{n}=\frac{\lambda (A_{n}\cap \cdot )}{\lambda (A_{n})}$. Rigorously $E$ should be defined on the projective tangent bundle, but as $\pi$ is one-to-one on $\mathbb{P}T\sigma _{*}$ there is no confusion. In the same way we see the sets $A_{n}$, $n\in \mathbb{N}$, as subsets of $A\subset \sigma _{*}$.

Any weak-∗ limit $\mu$ of $\mu _{n}^{F_{n}}:=\frac{1}{\sharp F_{n}}\sum _{k\in F_{n}}F_{*}^{k} \mu _{n}$ is invariant under $F$ and thus supported by Oseledec’s bundles. Let $\nu =\pi \mu$. By Lemma 4, $\mu$ is supported by the unstable bundle $E_{u}$ and $\phi _{*}(\hat{x})> \tau$ for $\mu$ a.e. $\hat{x}\in \mathbb{P}TM$. Note also that $\phi _{*}(\hat{x})\in \{\chi _{1}(\pi \hat{x}), \chi _{2}(\pi \hat{x}) \}$ for $\mu$-almost every $\hat{x}$. We claim that $\phi _{*}(\hat{x})=\chi _{1}(\pi \hat{x})$. If not $\nu$ would have an ergodic component with two positive exponents. It is well known such a measure is necessarily a periodic measure associated to a periodic source $S$ (see e.g. Proposition 4.4 in [36]). But there is an open neighborhood $U$ of the orbit of $S$ with $f^{-1}U\subset U$ and $\sigma _{*}\cap U=\emptyset$. In particular we have $\pi \mu _{n}^{F_{n}}(U)=0$ for all $n$ because $\pi \mu _{n}^{F_{n}}\left (\bigcup _{N\in \mathbb{N}}f^{N}\sigma _{*} \right )=1$ and $f^{N}\sigma _{*}\cap U=f^{N}(\sigma _{*}\cap f^{-N}U)\subset f^{N}( \sigma _{*}\cap U)=\emptyset$. By taking the limit in $n$ we would obtain $\nu (U)=\nu (S)=0$. Therefore $\phi _{*}(\hat{x})=\chi _{1}(\pi \hat{x})>\tau$ for $\mu$-almost every $\hat{x}$ and $\chi _{1}(x)>\tau >0\geq \chi _{2}(x)$ for $\nu$-almost every $x$.

We conclude the construction of an SRB measure by assuming the following proposition, whose proof is given in the next section.

Proposition 6

There exists an infinite sequence of positive real numbers $(\delta _{q})_{q}$ with $\delta _{q}\xrightarrow{q\rightarrow \infty} 0$ such that the property $(H)$ with respect to ℱ holds with respect to the additive cocycle on $\mathbb{P}TM$ associated to the observable $\psi _{q}-\delta _{q}$ for any $q\in \mathbb{N}^{*}$.

Then by Proposition 3 and Proposition 6 we obtain:

$$\begin{aligned} h(\nu )=h(\mu )&\geq \int \psi _{q}\, d\mu -\delta _{q}, \\ &\geq \int \phi \, d\mu - \frac{1}{r-1}\int \omega _{q}\, d\mu - \delta _{q}, \\ &\geq \chi _{1}(\nu )-\frac{1}{r-1}\left (\frac{1}{q}\int \log \|d_{x}f^{q} \|\, d\nu (x)-\chi _{1}(\nu )\right )-\delta _{q}. \end{aligned}$$

By a standard application of the subadditive ergodic theorem, we have

$$ \frac{1}{q}\int \log \|d_{x}f^{q}\|\, d\nu (x) \xrightarrow{q\rightarrow +\infty}\int \chi _{1}(x) \, d\nu (x)=\chi _{1}( \nu ). $$

Therefore $h(\nu )\geq \chi _{1}(\nu )$, since $\delta _{q}\xrightarrow{q\rightarrow \infty}0$. Then Ruelle’s inequality implies $h(\nu )= \chi _{1}(\nu )$. According to Ledrappier-Young characterization (Theorem 5), the measure $\nu$ is an SRB measure of $f$. Note also that any ergodic component $\xi$ of $\nu$ is also an SRB measure, therefore $h(\xi )=\chi _{1}(\xi )>\tau$. But by Ruelle inequality applied to $f^{-1}$, we get also $h(\xi )\leq -\chi _{2}(\xi )$. In particular we have $\chi _{1}(x)>\tau >0>-\tau >\chi _{2}(x)$ for $\nu$-almost every $x$.

5.4 Proof of the Følner Gibbs property (H)

In this subsection we prove Proposition 6. We will show that for any $\delta >0$ there is $q$ arbitrarily large and $\epsilon '_{q}>0$ such that we have for any partition $P$ of $\mathbb{P}TM$ with diameter less than $\epsilon '_{q}$:

$$ \exists n_{*} \ \forall x\in A_{n}\subset \sigma _{*} \textrm{ with } n_{*}< n \in \mathfrak{n}, \ \ \ \frac{1}{\lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A_{n}\right )} \geq e^{-\delta \sharp F_{n}}e^{\psi _{q}^{F_{n}}(\hat{x})}, $$

(5.1)

where we denote $\psi _{q}^{F_{n}}(\hat{x}):=\sum _{k\in F_{n}}\psi _{q}(F^{k}\hat{x})$ to simplify the notations.

For $G\subset \mathbb{N}$ we let $A^{G}$ be the set of points $x\in A$ with $G\subset E(x)$. When $G=\{ k\}$ or $\{k,l\}$ with $k,l\in \mathbb{N}$, we just let $A^{G}= A^{k}$ or $A^{k,l}$. We recall that $\partial F_{n}\subset E(x)$ for all $x\in A_{n}$, in others terms $A_{n}\subset A^{\partial F_{n}}$. We will show (5.1) for $A^{\partial F_{n}}$ in place of $A_{n}$.

Fix the error term $\delta >0$. Let $q$ be so large that $C_{r}^{1/q}< e^{\delta /3}$ and let $\epsilon '_{q}>0$ as in Lemma 8 (with $C_{r}$ being the universal constant in the same lemma). Without loss of generality we may assume $\epsilon '_{q}<\frac{\epsilon}{81}$. Recall that $\epsilon$ corresponds to the fixed scale in the definition of the geometric set $E$. We can also ensure that

$$ \forall \hat{x}, \hat{y} \in \mathbb{P}TM \text{ with } \hat{\mathrm{d}}( \hat{x}, \hat{y})< \epsilon '_{q}, \ \ |\phi (\hat{x})-\phi (\hat{y})|< \delta /3. $$

(5.2)

Let us remember some notations and definitions introduced just before Sect. 4.2. For $x\in \sigma _{*}$ the curve $D_{n}(x)$ denotes the image of $f^{n}\circ \gamma _{n}$ where $\gamma _{n}=\sigma \circ \theta _{n}$ is the subcurve of maximal length satisfying the following three items:

• $\theta _{n}:[-1,1]\circlearrowleft$ is affine,

• $\gamma _{n}$ is strongly $(n,\epsilon )$-bounded,

• $\gamma _{n}(0)=x$.

The integer $n$ is a $(\alpha , \epsilon )$-geometric time of $x$, when $\|d(f^{n}\circ \gamma _{n})(0)\|\geq \frac{3}{2} \alpha \epsilon$. We define the semi-length of $D_{n}(x)$ as the minimum of the lengths of $f^{n}\circ \gamma _{n}([0,1])$ and $f^{n}\circ \gamma _{n}([-1,0])$. The semi-length of $D_{n}(x)$ is larger than $\alpha \epsilon$ at a $(\alpha ,\epsilon )$-geometric time $n$.

In the next three lemmas we consider a strongly $\epsilon$-bounded curve $\sigma$.

Lemma 9

For any subset $N$ of $M$, any $k\in \mathbb{N}$ and any ball $B_{k}$ of radius less than $\epsilon '_{q}$, there exists a finite family $(y_{j})_{j\in J}$ of $A^{k}\cap f^{-k}B_{k}\cap N$ such that:

• $B_{k}\cap f^{k}( A^{k}\cap N)\subset \bigcup _{j\in J}D_{k}(y_{j})$,

• $D_{k}(y_{j})$, $j\in J$, are pairwise disjoint.

Proof

For $y, y'\in A^{k}\cap f^{-k}B_{k}\cap N$ we let $y\sim y'$ when $D_{k}(y)\cap D_{k}(y')\neq \emptyset$. We claim that $[y\sim y'] \Rightarrow [D_{k}(y)\cap B_{k}=D_{k}(y')\cap B_{k}]$. In particular ∼ defines an equivalence relation on $A^{k}\cap f^{-k}B_{k}\cap N$ (with finite quotient set). Then if $(y_{i})_{i}$ is a family of representatives, the curves $D_{k}(y_{j})$, $j\in J$, are pairwise disjoint and $B_{k}\cap f^{k}( A^{k}\cap N)\subset \bigcup _{j\in J}D_{k}(y_{j})$. It remains to show our claim. For $y,y'\in A^{k}\cap f^{-k}B_{k}\cap N$ with $D_{k}(y)\cap D_{k}(y')\neq \emptyset$, the curves $D_{k}(y)$, $D_{k}(y')$ and $D_{k}(y)\cup D_{k}(y')$ lie in a cone with opening angle $\pi /6$ by (4.2) and their length are larger than $\frac{4}{81}\epsilon >4\epsilon '_{q}$. By elementary Euclidean geometric arguments, the intersection of one of these curves with $2B_{k}$ is a curve crossing $2B_{k}$, i.e. its two endpoints lies in the boundary of $2B_{k}$ (see Figure 2). Two such subcurves of $B_{k}\cap (f^{k}\circ \sigma )_{*}$ if not disjoint are necessarily equal. Therefore $\left ( D_{k}(y)\cup D_{k}(y')\right )\cap 2B_{k}=D_{k}(y)\cap 2B_{k}=D_{k}(y') \cap 2B_{k}$.

Fig. 2

The curve $D_{k}(y)$ crossing the ball $2B_{k}$

 □

As the distortion is bounded on $D_{k}(y_{j})$, $j\in J$, by (4.3), we get:

$$\begin{aligned} \sum _{j\in J}\frac{4}{9}e^{-\phi _{k} (\hat{y}_{j})}\ell \left (D_{k}(y_{j}) \right )&\leq \sum _{j\in J}\ell \left (f^{-k}D_{k}(y_{j})\right ). \end{aligned}$$

The curves $\left (D_{k}(y_{i})\right )_{i}$, being pairwise disjoint, we have:

$$\begin{aligned} \sum _{j\in J}\ell \left (f^{-k}D_{k}(y_{j})\right )&\leq \ell \left ( \sigma _{*}\right )\leq 2\epsilon , \end{aligned}$$

therefore

$$\begin{aligned} \sum _{j\in J}\frac{4}{9}e^{-\phi _{k} (\hat{y}_{j})}\ell \left (D_{k}(y_{j}) \right )&\leq 2\epsilon . \end{aligned}$$

The semi-length of $D_{k}(y_{j})$ is larger than $\alpha \epsilon$ because $y_{j}$ belongs to $A^{k}$, so that we obtain finally:

$$\begin{aligned} \sum _{j\in J}e^{-\phi _{k} (\hat{y}_{j})} &\leq \frac{9}{4\alpha}. \end{aligned}$$

(5.3)

Below we consider the dynamical ball $B_{\sigma}^{F}(x,\epsilon '_{q},k)$ defined in (4.6).

Lemma 10

For any subset $N$ of $M$ and any dynamical ball $B_{[\!\![0,k]\!\!]}:=B_{\sigma}^{F}(x,\epsilon '_{q},k)$, there exists a finite family $(z_{i})_{i\in I}$ of $A^{k}\cap B_{[\!\![0,k]\!\!]}\cap N$ such that

• $f^{k}\left ( A^{k}\cap B_{[\!\![0,k]\!\!]}\cap N\right ) \subset \bigcup _{i\in I}D_{k}(z_{i})$,

• $D_{k}(z_{i})$, $i\in I$, are pairwise disjoint,

• $\sharp I \leq B_{q}e^{\delta k/3}e^{ \frac{\omega _{q}^{k}(\hat{x})}{r-1}}$ for some constant $B_{q}$ depending only on $q$.

Proof

As in the previous lemma we consider the subcurves $D_{k}(z)$ for $z \in A^{k}\cap B_{[\!\![0,k]\!\!]}\cap N$. By Lemma 8 we can reparametrize $B_{[\!\![0,k]\!\!]}$ by a family of strongly $(k, \epsilon '_{q})$-bounded curves with cardinality less than $B_{q}C_{r}^{\frac{k}{q}}e^{\frac{\omega _{q}^{k}(\hat{x})}{r-1}}$. Each of these curves is contained in some $f^{-k}D_{k}(z)$ with $z\in A^{k}\cap B_{[\!\![0,k]\!\!]}$. Arguing as in the proof of Lemma 9, such curves can be chosen pairwise disjoint. □

Lemma 11

For any dynamical ball $B_{[\!\![k,l]\!\!]}:=f^{-k}B_{f^{k}\circ \sigma}^{F}(f^{k}x, \epsilon '_{q},l-k)$, there exists a finite family $(y_{i})_{i\in I}$ of $A^{k,l}\cap B_{[\!\![k,l]\!\!]}$ and a partition $I=\coprod _{j\in J}I_{j}$ of $I$ with $j\in I_{j}$ for all $j\in J\subset I$ such that

• $f^{l}( A^{k,l}\cap B_{[\!\![k,l]\!\!]})\subset \bigcup _{i \in I}D_{l}(y_{i})$,

• $D_{l}(y_{i})$, $i\in I$, are pairwise disjoint,

• $\forall j\in J \ \forall i$, $i'\in I_{j}$, $D_{k}(y_{i})\cap B(f^{k}x, \epsilon '_{q})=D_{k}(y_{i'})\cap B(f^{k}x,\epsilon '_{q})$,

• $\forall j\in J$, $\sharp I_{j} \leq B_{q}e^{\delta (l-k)/3}e^{ \frac{\omega _{q}^{l-k}(F^{k}\hat{x})}{r-1}}$ for some constant $B_{q}$ depending only on $q$.

Proof

We first apply Lemma 9 to $\sigma$ and $N=A^{k,l}\cap B_{[\!\![k,l]\!\!]}$ to get the collection of strongly $\epsilon$-bounded curves $\left (D_{k}(y_{j})\right )_{j\in J}$. For $j\in J$ we let $\sigma _{j}^{k}$ be the strongly $\epsilon$-bounded curve $\sigma$ given by $D_{k}(y_{j})$. Then we apply Lemma 10 to each $\sigma _{j}^{k}$ for $j\in J$ and $N=f^{k}(B_{[\!\![k,l]\!\!]}\cap A_{k})$ to get a family $(z_{i})_{i\in I_{j}}$ of $D_{k}(y_{j})\cap A^{l-k}\cap f^{k}( B_{[\!\![k,l]\!\!]} \cap A_{k})$ satisfying:

• $f^{l-k}\left (D_{k}(y_{j})\cap A^{l-k}\cap f^{k}(B_{[\!\![k,l ]\!\!]}\cap A_{k})\right )\subset \bigcup _{j\in J}D_{k}(z_{i})$,

• $D_{l-k}(z_{i})$, $i\in I_{j}$, are pairwise disjoint,

• $\sharp I_{j} \leq B_{q} e^{\delta (l-k)/3}e^{ \frac{\omega _{q}^{l-k}(F^{k}\hat{x})}{r-1}}$.

For all $j\in J$ we can take $j\in I_{j}$ and $z_{j}=f^{k}(y_{j})$. We conclude the proof by letting $y_{i}=f^{-k}z_{i}\in A^{k,l}\cap B_{[\!\![k,l]\!\!]}$ for all $i\in I:=\coprod _{j\in J}I_{j}$. See figure 3.

Fig. 3

For $0\leq t< k$ the image of $f^{t}\circ \sigma$ in black may be large and the disks $D_{t}(y_{i})$ are scattered through the surface. For $t=k$, the sets $D_{k}(y_{j})$ for $j\in J$ are covering $(f^{k}\circ \sigma )_{*}\cap B_{k}$. For $t=l$, we drew in blue the sets $D_{l}(y_{i})\subset f^{l-k}D_{k}(y_{j})$ for $i\in I_{j}$ (Color figure online)

 □

We prove now $(H)$. Recall that $\lambda =\lambda _{\sigma}$ is the push-forward on $\mathbb{P}TM$ of the Lebesgue measure on $\sigma _{*}$. As $\sharp \partial F_{n}=o(n)$ and $\underline{d}_{\mathfrak {n}}(\mathcal {F})>0$ by Lemma 2, it is enough to show there is a constant $C$ such that for any strongly $\epsilon$-bounded curve $\sigma$ we have for $x\in A^{\partial F_{n}}$:

$$ \lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A^{\partial F_{n}} \right )\leq C^{\sharp \partial F_{n}}e^{2\delta \sharp F_{n}/3}e^{- \psi _{q}^{F_{n}}(\hat{x})}. $$

(5.4)

To prove (5.4) we argue by induction on the number of connected components of $F_{n}$. Let $[\!\![k,l[\!\![$, $0\leq k\leq l$, be the first connected component of $F_{n}$ and write $G_{n-l}= \mathbb{N}^{*}\cap (F_{n}-l)$. As $P$ has diameter less than $\epsilon '_{q}$, the set $\mathbb{P} T\sigma _{*} \cap P^{F_{n}}(\hat{x})$ is contained in the intersection of $\pi ^{-1}B_{[\!\![k,l]\!\!]}=\pi ^{-1}\left (f^{-k}B_{f^{k} \circ \sigma}^{F}(f^{k}x,\epsilon '_{q},l-k)\right )$ and $F^{-l} P^{G_{n-l}}(F^{l}\hat{x})$. Then with the notations of Lemma 11 we get:

$$\begin{aligned} &\lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A^{\partial F_{n}} \right )\\ &\quad \leq \lambda _{\sigma}\left ( \coprod _{i\in I}F^{-l}\left ( \pi ^{-1}A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l}\hat{x})\cap D_{l}(y_{i}) \right )\right ), \\ &\quad \leq \lambda _{\sigma}\left ( \coprod _{j\in J} F^{-k} \left ( \coprod _{i\in I_{j}}F^{-(l-k)}\left (\pi ^{-1}A^{\partial G_{n-l}} \cap P^{G_{n-l}}(F^{l}\hat{x})\cap D_{l}(y_{i})\right )\right )\right ) \end{aligned}$$

We recall that $\sigma _{j}^{k}$ denotes the strongly $\epsilon$-bounded curve $\sigma$ given by $D_{k}(y_{j})$ for $j\in J$. By the bounded distortion property (4.3) we get:

$$\begin{aligned} &\lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1} A^{\partial F_{n}} \right )\\ &\quad \leq 3\sum _{j\in J}e^{-\phi _{k}(\hat{y}_{j})}\lambda _{ \sigma _{j}^{k}}\left (\coprod _{i\in I_{j}}F^{-(l-k)}\left (\pi ^{-1} A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l}\hat{x})\cap D_{l}(y_{i}) \right )\right ). \end{aligned}$$

By using again the bounded distortion property (now between the times $k$ and $l$) we get with $\sigma _{i}^{l}$ being the curve associated to $D_{l}(y_{i})$:

$$\begin{aligned} &\lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A^{\partial F_{n}} \right )\\ &\quad \leq 9\sum _{j\in J}e^{-\phi _{k}(\hat{y}_{j})}\sum _{i\in I_{j}} e^{-\phi _{l-k}(F^{k}\hat{y}_{i})} \lambda _{\sigma _{i}^{l}}\left ( \pi ^{-1}A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l}\hat{x})\right ). \end{aligned}$$

We may assume that any $\hat{y}_{i}$, $i\in I$, lies in $P^{F_{n}}(\hat{x})$. In particular we have $|\phi _{l-k}(F^{k} \hat{y}_{i})-\phi _{l-k}(F^{k}\hat{x})|<(l-k) \delta /3$ by (5.2). Then

$$\begin{aligned} \lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A^{\partial F_{n}} \right )\leq {}&9\left (\sum _{j\in J}e^{-\phi _{k}(\hat{y}_{j})}\right ) e^{\delta (l-k)/3} e^{-\phi _{l-k}(F^{k}\hat{x})}\sup _{j} \sharp I_{j} \\ &{}\times \sup _{i\in I}\lambda _{\sigma _{i}^{l}}\left (\pi ^{-1}A^{ \partial G_{n-l}}\cap P^{G_{n-l}}(F^{l}\hat{x})\right ). \end{aligned}$$

By (5.3) and the last item of Lemma 11 we obtain:

$$\begin{aligned} &\lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}\partial A^{F_{n}} \right )\\ &\quad \leq \frac{81}{4\alpha}B_{q}e^{2\delta (l-k)/3}e^{-\phi _{l-k}(F^{k} \hat{x})+\frac{\omega _{q}^{l-k}(F^{k}\hat{x})}{r-1}}\sup _{i\in I} \lambda _{\sigma _{i}^{l}}\left (\pi ^{-1}A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l} \hat{x})\right ), \\ &\quad \leq \frac{81}{4\alpha}B_{q}e^{2\delta (l-k)/3}e^{-\psi _{q}^{ [\!\![k,l]\!\!]}(\hat{x})}\sup _{i\in I}\lambda _{\sigma _{i}^{l}} \left (\pi ^{-1}A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l}\hat{x}) \right ). \end{aligned}$$

By induction hypothesis (5.4) applied to $G_{n-l}$ for each $\sigma _{i}^{l}$, we have for all $i\in I$:

$$ \lambda _{\sigma _{i}^{l}}\left (\pi ^{-1}A^{\partial G_{n-l}}\cap P^{G_{n-l}}(F^{l} \hat{x})\right )\leq C^{\sharp \partial G_{n-l}} e^{2\delta \sharp G_{n-l}/3}e^{- \psi _{q}^{G_{n-l}}(F^{l}\hat{x})}. $$

Note that $\sharp \partial F_{n}=\sharp \partial G_{n-l}+2$ and $\sharp F_{n}=(l-k+1)+\sharp G_{n-l}$. We conclude by taking $C=\sqrt{ \frac{81B_{q}}{4\alpha}}$ that:

$$\begin{aligned} \lambda _{\sigma}\left (P^{F_{n}}(\hat{x})\cap \pi ^{-1}A^{\partial F_{n}} \right )&\leq \frac{81}{4\alpha}B_{q}e^{2\delta \sharp F_{n} /3}C^{ \sharp \partial G_{n-l}}e^{-\psi _{q}^{F_{n}}(\hat{x})}, \\ &\leq C^{\sharp \partial F_{n}} e^{2\delta \sharp F_{n}/3}e^{-\psi _{q}^{F_{n}}( \hat{x})}. \end{aligned}$$

This completes the proof of (5.1).

6 End of the proof of the Main Theorem

6.1 The covering property of the basins

For $x\in M$ the stable/unstable manifold $W^{s/u}(x)$ at $x$ are defined as follows:

$$ W^{s}(x):=\{y\in M, \ \limsup _{n\rightarrow +\infty}\frac{1}{n}\log d(f^{n} x,f^{n} y)< 0\}, $$

$$ W^{u}(x):=\{y\in M, \ \limsup _{n\rightarrow +\infty}\frac{1}{n}\log d(f^{-n} x,f^{-n} y)< 0\}. $$

For a subset $\Gamma$ of $M$ we let $W^{s}(\Gamma )=\bigcup _{x\in \Gamma}W^{s}(x)$. Let $\mathcal {R}^{*}$ be the set of regular points and $E_{s/u}$ be the subbundles of $TM$ as defined in Sect. 5.2. According to Pesin’s theory, there are a nondecreasing sequence of compact, a priori non-invariant, sets $(K_{n})_{n}$ (called the Pesin blocks) with $\mathcal {R}^{*}=\bigcup _{n} K_{n}$ and two families of embedded $C^{\infty}$ discs $(W^{s}_{loc}(x))_{x\in \mathcal {R}^{*}}$ and $(W^{u}_{loc}(x))_{x\in \mathcal {R}^{*}}$ (called the local stable and unstable manifolds) such that:

• $W^{s/u}_{loc}(x)$ are tangent to $E_{s/u}$ at $x$,

• the splitting $E_{u}(x)\oplus E_{s}(x)$ and the discs $W^{s/u}_{loc}(x)$ are continuous on $x\in K_{n}$ for each $n$.

For $\gamma >0$ and $x\in \mathcal {R}^{*}$ we let $W^{s/u}_{\gamma}(x)$ be the connected component of $B(x,\gamma )\cap W^{s/u}_{loc}(x)$ containing $x$.

Proposition 7

The set $\left \{\chi >\frac{R(f)}{r}\right \}$ is covered by the basins of ergodic SRB measures $\mu _{i}$, $i\in I$, up to a set of zero Lebesgue measure.

In fact we prove a stronger statement by showing that $\left \{\chi >\frac{R(f)}{r}\right \}$ is contained Lebesgue a.e. in $W^{s}(\Gamma )$ where $\Gamma$ is any $f$-invariant subset of $\bigcup _{i\in I}\mathcal {B}(\mu _{i})_{i\in I}$ with $\mu _{i}(\Gamma )=1$ for all $i\in I$.

So far we only have used the characterization of SRB measure in terms of entropy (Theorem 5). In the proof of Proposition 7 we will use the absolute continuity property of SRB measures. Let $\mu$ be a Borel measure on $M$. We recall a measurable partition $\xi$ in the sense of Rokhlin [37] is said to be $\mu$-subordinate to $W^{u}$ when $\xi (x)\subset W^{u}(x)$ and $\xi (x)$ contains an open neighborhood of $x$ in the topology of $W^{u}(x)$ for $\mu$-almost every $x$. The measure $\mu$ is said to have absolutely continuous conditional measures on unstable manifolds if for every measurable partition $\xi$ $\mu$-subordinate to $W^{u}$, the conditional measures $\mu _{x}^{\xi}$ of $\mu$ with respect to $\xi$ satisfy $\mu _{x}^{\xi }\ll Leb_{W^{u}(x)}$ for $\mu$-almost every $x$.

Proof

We argue by contradiction. Take $\Gamma$ as above. Assume there is a Borel set $B$ with positive Lebesgue measure contained in the complement of $W^{s}(\Gamma )$ such that we have $\chi (x)>b>\frac{R(f)}{r}$ for all $x\in B$. Then we follow the approach of Sect. 5.3. We consider a $C^{r}$ smooth disc $\sigma$ with $\chi (x,v_{x})>b$ for $x\in B'\subset B$, $\mathop{\mathrm{Leb}}_{\sigma _{*}}(B')>0$. One can then define the geometric set $E$ on a subset $B''$ of $B$ with $\mathop{\mathrm{Leb}}_{\sigma _{*}}(B'')>0$. We also let $\tau$, $\beta$, $\alpha$ and $\epsilon$ be the parameters associated to $E$. Recall that:

• $E$ is $\tau$-large with respect to the derivative cocycle $\Phi$,

• $\overline{d}(E(x))\geq \beta >0$ for $x\in B''$,

• $D_{k}(y)=f^{k}(H_{k}(y))$ has semi-length larger than $\alpha \epsilon$ when $k\in E(y)$, $y\in B''$.

Let $B'''$ be the subset of $B''$ given by density points of $B''$ with respect to $\mathop{\mathrm{Leb}}_{\sigma _{*}}$. In particular, we have

$$ \forall x\in B''', \ \ \frac{\textrm{Leb}_{\sigma _{*}}\left ( H_{k}(x)\cap B''\right ) }{\textrm{Leb}_{\sigma _{*}}(H_{k}(x))} \xrightarrow{k\rightarrow +\infty}1. $$

We choose a subset $A$ of $B'''$ with $\mathop{\mathrm{Leb}}_{\sigma _{*}}(A)>0$ such that the above convergence is uniform in $x\in A$. Then from this set $A$ and the geometric set $E$ on $A$ we may build $\mathfrak {n}$, $(F_{n})_{n\in \mathfrak {n}}$ and $(\mu _{n}^{F_{n}})_{n\in \mathfrak {n}}$ as in Sects. 2 and 3. As proved in Sect. 5.3 any limit measure $\mu$ of $\mu _{n}^{F_{n}}$ is supported on the unstable bundle and projects to an SRB measure $\nu$ with $\chi _{1}(x)\geq \tau >0>-\tau \geq \chi _{2}(x)$ for $\nu$ a.e. $x$. The measure $\nu$ is a barycenter of ergodic SRB measures with such exponents, in particular $\nu (\Gamma )=1$. Take $P=K_{N}$ a Pesin block with $\nu (P)>1-\frac{\beta}{2}$. We let $\theta$ and $l$ be respectively the minimal angle between $E_{u}$ and $E_{s}$ and the minimal length of the local stable and unstable manifolds on $P$.

Let $\xi$ be a measurable partition $\nu$-subordinate to $W^{u}$ with diameter less then $\gamma >0$. We have $\nu (P)=\nu (\Gamma \cap P)=\int \nu _{x}^{\xi}(\Gamma \cap P)\, d \nu (x)$ and $\nu _{x}^{\xi }\ll Leb_{W_{\gamma}^{u}(x)}$ for $\nu$ a.e. $x$. Then

$$\begin{aligned} \nu \left (x, \ \textrm{Leb}_{W_{\gamma}^{u}(x)}(\Gamma \cap P)=0 \right )\leq \nu \left (x, \ \nu _{x}^{\xi}(\Gamma \cap P)=0\right ), \\ \leq 1-\int \nu _{x}^{\xi}(\Gamma \cap P)\, d\nu (x), \\ \leq 1-\nu (P)< \frac{\beta}{2}. \end{aligned}$$

Therefore we get for some $c>0$:

$$ \nu \left (x, \ \textrm{Leb}_{W_{\gamma}^{u}(x)}(\Gamma \cap P)>c \right )> 1-\frac{\beta}{2}. $$

We let $G=\{x\in \Gamma \cap P, \ \mathop{\mathrm{Leb}}_{W_{\gamma}^{u}(x)}( \Gamma \cap P)>c\}$. Observe that:

$$ \nu (G)\geq \nu \left (x, \ \textrm{Leb}_{W_{\gamma}^{u}(x)}(\Gamma \cap P)>c\right )- \nu (M\setminus P)> 1-\beta . $$

For $x\in \sigma _{*}$ and $y\in P$ we use the following notations:

$$ \hat{x}_{\sigma}=(x,v_{x})\in \mathbb{P}T\sigma _{*} \ \ \ \ \ \hat{y}_{u}=(y,v_{y}^{u}) \in \mathbb{P}TM, $$

where $v_{y}^{u}$ is the element of $\mathbb{P}TM$ representing the line $E_{u}(y)$. Let $\hat{G}_{u}^{\gamma}$ be the open $\gamma /8$-neighborhood of $\hat{G}_{u}:=\{\hat{y}_{u}, \ y \in G \}$ in $\mathbb{P}TM$. Recall $E(x)$ denotes the set of geometric times of $x$. We let for $n\in \mathfrak {n}$:

$$ \zeta _{n}:=\int \frac{1}{\sharp F_{n} }\sum _{k\in E(x)\cap F_{n}} \delta _{F^{k} \hat{x}_{\sigma }} \, d\mu _{n}(\hat{x}_{\sigma }). $$

Observe that $\zeta _{n}(\mathbb{P}TM)\geq \inf _{x\in A_{n}} d_{n}(E(x)\cap F_{n})$. By the last item in Lemma 2, we have $\liminf _{n\in \mathfrak {n}}\inf _{ x\in A_{n}} d_{n}(E(x)\cap F_{n}) \geq \beta$. Therefore there is a weak-∗ limit $\zeta =\lim _{k}\zeta _{p_{k}}$ with $\zeta \leq \mu$ and $\zeta (\mathbb{P}TM) \geq \beta$. From $\mu (\hat{G}_{u}^{\gamma})\geq \mu (\hat{G}_{u})=\nu (G) >1-\beta$ we get $0<\zeta (\hat{G}_{u}^{\gamma})\leq \lim _{k} \zeta _{p_{k}}(\hat{G}_{u}^{ \gamma})$. Note also $\hat{A}_{\sigma}:=\{\hat{y}_{\sigma}, \ y\in A \}$ has full $\mu _{n}$-measure for all $n$. In particular, for infinitely many $n\in \mathbb{N}$ there is $(x^{n},v_{x^{n}})=\hat{x}^{n}_{\sigma }\in \hat{A}_{\sigma }$ with $F^{n}\hat{x}^{n}_{\sigma}\in \hat{G}_{u}^{\gamma}$ and $n\in E(x^{n})$. Let $\hat{y}^{n}_{u}=(y^{n},v^{u}_{y^{n}})\in \hat{G}_{u}$ which is $\gamma / 8$-close to $F^{n}\hat{x}^{n}_{\sigma}$. Then for $\gamma \ll \delta \ll \min (\theta , l,\alpha \epsilon )$ independent of $n$, the curve $D_{n}^{\delta}(x^{n}):=D_{n}(x^{n})\cap B(f^{n}x^{n},\delta )$ is transverse to $W^{s}(P\cap \Gamma \cap W^{u}_{\gamma} (y^{n}))$ and may be written as $\exp _{y^{n}}\left (\Gamma _{\psi}\right )$ where $\Gamma _{\psi}$ is the graph of a $C^{r}$ smooth function $\psi :E\subset E_{u}(y^{n})\rightarrow E_{s}(y^{n})$ with $\|d\psi \|< L$ for a universal constant $L$.

By Theorem 8.6.1 in [34] the associated holonomy map $h: W^{u}_{\gamma}(y^{n})\rightarrow D_{n}^{\delta}(x^{n})$, represented in Figure 4, is absolutely continuous and its Jacobian is bounded from below by a positive constant depending only on the Pesin block $P=K_{N}$ (not on $x^{n}$ and $y^{n}$). Since we have $\textrm{Leb}_{W_{\gamma}^{u}(x)}(\Gamma \cap P)>c$, we get for some constant $c'$ independent of $n$:

$$\begin{aligned} \mathrm{Leb}_{D_{n}(x^{n})}\left (W^{s}(\Gamma \cap P) \right )\geq c'. \end{aligned}$$

(6.1)

The distortion of $df^{n}$ on $H_{n}(x^{n})$ being bounded by 3, we get (recall $f^{n}H_{n}(x^{n})=D_{n}(x^{n})$):

$$ \frac{\mathop{\mathrm{Leb}}_{D_{n}(x^{n})}\left (D_{n}(x^{n})\setminus f^{n} B\right )}{\mathop{\mathrm{Leb}}_{D_{n}(x^{n})}\left (D_{n}(x^{n})\right )} \leq 9 \frac{\mathop{\mathrm{Leb}}_{H_{n}(x^{n})}\left (H_{n}(x^{n})\setminus B \right ) }{\mathop{\mathrm{Leb}}_{H_{n}(x^{n})}\left (H_{n}(x^{n})\right )} \xrightarrow{n\rightarrow \infty}0. $$

As $D_{n}(x^{n})$ is the image of a strongly $\epsilon$-bounded curve, its length is bounded from above by $2\epsilon$, so that we get:

$$\begin{aligned} (2\epsilon )^{-1}\mathrm{Leb}_{D_{n}(x^{n})}\left (D_{n}(x^{n}) \setminus f^{n} B\right )\leq \frac{\mathop{\mathrm{Leb}}_{D_{n}(x^{n})}\left (D_{n}(x^{n})\setminus f^{n} B\right )}{\mathop{\mathrm{Leb}}_{D_{n}(x^{n})}\left (D_{n}(x^{n})\right )} \xrightarrow{n\rightarrow \infty}0. \end{aligned}$$

(6.2)

Fig. 4

Holonomy of the local stable foliation between the transversals $D_{n}(x^{n})$ and $W^{u}_{\gamma}(y^{n})$ (Color figure online)

It follows from (6.1) and (6.2) that for $n$ large enough, there exists $x\in f^{n}B\cap W^{s}(\Gamma \cap P)$, in particular $B\cap f^{-n} W^{s}(\Gamma )=B\cap W^{s}(\Gamma )\neq \emptyset$. This contradicts the definition of $B$. □

6.2 The maximal exponent

In Proposition 7 we proved that Lebesgue almost every point $x$ with $\chi (x)>\frac{R(f)}{r}$ lies in the basin of an ergodic SRB measure $\mu$. To complete the proof of Theorem 1 it remains to show $\chi (x)=\chi (\mu )$ for a.e. such points $x$.

For uniformly hyperbolic systems, we have

$$ \Sigma \chi (x):=\max _{k}\Sigma ^{k} \chi (x)=\lim _{n}\frac{1}{n} \log \mathop{\mathrm{Jac}}(d_{x}f^{n}_{E_{u}})=\lim _{n} \int \log \mathop{\mathrm{Jac}}(d_{y}f_{E_{u}})\, d\delta _{x}^{n}. $$

As the geometric potential $y\mapsto \log \mathop{\mathrm{Jac}}(d_{y}f_{E_{u}})$ is continuous in this case, any point in the basin of an SRB measure $\mu$ satisfies $\Sigma \chi (x)=\int \Sigma \chi (y)\, d\mu (y)$. As the geometric potential is not continuous in our context, the proof of this last point is not straightforward.

As mentioned after Proposition 7, we proved in Sect. 6.1 that $\left \{\chi >\frac{R(f)}{r}\right \}$ is contained Lebesgue a.e. in $W^{s}(\Gamma )$ where $\Gamma$ is any $f$-invariant subset of $\bigcup _{i\in I}\mathcal {B}(\mu _{i})_{i\in I}$ with $\mu _{i}(\Gamma )=1$ for all $i$. For such a set $\Gamma$ we have $W^{s}(\Gamma )\subset \bigcup _{i\in I}\mathcal {B}(\mu _{i})_{i\in I}$ and therefore it is enough to find such a set $\Gamma$ satisfying $\chi (x)=\chi (\mu _{i})$ for $x\in W^{s}(\Gamma ) \cap \mathcal {B}(\mu _{i})$, $i\in I$.

Let $\mathcal {R}^{+*}$ denote the invariant subset of Lyapunov regular points $x$ of $(M,f)$ with $\chi _{1}(x)>0>\chi _{2}(x)$. Such a point admits so called regular neighborhoods (or $\epsilon$-Pesin charts):

Lemma 12

[33] For a fixed $\epsilon >0$ there exists a measurable function $q=q_{\epsilon}:\mathcal {R}^{+*}\rightarrow (0,1]$ with $e^{-\epsilon}< q(fx)/q(x)< e^{\epsilon}$ and a collection of embeddings $\Psi _{x}:B(0,q(x))\subset T_{x}M=E_{u}(x)\oplus E_{s}(x)\sim \mathbb{R}^{2}\rightarrow M$ with $\Psi _{x}(0)=x$ such that $f_{x}=\Psi _{fx}^{-1}\circ f\circ \Psi _{x}$ satisfies the following properties:

• $$ d_{0}f_{x} = \begin{pmatrix} a_{\epsilon}^{1}(x) &0 \\ 0& a_{\epsilon}^{2}(x), \end{pmatrix} $$

  with $e^{-\epsilon}e^{\chi _{i}(x)}< a_{\epsilon}^{i}(x) < e^{\epsilon}e^{ \chi _{i}(x)}$ for $i=1,2$,

• the $C^{1}$ distance between $f_{x}$ and $d_{0}f_{x}$ is less than $\epsilon$,

• there exists a constant $K$ and a measurable function $A=A_{\epsilon}:\mathcal {R}^{+*}\rightarrow \mathbb{R}$ such that for all $y,z\in B(0,q(x))$

  $$ Kd(\Psi _{x}(y),\Psi _{x}(z))\leq \|y-z\|\leq A(x)d(\Psi _{x}(y), \Psi _{x}(z)), $$

  with $e^{-\epsilon}< A(fx)/A(x)< e^{\epsilon}$.

For any $i\in I$ we let

$$ E_{i}:=\{x, \ \chi (x)=\chi (\mu _{i})\}. $$

The set $E_{i}$ has full $\mu _{i}$-measure by the subadditive ergodic theorem. Put $\Gamma _{i}=\mathcal {B}(\mu _{i})\cap E_{i}\cap \mathcal {R}^{+*}$ and $\Gamma =\bigcup _{i}\Gamma _{i}$. Clearly $\Gamma$ is $f$-invariant. We finally check that $\chi (x)=\chi (\mu _{i})$ for $x\in W^{s}(\Gamma _{i})$.

Lemma 13

If $y\in W^{s}(x)$ with $x\in \mathcal {R}^{+*}$, then $\chi (y)=\chi (x)$.

Proof

Fix $x\in \mathcal {R}^{+*}$ and $\delta >0$. We apply Lemma 12 with $\epsilon \ll \chi _{1}(x)$. For $\alpha >0$ we let $\mathcal {C}_{\alpha}$ be the cone $\mathcal {C}_{\alpha}=\{(u,v)\in \mathbb{R}^{2}, \ \alpha \|u\|\geq \|v \|\}$. We may choose $\alpha >0$ and $\epsilon >0$ so small that for all $k\in \mathbb{N}$ we have $d_{z}f_{f^{k}x}(\mathcal {C}_{\alpha})\subset \mathcal {C}_{\alpha}$ and $\|d_{z}f_{f^{k}x}(v)\|\geq e^{\chi _{1}(x)-\delta}$ for all $v\in \mathcal {C}_{\alpha}$ and all $z\in B(0, q_{\epsilon}(f^{k}x))$.

Let $y\in W^{s}(x)$. There is $C>0$ and $\lambda$ such that $d(f^{n}x,f^{n}y)< C\lambda ^{n}$ holds for all $n\in \mathbb{N}$. We can choose $\epsilon \ll \lambda$. In particular there is $N>0$ such that $f^{n}y$ belongs to $\Psi _{f^{n}x} B(0, q(f^{n}x))$ for $n\geq N$ since we have $A(f^{n}x)< e^{\epsilon n}A(x)$ and $q(f^{n}x)>e^{\epsilon n}q(x)$. Let $z \in B(0,q(f^{N}x))$ with $\Psi _{f^{N}x}(z)=y$. Then for all $v\in \mathcal {C}_{\alpha}$ and for all $n\geq N$ we have $\|d_{z}\left (\Psi _{f^{n-N}x}^{-1}\circ f^{n-N}\circ \Psi _{f^{N}x} \right )(v)\|\geq e^{(n-N)(\chi _{1}(x)-\delta )}$. As the conorm of $d_{f^{n-N}y}\psi _{f^{n}x}$ is bounded from above by $A(f^{n} x)^{-1}$ for all $n$ we get

$$\begin{aligned} \chi (y)&= \limsup _{n}\frac{1}{n}\log \|d_{y} f^{n-N}\|, \\ &= \limsup _{n}\frac{1}{n}\log \|d_{z} \left (f^{n-N}\circ \Psi _{f^{N}x} \right )\|, \\ & \geq \limsup _{n\rightarrow +\infty}\frac{1}{n}\log \left ( A(f^{n}x)^{-1} \left \|d_{z}\left (\Psi _{f^{n}x}^{-1}\circ f^{n}\circ \Psi _{f^{N}x} \right ) \right \|\right ), \\ &\geq \chi _{1}(x)-\delta -\epsilon . \end{aligned}$$

On the other hand we have

$$\begin{aligned} \left \|d_{z}\left (\Psi _{f^{n}x}^{-1}\circ f^{n}\circ \Psi _{f^{N}x} \right ) \right \|&\leq \prod _{k=N}^{n-1}\sup _{t\in B(0, q(f^{k}x))} \|d_{t}f_{f^{k}x}\|, \\ &\leq \left (e^{\chi _{1}(x)+\epsilon}+\epsilon \right )^{n-N}, \\ &\leq e^{(n-N)(\chi _{1}(x)+2\epsilon )}. \end{aligned}$$

Then it follows from $\|d_{f^{n-N}y}\psi _{f^{n}x}\|\leq K$:

$$\begin{aligned} \chi (y)&\leq \limsup _{n\rightarrow +\infty}\frac{1}{n}\log \left ( \left \|d_{z}\left (\Psi _{f^{n}x}^{-1}\circ f^{n}\circ \Psi _{f^{N}x} \right ) \right \|\right ), \\ &\leq \chi _{1}(x)+2\epsilon . \end{aligned}$$

As it holds for arbitrarily small $\epsilon$ and $\delta$ we get $\chi (y)=\chi _{1}(x)=\chi (x)$. □

We conclude with $\Lambda =\{\chi (\mu _{i}), \ i\in I\}$ that for Lebesgue a.e. point $x$, we have $\chi (x)\in \left ]-\infty , \frac{R(f)}{r}\right ]\cup \Lambda$ and that $\{\chi =\lambda \}\stackrel{o}{\subset}\bigcup _{i\in I, \, \chi ( \mu _{i})=\lambda}\mathcal {B}(\mu _{i})$ for all $\lambda \in \Lambda$. The proof of the Main Theorem is now complete. It follows also from Lemma 13, that the converse statement of Corollary 2 holds: if $(f,M)$ admits an SRB measure then $\mathop{\mathrm{Leb}}(\chi >0)>0$.

7 Nonpositive exponent in contracting sets

In this last section we show Theorem 2. For a dynamical system $(M,f)$ a subset $U$ of $M$ is said to be almost contracting when for all $\epsilon >0$ the set $E_{\epsilon}=\{k\in \mathbb{N}, \ \mathop{\mathrm{diam}}(f^{k} U)> \epsilon \}$ satisfies $\overline{d}(E_{\epsilon})=0$. In [21] the authors build subsets with historic behaviour and positive Lebesgue measure which are almost contracting but not contracting. We will show Theorem 2 for almost contracting sets.

We borrow the next lemma from [12] (Lemma 4 therein), which may be stated with the notations of Sect. 5 as follows:

Lemma 14

Let $f:M\circlearrowleft$ be a $C^{\infty}$ diffeomorphism and let $U$ be a subset of $M$ with $\mathop{\mathrm{Leb}}\left (\left \{\chi >a\right \}\cap U\right )>0$ for some $a>0$. Then for all $\gamma >0$ there is a $C^{\infty}$ smooth embedded curve $\sigma :[0,1]\rightarrow M$ and $I\subset \mathbb{N}$ with $\sharp I=\infty$ such that

$$ \forall n\in I, \ \mathrm{Leb}_{\sigma _{*}}\left (\{x\in U\cap \sigma _{*}, \, \| d_{x}f^{n}(v_{x})\|>e^{na} \} \right )>e^{-n\gamma}. $$

We are now in a position to prove Theorem 2 for almost contracting sets.

Proof of Theorem 2

We argue by contradiction by assuming $\mathop{\mathrm{Leb}}\left (\left \{\chi >a\right \}\cap U\right )>0$ for some $a>0$ with $U$ being a almost contracting set. By Yomdin’s Theorem on one-dimensional local volume growth for $C^{\infty}$ dynamical systems [43] there is $\epsilon >0$ so small that

$$ v^{*}(f,\epsilon ):=\sup _{\sigma}\limsup _{n\to \infty}\frac{1}{n} \sup _{x\in M}\log \mathrm{Leb}_{(f^{n}\circ \sigma )_{*}}\left (f^{n}B_{n}(x, \epsilon )\right )< a/2, $$

(7.1)

where the supremum holds over all $C^{\infty}$ smooth embedded curves $\sigma :[0,1]\rightarrow M$. As $U$ is almost contracting, there are subsets $(C_{n})_{n\in \mathbb{N}}$ of $M$ with $\lim _{n}\frac{\log \sharp C_{n}}{n}=0$ satisfying for all $n$:

$$ U\subset \bigcup _{x\in C_{n}}B_{n}(x,\epsilon ). $$

(7.2)

Fix an error term $\gamma \in ]0,\frac{a}{2}[$. Then by Lemma 14 there is a $C^{\infty}$-smooth curve $\sigma _{*}\subset U$ and an infinite subset $I$ of ℕ such that for all $n\in I$:

$$\begin{aligned} \sum _{x\in C_{n}}\mathrm{Leb}_{(f^{n}\circ \sigma )_{*}} \left ( f^{n}B_{n}(x, \epsilon )\right ) \geq &\mathrm{Leb}_{(f^{n}\circ \sigma )_{*}}( f^{n}(U \cap \sigma _{*})), \\ \geq & e^{na}\mathrm{Leb}_{\sigma _{*}} \left ( \{x\in U\cap \sigma _{*}, \, \| d_{x}f^{n}(v_{x})\|>e^{na} \}\right ), \\ \geq & e^{n(a-\gamma )} \text{ by (7.1),} \\ \sharp C_{n} \sup _{x\in M} \mathrm{Leb}_{(f^{n}\circ \sigma )_{*}} \left (f^{n}B_{n}(x,\epsilon )\right ) \geq & e^{n(a-\gamma )} \text{ by (7.2).} \end{aligned}$$

Therefore we get the contradiction $v^{*}(f,\epsilon )>a-\gamma >a/2$. □

Appendix

Let $\mathcal {A}=(A_{n})_{n\in \mathbb{N}}$ be a sequence in $M_{d}(\mathbb{R}^{d})$. For any $n\in \mathbb{N}$ we let $A^{n}=A_{n-1}\ldots A_{1}A_{0}$. We define the Lyapunov exponent $\chi (\mathcal {A})$ of $\mathcal{A}$ with respect to $v\in \mathbb{R}^{d}\setminus \{0\}$ as

$$ \chi (\mathcal {A},v):=\limsup _{n}\frac{1}{n}\log \|A^{n}(v)\|, $$

Lemma 15

$$ \sup _{v\in \mathbb{R}^{d}\setminus \{0\}}\chi (\mathcal {A},v)= \limsup _{n}\frac{1}{n}\log {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }. $$

Proof

The inequality ≤ is obvious. Let us show the other inequality. Let $v_{n}\in \mathbb{R}^{d}$ with $\|v_{n}\|=1$ and $\|A^{n}(v_{n})\|= {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }$. Then take $v=\lim _{k}v_{n_{k}}$ with $\lim _{k}\frac{1}{n_{k}}\log {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n_{k}} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }=\limsup _{n} \frac{1}{n}\log {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }$. We get

$$\begin{aligned} \|A^{n_{k}}(v)\|&\geq \|A^{n_{k}}(v_{k})\|- \|A^{n_{k}}(v-v_{k})\|, \\ &\geq {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n_{k}} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }(1-\|v-v_{k}\|), \\ \limsup _{k}\frac{1}{n_{k}}\log \|A^{n_{k}}(v)\|&\geq \limsup _{n} \frac{1}{n}\log {\left \vert \kern -0.25ex\left \vert \kern -0.25ex\left \vert A^{n} \right \vert \kern -0.25ex\right \vert \kern -0.25ex\right \vert }. \end{aligned}$$

 □