Thermodynamic Formalism for Random Non-uniformly Expanding Maps

Abstract

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $C^1$-potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carathéodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.

Appendices

Appendix A: Relative Pressure for Random Dynamical Systems

In this appendix we introduce a Carathéodory structure for random dynamical systems which allow us to define topological pressure for invariant sets that are not necessarily compact on the fibers and establish a variational principle for this pressure function, which is of independent interest. This result extends the previous variational principle for random bundle transformations, where compactness is assumed, proved by Kifer [25].

1.1 Relative pressure in random dynamical systems

Here we introduce the notion of relative pressure for random dynamical systems inspired by the work of Pesin and Pitskel in the deterministic setting. The Carathéodory structure introduced below are fiberwise versions of the ones present at the Appendix II of [41].

1.1.1 Relative pressure using coverings

Let $\Lambda \subset X$ be a measurable set which is positively F-invariant (i.e., $F(\Lambda )\subset \Lambda$) and set $\Lambda _\omega = \Lambda \cap X_\omega$. Let ${\mathcal {U}}$ be a finite open cover of M and set ${\mathcal {U}}_{\omega }:=\{U\cap X_{\omega }{:}\,U\in {\mathcal {U}}$}. Denote by ${{\mathcal {I}}}_\omega ^n$ the space of all n-strings ${i}(\omega )=(U_0, U_{1}, \ldots , U_{n-1})$ with $U_{i}\in {\mathcal {U}}_{\theta ^i\omega }$ and set $n( {i})=n$. For a given string denote by ${U}( {i})= \{x\in X_\omega {:}\, f_{\omega }^j(x)\in U_j\ \text { for } j=0, \ldots , n-1\}$. Furthermore, for every integer $N \geqslant 1$, let ${\mathcal {S}}_ N {\mathcal {U}}(\omega )$ be the space of all cylinders of depth at least N. Given $\alpha \in {\mathbb {R}}$ define

$$\begin{aligned} m_\alpha (\omega , f, \phi ,\Lambda ,{\mathcal {U}},N) = \inf _{{\mathcal {G}}} \Big \{\sum _{{{\underline{\mathrm{U}}}} \in {\mathcal {G}}_\omega } e^{-\alpha n({{\underline{\mathrm{U}}}})+ S_{n({{\underline{\mathrm{U}}}})} \phi _\omega ({{\underline{\mathrm{U}}}})} \Big \} \end{aligned}$$

(9.1)

where the infimum is taken over a class of families ${\mathcal {G}}$ of the form ${\mathcal {G}}_\omega \subset {\mathcal {S}}_ N {\mathcal {U}}(\omega )$ that cover $\Lambda _\omega$ such that (9.1) is measurable in $\omega$. In order to describe the class of families ${\mathcal {G}}$ first recall that $\Omega$ is a Lebesque space which implies that there exists a refining sequence of finite partitions $({\mathfrak {p}}_k)$ which generates ${\mathcal {F}}$ up to measure zero. In (9.1), we now take the infimum over all finite families of the following form: for any such ${\mathcal {G}}$ there exists $k\ge 1$ such that ${\mathcal {G}} \subset \{ V \times \{ {i} \}{:}\,V \in {\mathfrak {p}}_k, n( {i}) \ge N\}$, and ${\mathcal {G}}_\omega := \{ {U}( {i}){:}\,\text {for some } V \times \{ {i}\} \in {\mathcal {G}} \hbox { s.t. } \omega \in V\}$ is an open cover of $\Lambda _\omega$ for ${\mathbb {P}}$-almost every $\omega \in \Omega$. This construction is crucial in here as the cardinality of these class is bounded by the cardinality of finite subsets of a countable set, hence there are only countably many ${\mathcal {G}}$ of this type. In particular, $\Omega \ni \omega \mapsto m_\alpha (\omega , f, \phi ,\Lambda ,{\mathcal {U}},N)$ is measurable as it is the infimum of measurable functions over a countable family.

For fixed ${\mathcal {G}}$, it then immediately follows that $(\omega ,\alpha )\mapsto \sum _{{{\underline{\mathrm{U}}}} \in {\mathcal {G}}_\omega } e^{-\alpha n({{\underline{\mathrm{U}}}})+ S_{n({{\underline{\mathrm{U}}}})} \phi _\omega ({{\underline{\mathrm{U}}}})}$ is measurable in $\Omega$ and continuous in $\alpha$, a condition that ensures jointly measurability (see e.g. [15, Lemma 1.1]). Hence, by countability of $\{{\mathcal {G}}\}$, the infimum $(\omega ,\alpha )\mapsto m_\alpha (\omega , f, \phi ,\Lambda ,{\mathcal {U}},N)$ is jointly measurable and this guarantees the measurability of all the following quantities as a function of $\Omega$. The limit

$$\begin{aligned} m_\alpha (\omega , f, \phi , \Lambda , {\mathcal {U}})=\lim _{N \rightarrow \infty } m_\alpha (\omega , f,\phi ,\Lambda ,{\mathcal {U}},N) \end{aligned}$$

exists by monotonicity. Set also $\pi _\phi (\omega , f,\Lambda ,{\mathcal {U}})= \inf {\{\alpha {:}\,m_\alpha (\omega , f,\phi ,\Lambda ,{\mathcal {U}})=0\}}$ and

$$\begin{aligned} \pi _\phi (\omega , f,\Lambda )=\lim _{{\text {diam}}({\mathcal {U}})\rightarrow 0} \pi _\phi (\omega , f,\Lambda ,{\mathcal {U}}). \end{aligned}$$

A simple adaptation of the proof [12, Theorem 11.1] shows that the limit does exist and does not depend on the choice of coverings with diameter going to zero. We define the relative pressure $\pi _\phi (f,\Lambda )$ by

$$\begin{aligned} \pi _\phi (f,\Lambda )=\int _{\Omega }\pi _\phi (\omega , f,\Lambda )\, d{{\mathbb {P}}}. \end{aligned}$$

The relative entropy $h_\Lambda (f)$ is defined as the relative pressure $\pi _\phi (f,\Lambda )$ for $\phi \equiv 0$, and it can be related to topological pressure of a continuous potential $\phi$ by

$$\begin{aligned} \pi _\phi (f,\Lambda ) \leqslant h_\Lambda (f) + \int \sup \phi _\omega \;d{\mathbb {P}}. \end{aligned}$$

(9.2)

Indeed, let $\omega \in \Omega$ be a ${\mathbb {P}}$-typical point. Then, for every open covering ${\mathcal {U}}_\omega$ of $X_\omega$, any $\varepsilon >0$ and any large $N \geqslant 1$ (depending on $\omega$)

$$\begin{aligned} m_\alpha (\omega , f,\phi ,\Lambda ,{\mathcal {U}},N)&= \inf _{{\mathcal {G}}_\omega } \Big \{\sum _{{{\underline{\mathrm{U}}}} \in \,{\mathcal {G}}_\omega } e^{S_{n({{\underline{\mathrm{U}}}})} (-\alpha +\phi _\omega ({{\underline{\mathrm{U}}}})) } \Big \} \\&\leqslant \inf _{{\mathcal {G}}_\omega } \Big \{\sum _{{{\underline{\mathrm{U}}}} \in \,{\mathcal {G}}_\omega } e^{S_{n({{\underline{\mathrm{U}}}})} (-\alpha +\sup \phi _\omega ) +S_{n({{\underline{\mathrm{U}}}})} k_{{\mathcal {U}},\omega }} \Big \} \\&\leqslant \inf _{{\mathcal {G}}_\omega } \Big \{\sum _{{{\underline{\mathrm{U}}}} \in \,{\mathcal {G}}_\omega } e^{- n({{\underline{\mathrm{U}}}}) [\alpha -\int \sup \phi _\omega \,d{{\mathbb {P}}}- \int k_{{\mathcal {U}},\omega } \, d{{\mathbb {P}}}-\varepsilon ] } \Big \} \\&= m_{\alpha -\int \sup \phi _\omega \,d{{\mathbb {P}}}- \int k_{{\mathcal {U}},\omega } \, d{{\mathbb {P}}}-\varepsilon }\,(f,0,\Lambda ,{\mathcal {U}},N) \end{aligned}$$

where $k_{{\mathcal {U}},\omega }:=\sup \{|\varphi _{\omega }(x)-\varphi _{\omega }(y)|{:}\,y \in {\mathcal {U}}_\omega (x)\}$ tends to zero as ${\text {diam}}{\mathcal {U}}\rightarrow 0$ and the infimum is taken over all families ${\mathcal {G}}_\omega \subset {\mathcal {S}}_N {\mathcal {U}}(\omega )$ that cover $\Lambda _\omega$. Since $\varepsilon >0$ was chosen arbitrary, taking the limit as the diameter of ${\mathcal {U}}$ tends to zero we conclude that $\pi _\phi (f,\Lambda ) \leqslant h_\Lambda (f) + \int \sup \phi _\omega \;d{\mathbb {P}}$, as claimed.

1.1.2 Relative pressure using dynamic balls

Here we give a dual definition of relative pressure using dynamic balls. The proof that it coincides with the notion of Sect. 9 can follow a standard route, and is left as an exercise to the reader. Fix $\varepsilon >0$ and $\omega \in \Omega$ and consider the sets ${{\mathcal {I}}}_\omega ^n= X_\omega \times \{n\}$ and ${{\mathcal {I}}}_\omega =X_\omega \times {\mathbb {N}}$. For every $\alpha \in {\mathbb {R}}$ and $N\geqslant 1$, define

$$\begin{aligned} m_\alpha (\omega ,f,\phi ,\Lambda ,\varepsilon ,N) = \inf _{{\mathcal {G}}_\omega } \Big \{ \sum _{{(x,n)} \in {\mathcal {G}}_\omega } e^{-\alpha n(x)+ S_{n(x)} \phi _\omega ({{\mathrm {B}}_\omega (x,n(x),\varepsilon )})} \Big \}, \end{aligned}$$

(9.3)

where the infimum is taken over all finite or countable families ${\mathcal {G}}_\omega \subset \cup _{n \geqslant N}{{\mathcal {I}}}_\omega ^n$ such that the collection of sets $\{B_\omega (x,n,\varepsilon ){:}\,(x,n)\in {\mathcal {G}}_\omega \}$ cover $\Lambda _\omega$ and, as before, $S_{n} \phi _\omega ({{\mathrm {B}}_\omega (x,n,\varepsilon )}) =\sup \{ \sum _{j=0}^{n-1}\phi _{\theta ^j(\omega )} (f_\omega ^j(y)){:}\,y \in B_\omega (x,n,\varepsilon )\}.$ Using that the previous sequence is increasing on N, the limit

$$\begin{aligned} m_\alpha (\omega , f,\phi ,\Lambda , \varepsilon ) = \lim _{N\rightarrow +\infty }m_\alpha (\omega ,f,\phi ,\Lambda ,\varepsilon ,N) \end{aligned}$$

(9.4)

does exist. If $\alpha$ is such that (9.4) is finite and $\beta >\alpha$ then $m_\beta (f,\phi ,\Lambda ,\varepsilon ,N) \leqslant e^{-(\beta -\alpha )N} m_\alpha (f,\phi ,\Lambda ,\varepsilon ,N)$ tends to zero as $N\rightarrow \infty$, hence $m_\beta (f,\phi ,\Lambda , \varepsilon )=0$. For that reason we define

$$\begin{aligned} \pi _\phi (\omega ,f,\Lambda ,\varepsilon ) = \inf {\{\alpha {:}\,m_\alpha (\omega ,f,\phi ,\Lambda , \varepsilon )=0\}}, \end{aligned}$$

which is decreasing on $\varepsilon$. The (random) relative pressure of the set $\Lambda$ with respect to $(f,\phi )$ is defined by

$$\begin{aligned} \pi _\phi (\omega ,f,\Lambda ) = \lim _{\varepsilon \rightarrow 0} \pi _\phi (\omega ,f,\Lambda ,\varepsilon ) . \end{aligned}$$

(9.5)

Remark 9.1

In the case that $\Lambda$ is compact and invariant, it follows from the general results on Carathéodory structures in [41] that the limit (9.3) can be computed using dynamic balls of the same length as

$$\begin{aligned} \pi _\phi (\omega ,f,\Lambda )=\limsup _{n\rightarrow \infty }\frac{1}{n} \log \Big [ \inf _{{\mathcal {G}}_\omega } \Big \{ \sum _{{(x,n)} \in {\mathcal {G}}_\omega } e^{S_{n} \phi _\omega ({{\mathrm {B}}_\omega (x,n,\varepsilon )})} \Big \} \Big ] \end{aligned}$$

Under this assumption and ergodicity of the measure ${\mathbb {P}}$, Bogenschütz and Ochs [10] proved that $\pi _\phi (f,\Lambda )= \pi _\phi (\omega ,f,\Lambda )$ for ${\mathbb {P}}$-almost every $\omega$.

Proposition 9.2

Let $(X_\omega )_\omega$ be a family of compact sets, $f=(f_\omega )_\omega$ be a family of continuous maps $f_\omega {:}\,X_\omega \rightarrow X_{\theta \omega }$, $\phi =(\phi _\omega )_\omega$ be a continuous potential and $\Lambda \subset X$ be an F-invariant set. Then $\pi _{S_\ell \phi }(f^\ell ,\Lambda )=\ell \pi _\phi (f,\Lambda )$ for every $\ell \geqslant 1$, where $f^\ell =(f_\omega ^\ell )_\omega$.

Proof

Fix $\ell \geqslant 1$. By continuity of each $f_\omega$ and compactness of the fibers $X_\omega$, given any $\rho >0$ there exists $\varepsilon (\omega ,\ell ,\rho )>0$ (depend) such that $d(x,y) < \varepsilon$ implies $d(f_\omega ^j(x),f_\omega ^j(y))<\rho$ for every $0 \leqslant j < \ell$ and every $0<\varepsilon \leqslant \varepsilon (\omega ,\ell ,\rho )$. It follows that

$$\begin{aligned} B_{\omega ,f}(x,\ell n,\varepsilon ) \subset B_{\omega , f^\ell }(x, n,\varepsilon ) \subset B_{\omega ,f}(x,\ell n, \rho ) \end{aligned}$$

(9.6)

for each $0<\varepsilon \leqslant \varepsilon (\omega ,\ell ,\rho )$ and $x\in X_\omega$, where $B_{\omega ,g}(x,n,\varepsilon )$ denotes the dynamic ball centered at x, of radius $\varepsilon$ and size n under the iteration of the map $g_{\theta ^{n-1} \omega } \dots g_{\theta \omega } g_\omega$.

First we prove the $\geqslant$ inequality. Fix $\omega \in \Omega$. Given $N\geqslant 1$ and any family ${\mathcal {G}}_\omega (\ell ) \subset \cup _{n \geqslant N} {{\mathcal {I}}}_n$ such that the balls $B_{\omega ,f^\ell }(x,j,\varepsilon )$ with $(x,j)\in {\mathcal {G}}_\omega (\ell )$ cover $\Lambda _\omega$, denote

$$\begin{aligned} {\mathcal {G}}_\omega :=\{(x,j\ell ){:}\,(x,j)\in {\mathcal {G}}_\omega (\ell )\}. \end{aligned}$$

The second inclusion in (9.6) ensures that the dynamic balls $B_{\omega ,f}(x,k,\rho )$ with $(x,k)\in {\mathcal {G}}_\omega$ cover $\Lambda _\omega$. In particular

$$\begin{aligned} \sum _{(x,j)\in {\mathcal {G}}_\omega (\ell )}&e^{-\alpha \ell j + \sum _{i=0}^{j-1} S_\ell \phi _\omega (f_\omega ^{i\ell }(\,B_{\omega ,f^\ell }(x,j,\varepsilon )\,))}\\&\qquad \geqslant \sum _{(x,k)\in {\mathcal {G}}_\omega } e^{-\alpha k + \sum _{i=0}^{k-1} \phi _{\theta ^i\omega }(f_\omega ^i(\,B_{\omega ,f}(x,k,\rho )\,)) - \sum _{i=0}^{k-1} \kappa _\rho (\theta ^i\omega )} \end{aligned}$$

where $\kappa _\rho (\omega )=\sup \{|\varphi _{\omega }(x)-\varphi _{\omega }(y)|{:}\,d(x,y)<\rho \}$. Ergodicity and Birkhoff’s ergodic theorem ensures that $\frac{1}{k}\sum _{i=0}^{k-1} \kappa _\rho \circ \theta ^i$ is almost everywhere convergent to the constant $\kappa _{\rho }^*=\int \kappa _\rho (\omega )\,d{{\mathbb {P}}}$. It is not hard to check that $\lim _{\rho \rightarrow 0} \kappa ^*_\rho =0$. Hence, given $\zeta >0$ it holds that

$$\begin{aligned} \sum _{(x,j)\in {\mathcal {G}}_\omega (\ell )}&e^{-\alpha \ell j + \sum _{i=0}^{j-1} S_\ell \phi _\omega (f_\omega ^{i\ell }(\,B_{\omega ,f^\ell }(x,j,\varepsilon )\,))}\\&\quad \geqslant \sum _{(x,k)\in {\mathcal {G}}_\omega } e^{-(\alpha +\kappa ^*_\rho -\zeta ) k + \sum _{i=0}^{k-1} \phi _{\theta ^i\omega }(f_\omega ^i(\,B_{\omega ,f}(x,k,\rho )\,))} \end{aligned}$$

provided that $N\geqslant 1$ is large enough. In other words, as ${\mathcal {G}}_\omega$ is arbitrary this proves

$$\begin{aligned} m_{\alpha \ell }(\omega , f^\ell ,S_\ell \phi ,\Lambda ,\varepsilon ,N) \geqslant m_{\alpha +\kappa ^*_\rho -\zeta }(\omega , f,\phi ,\Lambda ,\rho ,N\ell ), \end{aligned}$$

for every large $N\geqslant 1$. Since $\varepsilon (\omega ,\ell ,\rho ) \rightarrow 0$ as $\rho \rightarrow 0$ and $\zeta$ was taken arbitrary the latter proves the desired inequality.

For the $\leqslant$ inequality, we need to show the relative pressure is not affected if one restricts the infimum to families ${\mathcal {G}}_\omega$ of pairs (x, k) such that k is always a multiple of $\ell$. More precisely, let $m_\alpha ^\ell (\omega , f,\phi ,\Lambda ,\varepsilon , N)$ be the infimum over this subclass of families, and let $m_\alpha ^\ell (f,\phi ,\Lambda ,\varepsilon )$ be its limit as $N\rightarrow \infty$. $\quad \square$

Lemma 9.3

We have $m_\alpha ^\ell (\omega , f,\phi ,\Lambda ,\varepsilon ) \leqslant m_{\alpha -\rho }(\omega , f,\phi ,\Lambda ,\varepsilon )$ for every $\rho >0$.

Proof

Fix $\rho >0$. It is enough to prove that $m_\alpha ^\ell (f,\phi ,\Lambda ,\varepsilon ,N) \leqslant m_{\alpha -\rho }(f,\phi ,\Lambda ,\varepsilon ,N)$ for every large N. Let $N\geqslant 1$ be large so that $N\rho > \ell (\alpha + \sup |\phi _\omega |)$. Given any ${\mathcal {G}}_\omega \subset \cup _{n \geqslant N} {{\mathcal {I}}}_n$ such that the balls $B_{\omega ,f}(x,k,\varepsilon )$ with $(x,k)\in {\mathcal {G}}_\omega$ cover $\Lambda _\omega$, define ${\mathcal {G}}'_\omega$ to be the family of all $(x,k')$, $k'=\ell [k/\ell ]$ such that $(x,k)\in {\mathcal {G}}_\omega$. Notice that

$$\begin{aligned} -\alpha k' + S_{k'}\phi _\omega (x) \leqslant -\alpha k + \alpha \ell + S_k\phi _\omega (x) + \ell \sup |\phi _\omega | \leqslant (-\alpha +\rho ) k + S_k\phi _\omega (x) \end{aligned}$$

given that $k\geqslant N$. The lemma follows immediately. $\quad \square$

Let us resume the proof of the proposition. Take an arbitrary $0<\varepsilon \leqslant \varepsilon (\omega ,\ell ,\rho )$. Let ${\mathcal {G}}'_\omega$ be any family of pairs (x, k) with $k\geqslant N\ell$ and such that every k is a multiple of $\ell$. Define ${\mathcal {G}}_\omega (\ell )$ to be the family of pairs (x, j) such that $(x,j\ell )\in {\mathcal {G}}'_\omega$. The first inclusion in (9.6) ensures that if the balls $B_{\omega ,f}(x,k,\varepsilon )$ with $(x,k)\in {\mathcal {G}}'_\omega$ cover $\Lambda _\omega$ then so do the balls $B_{\omega ,f^\ell }(x,j,\varepsilon )$ with $(x,j)\in {\mathcal {G}}_\omega (\ell )$. Then

$$\begin{aligned} \sum _{(x,k)\in {\mathcal {G}}'_\omega }&e^{-\alpha k + \sum _{i=0}^{k-1} \phi _{\theta ^i\omega }(f_\omega ^i(\, B_{\omega ,f}(x,k,\varepsilon )\,))} \\&\quad \geqslant \sum _{(x,j)\in {\mathcal {G}}_\omega (\ell )} e^{-\alpha \ell j + \sum _{i=0}^{j-1} S_\ell \phi _\omega (f_\omega ^{i\ell }(\, B_{\omega ,f^\ell }(x,j,\varepsilon )\,))- \sum _{i=0}^{\ell j-1} \kappa _\rho (\theta ^i\omega )}. \end{aligned}$$

Since ${\mathcal {G}}_\omega (\ell )$ is arbitrary, an argument identical to the previous one ensures that

$$\begin{aligned} m_\alpha ^\ell (\omega , f,\phi ,\Lambda ,\varepsilon ,N\ell ) \geqslant m_{\alpha \ell }(\omega ,f^\ell ,S_\ell \phi ,\Lambda ,\varepsilon ,N). \end{aligned}$$

Taking the limit when $N\rightarrow \infty$ and using Lemma 9.3,

$$\begin{aligned} m_{\alpha -\rho }(\omega , f,\phi ,\Lambda ,\varepsilon ) \geqslant m_\alpha ^\ell (\omega , f,\phi ,\Lambda ,\varepsilon ) \geqslant m_{\alpha \ell }(\omega , f^\ell ,S_\ell \phi ,\Lambda ,\varepsilon ). \end{aligned}$$

It follows that $\ell \big (\pi _\phi (\omega , f,\Lambda ,\varepsilon ) + \rho \big ) \geqslant \pi _{S_\ell \phi }(f^\ell , \Lambda ,\varepsilon )$. Since $\rho$ is arbitrary and $\varepsilon (\omega ,\ell ,\rho )$ tends to zero as $\rho \rightarrow 0$ we conclude that $\pi _{S_\ell \phi }(f^\ell ,\Lambda ) \geqslant \ell \,\pi _\phi (f, \Lambda )$. $\quad \square$

1.2 A variational principle for relative pressure

The main results of this appendix are Theorems 9.4 and 9.8 which, combined, provide a variational principle for the relative pressure of random dynamical systems.

Theorem 9.4

Let $\phi \in L_{X}^1(\Omega , C(M))$ and let $\Lambda \subset X$ be a measurable set so that $F(\Lambda )\subseteq \Lambda$. Then

$$\begin{aligned} \pi _\phi (f,\Lambda )\geqslant \sup _{\mu (\Lambda )=1} \Big \{h_{\mu }(f)+\int _\Lambda \phi \, d\mu \Big \}. \end{aligned}$$

Proof

Let $\mu \in {\mathcal {M}}(X,f)$ with $\mu (\Lambda )=1$ and denote by $\mu =(\mu _\omega )_{\omega }$ the disintegration of $\mu$. Assume, without loss of generality, that $\mu$ is ergodic. In fact, Lemma 6.19 in [15] ensures that $\zeta \in {\mathcal {M}}(X,f)$ is an extreme point of ${\mathcal {M}}(X,f)$ if and only if $\zeta$ is ergodic. This, together with Choquet representation theorem, provides an ergodic decomposition theorem for all measures in $\zeta \in {\mathcal {M}}(X,f)$. Since $\mu (\Lambda )=1$, we can obtain the ergodic decomposition on some partition of $\Lambda$. Let $\eta$ be the partition of $\Lambda$ which induces ergodic components $\Lambda _s$ for $s\in S$, where S denotes some index set. Let us denote by $\mu _s$ the probability measure on $\Lambda _s$ and by $\nu$ the probability measure on the quotient space ${{\tilde{\Lambda }}}=\Lambda /\eta$. By convexity of the maps $\mu \mapsto h_\mu (f)$ and $\mu \mapsto \int \phi \,d\mu$,

$$\begin{aligned} h_\mu (f)=\int _{{{\tilde{\Lambda }}}}h_{\mu _s}(f)d\nu (s),\ \ \ \int _{\Lambda }\phi \,d\mu =\int _{\tilde{\Lambda }}\Bigl (\int _{\Lambda _s}\phi d\mu _s\Bigr ) \,d\nu (s). \end{aligned}$$

In consequence there is an ergodic component $\mu _s$ such that $h_{\mu _s}(f)+\int _{\Lambda _s}\phi \, d\mu _s\geqslant h_\mu (f)+\int _{\Lambda }\phi \, d\mu$, proving our claim.

Throughout, assume that $\mu \in {\mathcal {M}}(X,f)$ is F-invariant and ergodic. Denote by $\mu _{\Omega }(\cdot )=\mu (\Omega \times \cdot )$ the probability measure obtained as marginal of $\mu$ on M. Since the measure $\mu _{\Omega }$ is regular, we can verify the following lemma as in the deterministic case (see [42, Lemma 1]). $\quad \square$

Lemma 9.5

For every $\varepsilon >0$, there are a positive number $0<\delta <\varepsilon$, a Borel partition $\xi =\{C_1, \ldots , C_m\}$ of M, and a finite open covering ${\mathcal {U}}=\{U_1,\ldots , U_k\}$ of M with $k\geqslant m$ such that

• (1) ${\text {diam}}(U_i), {\text {diam}}(C_j)\leqslant \varepsilon$ for $i=1,\ldots ,k$ and $j=1,\ldots ,m$.

• (2) ${\bar{U}}_i\subset C_i$ for $i=1, \ldots , m$.

• (3) $\mu _{\Omega }(C_i{\setminus } U_i)<\delta$ for $i=1,\ldots ,m$

• (4) $\mu _{\Omega }(\bigcup _{i=m+1}^{k}U_i)<\delta$.

• (5) $2\delta \log m <\varepsilon$.

Now, fix $\varepsilon >0$ and let $\delta >0$, the partition $\xi$ and the covering ${\mathcal {U}}$ be given by Lemma 9.5. By Birkhoff’s ergodic theorem, for ${{\mathbb {P}}}$-a.e. $\omega \in \Omega$, there are $N_1(\omega )\geqslant 1$ and $A_1(\omega )\subset X_\omega$ with $\mu _\omega (A_1(\omega ))>1-\delta$ such that

$$\begin{aligned} \frac{1}{n}\#\Big \{ 0\leqslant l \leqslant n-1 {:}\, f_\omega ^l(y)\in \bigcup _{i=m+1}^{k}U_i \Big \} <2\delta \end{aligned}$$

(9.7)

for every $x\in A_1(\omega )$ and every $n\geqslant N_1(\omega )$.

Given $\omega \in \Omega$, $n\geqslant 1$ and $x\in M$, consider the partition

$$\begin{aligned} \xi _{n}(\omega )=\xi _\omega \bigvee f_{\omega }^{-1}\xi _{\theta \omega }\bigvee \cdots \bigvee (f_{\omega }^{n-1})^{-1}\xi _{\theta ^{n-1}\omega }, \end{aligned}$$

and let $\xi _{n}(\omega )(x)$ denote the element of the partition $\xi _{n}(\omega )$ which contains x. By the Shannon–Macmillan–Breiman theorem for random dynamical systems (cf. Proposition 2.1 in [50]), for ${{\mathbb {P}}}$-a.e. $\omega \in \Omega$, there are $N_2(\omega )\geqslant 1$ and $A_2(\omega )\subset X_\omega$ with $\mu _\omega (A_2(\omega ))>1-\delta$ so that

$$\begin{aligned} \mu _\omega (\xi _{n}(\omega )(x)) \leqslant \exp \Bigl (-(h_\mu (f,\xi )-\delta )n\Bigr ) \end{aligned}$$

(9.8)

for every $x\in A_2(\omega )$ and $n\geqslant N_2(\omega )$. Using once more Birkhoff’s ergodic theorem, for ${{\mathbb {P}}}$-a.e. $\omega \in \Omega$ there are $N_3(\omega )\geqslant 1$ and $A_3(\omega )\subset X_\omega$ with $\mu _\omega (A_3(\omega ))>1-\delta$ such that if $x\in A_3(\omega )$ then

$$\begin{aligned} \Bigl |\frac{1}{n}\sum _{i=0}^{n-1}\phi _{\theta ^n\omega }(f_{\omega }^i(x))-\int _\Lambda \phi \,d\mu \Bigr |<\delta \quad \text {and}\quad \Bigl |\frac{1}{n}\sum _{i=0}^{n-1}\kappa _{\varepsilon }(\theta ^i\omega )-\int _\Omega \kappa _{\varepsilon }^*(\omega ){{\mathbb {P}}}\Bigr |<\delta \end{aligned}$$

(9.9)

for every $n\geqslant N_3(\omega )$, where $\kappa _\varepsilon (\omega )=\sup \{|\varphi _{\omega }(x)-\varphi _{\omega }(y)|{:}\,d(x,y)<\varepsilon \}$ and the function $\kappa _{\varepsilon }^*\in L^1({{\mathbb {P}}})$ is $\theta$-invariant function and satisfies $\int _\Omega \kappa _{\varepsilon }^*(\omega )d{{\mathbb {P}}}=\int _\Omega \kappa _{\varepsilon }(\omega )d{{\mathbb {P}}}$.

Remark 9.6

We note that $0\leqslant \kappa _{\varepsilon }\leqslant \kappa _{\varepsilon '}$ whenever $0<\varepsilon \leqslant \varepsilon '$. This implies that $0\leqslant \kappa _{\varepsilon }^*\leqslant \kappa _{\varepsilon '}^*$ if $0<\varepsilon \leqslant \varepsilon '$. It is not hard to check that $\lim _{n\rightarrow \infty }\int _{\Omega }\kappa _{\varepsilon }^*(\omega )d{{\mathbb {P}}}=0$, since $\lim _{\varepsilon \rightarrow 0}\kappa _{\varepsilon }(\omega )=0$ for ${{\mathbb {P}}}$-a.e. $\omega$ and $\int _\Omega \kappa _{\varepsilon }^*(\omega )d{{\mathbb {P}}}=\int _\Omega \kappa _{\varepsilon }(\omega ) d{{\mathbb {P}}}$.

Now, if $N(\omega )=\max _{1\leqslant i \leqslant 3}\{N_i(\omega )\}$ and $A_\omega =\bigcap _{i=1}^{3}A_i(\omega )$, then $\mu _\omega (A_\omega )\geqslant 1-3\delta$. Moreover, taking $\alpha <h_\mu (f,\xi )+\int _{\Lambda }\phi \,d\mu -\int _\Omega \kappa _{\varepsilon }^* d{{\mathbb {P}}}-3\delta -\varepsilon$ and $N\geqslant N(\omega )$, by (9.1) there exists a finite covering ${\mathcal {G}}_\omega \subset {\mathcal {S}}_N {\mathcal {U}}(\omega )$ of $\Lambda _\omega$ so that $n( {U})\geqslant N$ for ${U}\in {\mathcal {G}}_\omega$ and

$$\begin{aligned} \Bigl |\sum _{ {U}\in {\mathcal {G}}_\omega }\exp \Bigl (-\alpha n( {U})+S_{n( {U})}\phi _\omega ( {U})\Bigr )-m_{\alpha }(f,\phi ,\Lambda ,{\mathcal {U}},\omega )\Bigl |<\delta . \end{aligned}$$

Let ${\mathcal {G}}_\omega (l)$ be the subset of ${\mathcal {G}}_\omega$ each of whose elements ${U}$ satisfies $n( {U})=l$ and ${U}\cap A_\omega \ne \emptyset$, and set $Y_\omega (l)=\bigcup _{ {U}\in {\mathcal {G}}_\omega } {U} \subset X_\omega$. A standard application of Shannon–Macmillan–Breiman’s theorem, entirely analogous to Lemma 2 in [41, Appendix II], guarantees that

$$\begin{aligned} \sharp {\mathcal {G}}_\omega (l)\geqslant \mu _{\omega }(Y_\omega (l)\cap A_\omega ) \cdot \exp \Bigl ((h_\mu (f,\xi )-\delta -2\delta \log \#\xi )l\Bigr ) \end{aligned}$$

(9.10)

for every $l\geqslant N$. Together with the estimates in (9.9), the fact that every ${\underline{U}}\cap A_\omega \ne \emptyset$ for every ${\underline{U}}\in {\mathcal {G}}_\omega$ and that all elements in ${\mathcal {U}}$ have diameter smaller or equal to $\varepsilon$, this yields

$$\begin{aligned} \begin{aligned} m_\alpha (\omega , f,\phi ,\Lambda ,{\mathcal {U}},N) +\delta&>\sum _{ {U}\in {\mathcal {G}}_\omega }\exp \Bigl (-\alpha n( {U})+S_{n( {U})}\phi _\omega ( {U})\Bigr )\\&= \sum _{l=N}^{\infty }\sum _{ {U}\in {\mathcal {G}}_\omega (l)}\exp \Bigl (-\alpha l+S_{l}\phi _\omega ( {U}) \Bigr ) \\&\geqslant \sum _{l=N}^{\infty }\sum _{ {U}\in {\mathcal {G}}_\omega (l)} \exp \Bigl \{\Bigl (-\alpha + \int _{\Lambda }\phi \, d\mu -\int _{\Omega }\kappa _\varepsilon ^*(\omega )d{{\mathbb {P}}}-2\delta \Bigr )l\Bigr \}. \end{aligned} \end{aligned}$$

Using (9.10), item (5) in Lemma 9.5, that $0<\delta <\varepsilon$ and the choice of $\alpha$ we get

$$\begin{aligned} \begin{aligned} m_\alpha&(\omega , f,\phi ,\Lambda ,{\mathcal {U}},N) +\delta \\&\geqslant \sum _{l=N}^{\infty } \mu _\omega (Y_\omega (l)\cap A_\omega ) \exp \Bigl \{\Bigl (-\alpha + h_\mu (f,\xi )+\int _{\Lambda }\phi \,d\mu -\int _{\Omega }\kappa _\varepsilon ^*(\omega )d{{\mathbb {P}}}-3\delta -\varepsilon \Bigr )l\Bigr \} \\&\geqslant \sum _{l=N}^{\infty }\mu _\omega (Y_\omega (l)\cap A_\omega )\geqslant \mu _\omega (A_\omega ) \geqslant 1-3\delta . \end{aligned} \end{aligned}$$

Since $m_\alpha (\omega , f,\phi ,\Lambda ,{\mathcal {U}},N) \geqslant 1-4\delta$ for every large N it follows that

$$\begin{aligned} \pi _{\phi }(f,\Lambda ,{\mathcal {U}},\omega )\geqslant h_\mu (f,\xi )+\int _{\Lambda }\phi \,d\mu -\int _{\Omega }\kappa _{\varepsilon }^*(\omega )d{{\mathbb {P}}}-4\varepsilon . \end{aligned}$$

Taking the limit as $\varepsilon$ tends to zero (choosing appropriate coverings ${\mathcal {U}}_\varepsilon$ and recalling Remark 9.6) we get $\pi _{\phi }(\omega , f,\Lambda )\geqslant h_\mu (f)+\int _{\Lambda }\phi \,d\mu$, which proves the theorem. $\quad \square$

Observe that it may occur that $\pi _\phi (f,\Lambda ) > \sup _{\mu (\Lambda )=1} \big \{h_{\mu }(f)+\int _\Lambda \phi \, d\mu \big \}$ (we refer the reader to [41] for such examples in the deterministic context of subshifts of finite type). Hence, the converse inequality in Theorem 9.4 may fail. Nevertheless, the previous partial variational principle is sufficient to prove that topological pressure can be computed as the maximum of the relative pressure of an invariant set and its complement. More precisely:

Proposition 9.7

Assume that $\Lambda \subset \Gamma$ are F-invariant measurable subsets of X. Then the following properties hold:

• (1) $\pi _{\phi }(\omega , f,\Lambda , {\mathcal {U}}) \leqslant \pi _{\phi }(\omega , f,\Gamma ,{\mathcal {U}})$ for every finite covering ${\mathcal {U}}$ of M,

• (2) $\pi _{\phi }( f,\Lambda ) \leqslant \pi _{\phi }(f,\Gamma )$,

• (3) $\pi _\phi (f,X)=\max \{\pi _\phi (f,\Lambda ), \pi _\phi (f,\Lambda ^c) \}$.

Proof

Assume that $\Lambda \subset \Gamma \subset X$ are measurable sets and ${\mathcal {U}}$ is a finite covering of M. Item (1) is a direct consequence of the monotonicity condition of the exterior measures $m_\alpha (\omega , f, \phi ,\Lambda ,{\mathcal {U}},N) \leqslant m_\alpha (\omega , f, \phi ,\Gamma ,{\mathcal {U}},N)$. Item (2) is a direct consequence of item (1). Finally, on the one hand item (2) implies $\pi _\phi (f,X) \geqslant \max \{\pi _\phi (f,\Lambda ), \pi _\phi (f,\Lambda ^c) \}$. On the other hand, since each $X_\omega$ is compact, the variational principle in (4.4) together with Theorem 9.4 ensures that

$$\begin{aligned} \pi _\phi (f)=&\sup _{\mu \in {\mathcal {M}}(X, f)} \Big \{h_{\mu }(f)+\int \phi \,d\mu \Big \} = \sup _{\mu \in {\mathcal {M}}_e(X, f)} \Big \{h_{\mu }(f)+\int \phi \,d\mu \Big \} \\&= \max \Big \{ \sup _{\mu (\Lambda )=1} \Big \{h_{\mu }(f)+\int \phi \,d\mu \Big \}, \; \sup _{\mu (\Lambda ^c)=1} \Big \{h_{\mu }(f)+\int \phi \,d\mu \Big \} \Big \}\\&\leqslant \max \{\pi _\phi (f,\Lambda ), \pi _\phi (f,\Lambda ^c) \}. \end{aligned}$$

This proves item (3) and finishes the proof of the proposition. $\quad \square$

Finally, although not needed here, we finish this section with the following converse of Theorem 9.4 for product measures, yielding a variational principle for the relative entropy of random dynamical systems.

Let $\Lambda \subset \Omega \times X$ be a measurable set so that $F(\Lambda )\subseteq \Lambda$. For each $(\omega ,x)\in \Lambda$ and $n\geqslant 1$ consider the empirical measures $\mu _n(\omega ,x)=\frac{1}{n}\sum _{j=0}^{n-1} \delta _{F^j (\omega ,x)},$ and denote by $V(\omega ,x) \subset M(X,f)$ the set of F-invariant probability measures obtained as weak$^*$ accumulation points of $(\mu _n(\omega ,x))_{n\geqslant 1}$. As these measures may not be supported on $\Lambda$, consider the set

$$\begin{aligned} {\mathcal {E}}(\Lambda )=\{(\omega ,x)\in \Lambda {:}\, V(\omega ,x) \cap \mathcal M(\Lambda ,f)\ne \emptyset \} \end{aligned}$$

of points whose empirical measures have some accumulation measure supported on $\Lambda$. We have the following variational principle:

Theorem 9.8

Let $\phi \in L_{X}^1(\Omega , C(M))$ and let $\mu \in M(X,f)$ be a probability measure with $\mu (\Lambda )=1$. Then

$$\begin{aligned} \pi _{\phi }(f, \Lambda ) \geqslant \pi _{\phi }(f, {\mathcal {E}}(\Lambda )) = \sup \Big \{ h_\mu (f) + \int \phi \, d\mu {:}\, \mu (\Lambda )=1\Big \}. \end{aligned}$$

Proof

The inequalities $\pi _{\phi }(f, \Lambda ) \geqslant \pi _{\phi }(f, {\mathcal {E}}(\Lambda )) \geqslant \sup \Big \{ h_\mu (f) + \int \phi \, d\mu {:}\, \mu (\Lambda )=1\Big \}$ follow immediately from Proposition 9.7 (2) and Theorem 9.4. Thus we are left to prove that $\pi _{\phi }(f, {\mathcal {E}}(\Lambda )) \leqslant \sup \Big \{ h_\mu (f) + \int \phi \, d\mu {:}\, \mu (\Lambda )=1\Big \}.$ Actually, we will prove that there exists a ${{\mathbb {P}}}$-full measure subset $\Omega _0$ so that

$$\begin{aligned} \pi _{\phi }(f, {\mathcal {E}}(\Lambda )\cap (\Omega _0\times M))\leqslant h_{\mu }(f)+\int _{\Omega }\phi \ d{{\mathbb {P}}}. \end{aligned}$$

(9.11)

We note that, by the definition of topological pressure, $\pi _{\phi }(f,Y)=\int _{\Omega }\pi _{\phi }(f, Y,\omega ) d{{\mathbb {P}}}$. Hence, (9.11) is sufficient to prove the theorem as removing a set of the form $Z\times M$ from Y, where Z is a zero set with respect to ${{\mathbb {P}}}$, will not change its value.

The strategy is a modification of the arguments in [42, Theorem 2]. For the proof, we need two auxiliary results. Let E be a finite set and ${i}=(i_0, \dots , i_{k-1})\in E^k$. Denote by $\mu _{ {i}}$ the measure on E given by

$$\begin{aligned} \mu _{ {i}}(e)=\frac{1}{k}\#\{0\leqslant j \leqslant k-1{:}\, i_j=e\}. \end{aligned}$$

Put

$$\begin{aligned} H(\mu _{ {i}})=-\sum _{e\in E}\mu _{ {i}}(e)\log \mu _{ {i}}(e) \quad \text {and}\quad R(k,h,E)=\{ {i}\in E^k:\ H(\mu _{ {i}}) \leqslant h\}. \end{aligned}$$

Some combinatorial arguments yield the following lemma: $\quad \square$

Lemma 9.9

[12, Lemma 2.16]. We have

$$\begin{aligned} \limsup _{k\rightarrow \infty }\frac{1}{k}\log \# R(k,h,E)\leqslant h. \end{aligned}$$

Let ${\mathcal {U}}=\{U_1,\dots , U_r\}$ be an open cover of M with $\delta :={\text {diam}}{\mathcal {U}}>0$ and fix $\varepsilon >0$. As before, for each $\omega \in \Omega$ and every string ${a}=(U_0, U_{1}, \ldots , U_{N-1})$ with $U_{i}\in {\mathcal {U}}$, we set ${U}( {a})= \{x\in X_\omega {:}\,f_{\omega }^j(x)\in U_j\ \text {for}\, j=0, \ldots , N-1\}$ and $n( {U}( {a}))=N$ (we omit the dependence on $\omega$ for notational simplicity), and recall that $\kappa _\delta (\omega )=\sup \{|\varphi _{\omega }(x)-\varphi _{\omega }(y)|{:}\,d(x,y)<\delta \}$. Endow the space $(\Omega ,{\mathcal {F}},{{\mathbb {P}}})$ with the topology via the definition of the Lebesgue space.

Proposition 9.10

There exists a full ${{\mathbb {P}}}$-measure subset $\Omega _1\subset \Omega$ such that for each $(\omega ,x)\in {\mathcal {E}}(\Lambda ) \cap (\Omega _1\times M)$ and $\mu \in V(\omega ,x)\cap M(\Lambda ,f)$ there exists an integer $m=m(\omega ,x,\mu )\geqslant 1$ satisfying that: for every positive integer $n \geqslant 1$ there are $N\geqslant n$ and a string ${a}$ with $n( {U}( {a}))=N$ such that:

• (1) $x\in {U}( {a})$,

• (2) $\sup _{z\in {U}( {a})}\sum _{k=0}^{N-1}\phi _{\theta ^k\omega }(f_{\omega }^k(z)) \leqslant N\Bigl (\int _{\Omega \times M} \phi \,d\mu +\frac{1}{N}\sum _{i=0}^{N-1}\kappa _\delta (\theta ^i\omega ) +\varepsilon \Bigr )$

• (3) the string ${a}$ contains a substring ${a}'$ with $n( {U}( {a}'))=km\geqslant N-m$ such that

  $$\begin{aligned} \frac{1}{m}H( {a}')\leqslant h_{\mu }(f)+\varepsilon . \end{aligned}$$

Proof

Take a Borel partition $\zeta =\{C_1,\dots , C_r\}$ of M such that $\overline{C_i}\subset U_i$ and consider the partition of $\Omega \times M$ given by $\Omega \times \zeta =\{\Omega \times C_1,\dots ,\Omega \times C_r\}$. Notice that the supremum in the definition of the random measure theoretic entropy $h_{\mu }(f)=\sup _{\xi }h_{\mu }(f,\xi )$ can be taken over a finite partition whose elements are of the form $\Omega \times \xi$, where $\xi$ is a partition of M. This, together with (4.1), ergodicity and invertibility of $(\theta ,{{\mathbb {P}}})$ and Kingman’s subadditive ergodic theorem, implies that there exists such a partition $\zeta$ satisfying

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{1}{n} H_{\mu _\omega } \Big (\bigvee _{k=0}^{n-1}(f^k_\omega )^{-1}\zeta \Big ) = h_{\mu }(f,\zeta ) \leqslant h_\mu (f)+\frac{\varepsilon }{2} \qquad {{\mathbb {P}}}-\text {a.e.} \, \omega . \end{aligned}$$

(see e.g. proof of Proposition 2.2 in [50]). In particular, there is a measurable set $\Omega _1\subset \Omega$ with ${{\mathbb {P}}}(\Omega _1)=1$ so that, for every $\omega \in \Omega _1$, there exists $m=m(\omega )\geqslant 1$ satisfying

$$\begin{aligned} \frac{1}{m}H_{\mu _\omega }\bigl (\zeta ^{(m)}(\omega )\bigr ) \leqslant h_\mu (f)+\frac{\varepsilon }{2}, \end{aligned}$$

(9.12)

where $\zeta ^{(m)}(\omega )=\bigvee _{i=0}^{m-1}(f_\omega ^i)^{-1}\zeta$. Now, fix $(\omega ,x)\in \Lambda \cap (\Omega _1\times M$) and $\mu \in V(\omega ,x)\cap M(\Lambda ,f)$. Take a positive integer sequence $n_i\rightarrow \infty$ such that $\mu _{n_i}(\omega ,x)\rightarrow \mu$ in the weak * topology. We may assume without loss of generality that $n_j=mk_j$, where $m=m(\omega )$ and $k_j\geqslant 1$.

Now, fix $\beta >0$ arbitrary. By regularity of $\mu$ (hence of $\mu _\Omega (\cdot ):=\mu (\Omega \times \cdot ))$, there exists a compact set $K_i\subset C_i$ with $\mu (\Omega \times (C_i{\setminus } K_i)) < \beta \mu (\Omega \times C_i)$. Choosing an element $B_i$ of the open cover $\bigvee _{i=0}^{m-1}(f_\omega ^i)^{-1}{\mathcal {U}}$ which contains $K_i$, one can find a Borel set $V_i^*$ such that $K_i\subset V_i^*\subset B_i$ and $\{V_i^*\}$ is a Borel partition of M. For each $1\leqslant i \leqslant r$, every $n_j=mk_j$ and $0\leqslant q\leqslant m-1$ we set

$$\begin{aligned} M_i^{(j)}=\# \{0\leqslant s \leqslant n_{j}-1{:}\, f_{\omega }^s(x)\in V_{i}^*\} \end{aligned}$$

and

$$\begin{aligned} M_{i,q}^{(j)}=\#\{0\leqslant s\leqslant n_{j}-1{:}\, f_{\omega }^s(x)\in V_{i}^*, s=q\ (\text {mod}\ m)\}. \end{aligned}$$

The latter describe the frequency of visits of the random orbit to a pre-determined sequence of Borel sets considered above. Set also

$$\begin{aligned} p_{i}^{(j)}=M_{(i_1,i_2, \dots , i_{n_j})}^{(j)}/n_j \quad \text {and}\quad p_{i,q}^{(j)}=M_{(i_1,i_2, \dots , i_{n_j}),q}^{(j)}/k_j. \end{aligned}$$

As $\mu _{n_j}(\omega ,x)\rightarrow \mu$ as $j\rightarrow \infty$ we deduce that, taking continuous bump functions supported on $V_i^*$,

$$\begin{aligned} \liminf _{j\rightarrow \infty } p_{i}^{(j)}&= \liminf _{j\rightarrow \infty } \frac{1}{n_j}\sum _{k=0}^{n_j-1} \chi _{V_i^*} (F^k(\omega ,x)) \geqslant \mu (\Omega \times K_i) \\&\geqslant (1-\beta ) \, \mu (\Omega \times C_i). \end{aligned}$$

Similarly,

$$\begin{aligned} \limsup _{j\rightarrow \infty } p_{i}^{(j)} \leqslant \int _{\Omega }\mu _{\omega }(K_i)d{{\mathbb {P}}}+\sum _{i\ne j}\int _\Omega \mu _{\omega }(C_j{\setminus } K_j) \leqslant \mu (\Omega \times C_i)+r \beta . \end{aligned}$$

Therefore, taking j sufficiently large and $\beta >0$ sufficiently small, we have

$$\begin{aligned} -\frac{1}{m}\sum _{i}p_i^{(j)}\log p_i^{(j)} \leqslant -\frac{1}{m}\sum _{i}\mu _{\Omega }(C_i)\log \mu _{\Omega }(C_i)+\frac{\varepsilon }{2}\leqslant h_{\mu }(f)+\varepsilon . \end{aligned}$$

The rest part of the proof of the first and the third statement is the same as in the proof of Lemma 2.15 in [12]. By convexity of the function $c(x)=-x\log x$, we have

$$\begin{aligned} c(p_i^{(j)})\geqslant \frac{1}{m}\sum _{q=0}^{m-1} c(p_{i,q}^{(j)}) \end{aligned}$$

and hence $\sum _{i=1}^{t}c(p_i^{(j)})\geqslant \frac{1}{m}\sum _{q=0}^{m-1} \sum _{i=1}^{t}c(p_{i,q}^{(j)}).$ This ensures that there exists $0\leqslant q\leqslant m-1$ such that

$$\begin{aligned} \frac{1}{m}\sum _{i=1}^{t}c(p_{i,q}^{(j)})\leqslant \frac{1}{m}\sum _{i=1}^{t}c(p_i^{(j)})\leqslant h_\mu (f)+\varepsilon . \end{aligned}$$

Set $N=n_j+q$. For $s<q$ we chose $U_s\in {\mathcal {U}}$ so that $f_\omega ^s(x)\in U_s$. For every $V_{i}^{*}$, we take a string ${a}_i$ so that $V^{*}_{i} \subset B_i= {U}( {a}_i)$. For $s\geqslant q$ we write $s=q+mp+e$ with $p\geqslant 0$ and $0\leqslant e<m$ and set $U_s=U_{e,i}$, where i is chosen such that $f_\omega ^{q+mp}(x)\in V_i^*$. Set ${a}_p=U_{0,i}\dots U_{m-1, i}$ and take the substring ${a}$ as $U_0\dots U_{q-1} {a}_0\ldots {a}_{k_j-1}$. Then for ${a}'= {a}_0\dots {a}_{k_j-1}$ the measure $\mu _{ {a}'}$ is given by the probabilities $p_{i,q}^{(j)}$ and it satisfies

$$\begin{aligned} \frac{1}{m} H( {a}')=\frac{1}{m} \sum _{i=0}^{t} c(p_{i,q}^{(j)})\leqslant h_{\mu }(f)+\varepsilon , \end{aligned}$$

which ensures the first and the third statements. The second statement follows from the weak$^*$ convergence $\mu _{n_j}(\omega ,x)\rightarrow \mu$ as $j\rightarrow \infty$ and the definition of the function $\kappa _\delta (\omega )$. $\quad \square$

We are now in a position to complete the proof of Theorem 9.8, which was reduced to the proof of the claim in (9.11). In what follows, take

$$\begin{aligned} P:=\sup \Big \{ h_\mu (f) + \int \phi \, d\mu {:}\, \mu (\Lambda )=1\Big \}. \end{aligned}$$

By ergodicity of $(\theta , {{\mathbb {P}}})$, we can take a measurable set $\Omega _2\in {\mathcal {F}}$ with ${{\mathbb {P}}}(\Omega _2)=1$ satisfying that for $\varepsilon >0$ and $\omega \in \Omega _2$ there is an integer $n(\varepsilon ,\omega )\geqslant 1$ such that

$$\begin{aligned} \Big |\frac{1}{n}\sum _{i=0}^{n-1}\kappa _{\delta }(\theta ^{i} \omega )-\int _\Omega \kappa _{\delta }(\omega )\ d{{\mathbb {P}}}\Big |\leqslant \varepsilon \end{aligned}$$

for $n\geqslant n(\varepsilon ,\omega )$. Set also $\Omega _0=\Omega _1\cap \Omega _2$ [recall (9.12)].

For each $m\geqslant 1$, denote by $G_m$ the set of all points $(\omega ,x)\in {\mathcal {E}}(\Lambda ) \cap (\Omega _0\times M)$ for which there exists $\mu \in V(\omega ,x)\cap M(\Lambda ,f)$ and the statement in Proposition 9.10 holds for the positive integer m. Moreover, taking $u\in {\mathbb {R}}$, put

$$ \begin{aligned} G_{m,u} =\Big \{ (\omega ,x)\in G_m {:}\, \exists \mu \in V(\omega ,x)\cap M(\Lambda ,f) \; \& \; \int _{\Lambda }\phi \ d\mu \in [u-\varepsilon ,u+\varepsilon ] \Big \}. \end{aligned}$$

Observe that for each $(\omega ,x)\in G_{m,u}$ and probability $\mu \in V(\omega ,x)\cap M(\Lambda ,f)$ satisfying $\int _{\Lambda }\phi \ d\mu \in [u-\varepsilon ,u+\varepsilon ]$ one has that $h_\mu (f) \le P - u + \varepsilon$. Moreover, taking $c=\int _\Omega |\phi _\omega |_{\infty }\ d{{\mathbb {P}}}$ and an $\varepsilon$-dense subset $\{u_1,\dots ,u_s\}$ of the interval $[-c,c]$ it follows, by construction, that

$$\begin{aligned} {\mathcal {E}}(\Lambda ) \cap (\Omega _0\times M)=\bigcup _{m\geqslant 1}\bigcup _{i=1}^{s}G_{m,u_i} \end{aligned}$$

(9.13)

Since the quantity $\pi _\phi (\omega ,f,{\mathcal {E}}(\Lambda ) \cap (\Omega _0\times M))$ is generated by some Carathéodory structure then

$$\begin{aligned} \pi _\phi (\omega ,f,{\mathcal {E}}(\Lambda ) \cap (\Omega _0\times M),{\mathcal {U}})=\sup _{i,m} \pi _\phi (\omega ,f,G_{m,u_i},{\mathcal {U}}) \end{aligned}$$

(9.14)

(see e.g. [41]).

We now fix $m\geqslant 1$ and $u\in {\mathbb {R}}$ and proceed to give an upper bound for the pressure of any non-empty set $G_{m,u}$. For each $N\geqslant 1$, denote by ${\mathcal {G}}_{m,u}(\omega ,N)$ the set of all strings ${\underline{a}}'$ given in Proposition 9.10 for the pairs $(\omega ,x) \in G_{m,u}$ and $\mu \in V(\omega ,x) \cap M(\Lambda , f)$ having length $km \in [N-m, N]$. Item (3) in Proposition 9.10 ensures that

$$\begin{aligned} \#{\mathcal {G}}_{m,u}(\omega ,N)&\leqslant (\#{\mathcal {U}})^m \; \# \{ {\underline{a}}' \in (\#{\mathcal {U}})^{km} {:}\, H({\underline{a}}') \leqslant m(h_\mu (f)+\varepsilon ) \} \\&\leqslant (\#{\mathcal {U}})^m \; \#R(km, m(h+\varepsilon ),\#{\mathcal {U}}), \end{aligned}$$

where $h= P- u + \varepsilon$. As $km\leqslant N < (k+1)m$, by Lemma 9.9 we obtain

$$\begin{aligned} \limsup _{N\rightarrow \infty }\frac{1}{N} \log \#{\mathcal {G}}_{m,u}(\omega ,N) \leqslant h+\varepsilon . \end{aligned}$$

Altogether we deduce that

$$\begin{aligned} m_\alpha (\omega ,f,\phi ,G_{m,u},{\mathcal {U}},N)&\leqslant \sum _{k=N}^{\infty } \sum _{{\underline{U}} \in {\mathcal {G}}_{m,u}(\omega ,k)} \# {\mathcal {G}}_{m,u}(\omega ,k) \exp (-\alpha k +S_{k}\phi _\omega ( {U})) \\&\leqslant \sum _{k=N}^{\infty }\exp \Big (-\alpha k +k(h+2\varepsilon ) +k\big [u+2\varepsilon + \frac{1}{k}\sum _{i=0}^{k-1}\kappa _{\delta }(\theta ^i\omega )\big ]\Big ) \\&\leqslant \sum _{k=N}^{\infty } \exp \Big (-k\Big (\alpha -\Big [ P+ \int _\Omega \kappa _{\delta }\ d{{\mathbb {P}}}+5\varepsilon \Big ]\Big )\Big ) \end{aligned}$$

for every sufficiently large N. Therefore, if $\alpha \geqslant P+\int _\Omega \kappa _{\delta }(\omega )\ d{{\mathbb {P}}}+5\varepsilon$, then $m_\alpha (\omega ,f,\phi ,G_{m,u},{\mathcal {U}})=\lim _{N\rightarrow \infty }m_\alpha (\omega ,f,\phi ,G_{m,u},{\mathcal {U}},N)=0$, which implies that

$$\begin{aligned} \pi _\phi (\omega ,f,G_{m,u},{\mathcal {U}})&\leqslant P+\int _\Omega \kappa _{\delta }(\omega )\ d{{\mathbb {P}}}+5\varepsilon . \end{aligned}$$

This, together with  (9.14), ensures that

$$\begin{aligned} \pi _\phi (\omega ,f,{\mathcal {E}}(\Lambda ) \cap (\Omega _0\times M),{\mathcal {U}}) \leqslant P+\int _\Omega \kappa _{\delta }(\omega )\ d{{\mathbb {P}}}+5\varepsilon . \end{aligned}$$

Taking $\delta ={\text {diam}}\ {\mathcal {U}}\rightarrow 0$ and $\varepsilon \rightarrow 0$ we prove that (9.11) holds, thus completing the proof of the theorem. $\quad \square$

Appendix B: Ledrappier–Young Formula for Random Dynamics

The main goal of this “Appendix” is to present a mainly self-contained characterization of non-uniformly expanding equilibrium states (i.e. equilibrium states having only positive Lyapunov exponents) as those invariant probabilities which are absolutely continuous with respect to an expanding conformal measure. Let us first set the abstract framework.

1.1 Setting and main statement

Let (M, d) be a compact metric space and let ${\mathcal {B}}$ be the Borel $\sigma$-algebra. Assume that $(\Omega , {\mathcal {F}}, {{\mathbb {P}}})$ is a Lebesgue space and that $\theta {:}\,\Omega \rightarrow \Omega$ is an invertible ${{\mathbb {P}}}$-preserving measurable transformation and let $X\subset \Omega \times M$ be a measurable compact subset. Consider a family $f=(f_\omega )_\omega$ of continuous maps $f_\omega {:}\,X_\omega \rightarrow X_{\theta \omega }$ and consider the skew-product $F{:}\, X \rightarrow X$ defined by $F(\omega ,x)=(\theta (\omega ), f_\omega (x)).$ The main assumption is the existence of an expanding conformal measure. More precisely, assume there exists a probability $\nu =(\nu _\omega )_\omega$ on X such that ${\mathcal {L}}_\omega ^*\nu _\omega =\lambda _\omega \nu _{\theta \omega }$ for ${{\mathbb {P}}}$-a.e. $\omega$, an F-invariant subset $H\subset X$ satisfying $\nu (H)=1$ and $\alpha >0$ so that

$$\begin{aligned} \liminf _{n\rightarrow \infty } \frac{1}{n} \sum _{i=0}^{n-1} \log \Vert Df_{\theta ^i\omega }(f_\omega ^i(x))^{-1}\Vert ^{-1} \geqslant \alpha . \quad \text { for every }(\omega ,x)\in H. \end{aligned}$$

(10.1)

Remark 10.1

Khanin and Kifer [22, 26] considered a class of random average expanding maps where for ${{\mathbb {P}}}$-a.e. $\omega$ there exist a sequence $(n_k(\omega ))_k$ of instants at which all points in $X_\omega$ have a hyperbolic time property. In particular, any partition of small diameter (in the fibers) is generating for the random dynamics. This is useful as the Kolmogorov–Sinai theorem allows to reduce the computation of the measure theoretical entropy to the entropy of a partition. Under the weaker assumption (10.1) it may occur that for every $\omega \in \Omega$ there are points with no expanding behavior, hence natural generating partitions may fail to exist.

The previous remark justifies the need to construct suitable partitions adapted to the non-uniformly expanding nature of the random dynamics, a main step to obtain a Rohklin formula for entropy hereafter. Most known results deal with the case of SRB measures (i.e., absolutely continuous along the unstable manifolds) in which case Rohklin’s formula is known as Pesin’s formula. In the deterministic context of $C^2$-diffeomorphisms, the construction of suitable partitions and Rohklin’s formula for the entropy were achieved by Ledrappier and Young [28]. An adaptation of their construction in the case of non-uniformly expanding maps appeared in [49]. In the random dynamical systems context there have been several contributions to the theory characterizing SRB measures as those satisfying Pesin’s formula, justified by the need of dealing with invertible or noninvertible randomness and fiber dynamics (we provide more details after the statement of the main result below). Unfortunately, most of these results seem not to adapt to our invertible base, non-invertible fiber and mild hyperbolicity assumptions, which reinforces the need of the following main result of this “Appendix”.

Theorem E

Let $(\Omega ,{{\mathbb {P}}})$ be a probability space, $f=(f_\omega )_\omega$ be a family of $C^1$-local diffeomorphisms $f_\omega {:}\,X_\omega \rightarrow X_{\theta (\omega )}$, and $\phi =(\phi _\omega )_\omega \in L^1_X(\Omega ,C^\beta (M))$, for some $\beta >0$. Assume that $\nu =(\nu _\omega )_\omega$ is a conformal measure satisfying (10.1) with Jacobians $J_{\nu _\omega } f_\omega =\lambda _\omega e^{-\phi _\omega }$ for ${{\mathbb {P}}}$-a.e $\omega$. If, in addition, $\pi _\phi (f,X)=\int \log \lambda _\omega \, d{{\mathbb {P}}}$ then every equilibrium state $\eta$ for f with respect to $\phi$ such that $\eta (H\cap {\text {supp}}\nu )=1$ is absolutely continuous with respect to $\nu$, meaning that $\eta =(\eta _\omega )_\omega$ and $\eta _\omega \ll \nu _\omega$ for ${{\mathbb {P}}}$-a.e. $\omega$. In particular, if the random dynamical system is topologically transitive on ${\text {supp}}\nu$ then there exists at most one expanding equilibrium state for f with respect $\phi$ on ${\text {supp}}\nu$.

Some remarks are in order. In the special case that $\nu _\omega$ denotes the Lebesgue measure on $X_\omega$, the characterization of expanding equilibrium states as probabilities absolutely continuous with respect to the Lebesgue measure corresponds to Pesin’s formula for random dynamical systems and the family of potentials $\phi _\omega =-\log |\det Df_\omega |$, $\omega \in \Omega$. This topic has been extensively studied and several contributions were given which considered, separately, the cases where the randomness is independent and identically distributed, both invertible and non-invertible map $\theta$ and both the case of $C^2$ diffeomorphisms or endomorphisms (cf. [9, 29, 31,32,33] and references therein). In comparison to the deterministic context [40], it is worth mentioning that, in a random dynamical systems setting, the $C^2$-differentiability assumption may not be dropped to $C^{1+\alpha }$ in general (cf. Remark 2.4 in [33]). Our requirements on the $C^1$-smoothness of the maps in Theorem E are enough because we deal with the case that all Lyapunov exponents have positive sign (hence we need no absolutely continuous stable foliation nor to control angles between stable and unstable Pesin subbundles). Furthermore, it seems that Theorem E produces new results even in this context of SRB measures, as we could not find any such work dealing with invertible base and non-invertible and non-uniformly expanding fiber dynamics.

In the context of general equilibrium states, other versions of Theorem E for random dynamical systems appeared in [8, 22, 24, 26, 36], most of the times associated to random dynamical systems associated to $C^2$-expanding maps and making use of some Markov structure. Our approach is inspired by the arguments in [32, Chapter VI], from which we borrow some information concerning Lyapunov exponents and the theory of invariant manifolds.

1.2 Natural extensions and lifts

Some of the reasons to consider natural extensions are that the dynamics generating the random dynamics are not invertible and that, as mentioned in Remark 10.1, there may be points $(\omega ,x), (\omega ,y) \in X$ so that $f_\omega (x)=f_\omega (y)$ but which behave differently, meaning that the refined partitions show a different behavior (in terms of contraction and non-contraction) in neighborhoods of $(\omega ,x)$ and $(\omega ,y)$.

The natural extension of the skew-product F is the map ${{\tilde{F}}}{:}\,{{\tilde{X}}} \rightarrow {{\tilde{X}}}$ defined on

$$\begin{aligned} {{\tilde{X}}} = \Big \{ (\dots , (\omega _{-1},x_{-1}), (\omega _0,x_0)) \in X^{\mathbb {N}}{:}\,F(\omega _i,x_i)=(\omega _{i+1},x_{i+1}), \;\forall i < 0\Big \}, \end{aligned}$$

and given as usual by

$$\begin{aligned} {{\tilde{F}}} (\,(\dots , (\omega _{-1},x_{-1}), (\omega _0,x_0))\,) = (\dots , (\omega _{-1},x_{-1}), (\omega _0,x_0), F(\omega _0,x_0)) \end{aligned}$$

(10.2)

Notice that ${{\tilde{X}}}\subset X^{\mathbb {N}} \subset (\Omega \times M)^{{\mathbb {N}}}$. Based on the invertibility of $\theta$, we may consider a simplification of the natural extension. Indeed, consider alternatively the map

$$\begin{aligned} {{\tilde{F}}} {:}\, \bigcup _{\omega \in \Omega } \{\omega \} \times {{\tilde{X}}}_\omega \rightarrow \{\omega \} \times \bigcup _{\omega \in \Omega } {{\tilde{X}}}_\omega \end{aligned}$$

where

$$\begin{aligned} {{\tilde{F}}} (\omega , \, (\dots , x_{-2}, x_{-1}, x_0)) = (\theta \omega ,\, (\dots , x_{-2}, x_{-1}, x_0, f_\omega (x_0))) \end{aligned}$$

(10.3)

and

$$\begin{aligned} {{\tilde{X}}}_\omega = \Big \{ (\dots , x_{-2}, x_{-1}, x_0) \in \prod _{i=-\infty }^0 X_{\theta ^i\omega } {:}\, f_{\theta ^{-i}\omega }(x_{-i})=x_{-i+1}, \; \forall i\geqslant 0 \Big \}. \end{aligned}$$

The space ${{\tilde{X}}}_\omega$ can be thought as the natural extension for a single realization of the random dynamics and, clearly, $\tilde{X}:=\{\omega \} \times \bigcup _{\omega \in \Omega } {{\tilde{X}}}_\omega \subset \Omega \times M^{{\mathbb {N}}}$. Moreover, the maps defined by (10.2) and (10.3) are measurable, invariant and make the following diagrams commute.

$$\begin{aligned} \begin{array}{cccc} ((\omega _{-i},x_{-i}))_{i\geqslant 0} &{}\quad \xrightarrow []{\quad {{\tilde{F}}}\quad } &{}\quad (\,\dots , (\omega _{-1},x_{-1}), (\omega _0,x_0), (\theta \omega _0, f_{\omega _0}(x_0))\,) \\ \downarrow {{{\tilde{\pi }}}_0} &{}\quad &{}\quad \downarrow {{{\tilde{\pi }}}_0} \\ (\omega _0,x_0) &{}\quad \xrightarrow []{\quad F\quad } &{}\quad (\, \theta \omega _0, f_{\omega _0}(x_0)\,) \\ \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{cccc} ((\omega _{-i},x_{-i}))_{i\geqslant 0} &{}\quad \xrightarrow []{\quad {{\tilde{F}}}\quad } &{}\quad (\,\dots , (\omega _{-1},x_{-1}), (\omega _0,x_0), (\theta \omega _0, f_{\omega _0}(x_0))\,) \\ \downarrow {\pi _0^1} &{}\quad &{}\quad \downarrow {\pi _0^1} \\ (\omega _0, (x_{-i})_{i\geqslant 0}) &{}\quad \xrightarrow []{\quad {{\tilde{F}}}\quad } &{}\quad (\,\theta {\omega _0}\, (\dots , x_{-2}, x_{-1}, x_0, f_\omega (x_0)) \,) \\ \downarrow {{{\tilde{\pi }}}_0} &{}\quad &{}\quad \downarrow {{{\tilde{\pi }}}_0} \\ (\omega _0,x_0) &{}\quad \xrightarrow []{\quad F\quad } &{}\quad (\, \theta \omega _0, f_{\omega _0}(x_0)\,) \\ \end{array} \end{aligned}$$

where $\pi _0^1{:}\,(\Omega \times M)^{{\mathbb {N}}} \rightarrow \Omega \times M^{{\mathbb {N}}}$, ${{\tilde{\pi }}}_0{:}\,\Omega \times M^{{\mathbb {N}}} \rightarrow \Omega \times M$ and ${{\tilde{\pi }}}_0{:}\,(\Omega \times M)^{{\mathbb {N}}} \rightarrow \Omega \times M$ are the natural projections on the corresponding first coordinates. Since $\pi _0^1$ is a bijection, the map ${{\tilde{F}}}$ seems a more suitable concept than the natural extension ${{\tilde{F}}}$. By a slight abuse of notation we shall refer to the latter when mentioning the natural extension of F. For each $\omega \in \Omega$ we denote by ${{\tilde{f}}}_\omega {:}\,{{\tilde{X}}}_\omega \rightarrow {{\tilde{X}}}_{\theta \omega }$ the map

$$\begin{aligned} (\dots , x_{-2}, x_{-1}, x_0) \mapsto (\dots , x_{-2}, x_{-1}, x_0, f_\omega (x_0)), \end{aligned}$$

where each fiber ${{\tilde{X}}}_\omega$ is endowed with the metric $\tilde{d}({\underline{x}},{\underline{y}})= \sum _{i \geqslant 0} 2^{-i}d(x_{-i},y_{-i})$ and the natural sigma-algebra inherited from $M^{{\mathbb {N}}}$ .

The non-uniform expanding structure of the fiber dynamics can now be lifted to the natural extension. This can be made both for the relevant sets as well as for the invariant probability measures.

Remark 10.2

It is well known that for every F-invariant probability $\eta$ there exists a unique ${{\tilde{F}}}$-invariant probability ${{\tilde{\eta }}}$ so that $({{\tilde{\pi }}}_0)_*{{\tilde{\eta }}} =\eta$ and, moreover, $h_{{{\tilde{\eta }}}}({{\tilde{F}}})=h_\eta (F)$ (cf. [45]). By construction, the probability ${{\tilde{\eta }}} := (\pi _0^1)_*{{\tilde{\eta }}}$ is ${{\tilde{F}}}$-invariant, $({{\tilde{\pi }}}_0)_*{{\tilde{\eta }}}=\eta$ and $({{\tilde{\pi }}}_\Omega )_*{{\tilde{\eta }}}={\mathbb {P}}$, where $\tilde{\pi }_\Omega {:}\,\Omega \times M^{{\mathbb {N}}} \rightarrow \Omega$ is the natural projection on $\Omega$. In particular, by the semiconjugacy it follows that $h_{{{\tilde{\eta }}}}({{\tilde{F}}})=h_\eta (F)$. Furthermore, if $\eta (H)=1$ then the ${{\tilde{F}}}$-invariant set $\tilde{H}=({{\tilde{\pi }}}_0)^{-1}(H)$ has full ${{\tilde{\eta }}}$-measure.

Remark 10.3

Although the conformal measure $\nu =(\nu _\omega )_\omega$ is not necessarily invariant, one can also construct a natural family of lifts associated to it. Indeed, if $(\omega ,x)\in H$ is an Oseledets typical point then the map

$$\begin{aligned} {{\tilde{\pi }}}_0 \mid _{{{\widetilde{W}}}^u_\text {loc}(\omega , {\underline{x}})} :{{{\widetilde{W}}}^u_\text {loc}(\omega , {\underline{x}})} \rightarrow W^u_\text {loc}(\omega , x) \end{aligned}$$

is a bijection and the measure $\nu _\omega$ (restricted to any subset of $W^u_\text {loc}(\omega , x)$) can be lifted to ${{\tilde{X}}}$. This will be important later on in the proof that all expanding equilibrium states satisfy the Rohklin property.

1.3 Oseledets’s theorem and local unstable manifolds

The following is a version of the Oseledets theorem for random dynamical systems with not necessarily invertible $C^1$-fiber maps. We refer the reader to [2] and [32, Chapter III] for more details and proofs.

Theorem 10.4

Let $\theta$ be an ergodic map on a probability space $(\Omega , {{\mathbb {P}}})$, M be a compact Riemannian manifold, $f=(f_\omega )_\omega$ be a family of $C^1$-maps $f_\omega {:}\,X_\omega \subset M\rightarrow X_{\theta (\omega )} \subset M$ and let F be the natural skew-product. Suppose that $\eta$ is an F-invariant probability such that $\pi _*\eta ={{\mathbb {P}}}$ and that $\int \log ^+ \Vert Df_\omega (x)\Vert \, d\eta (\omega ,x)<+\infty$. There exists an F-invariant subset $\Sigma \subset X$ satisfying $\eta (\Sigma )=1$ and so that for every $(\omega ,x)\in \Sigma$ there exists $1\leqslant k(x) \leqslant \dim M$, a filtration of linear subspaces of $T_x M$

$$\begin{aligned} \{0\}=V^0_{\omega ,x} \subset V^1_{\omega ,x} \subset \dots \subset V^{k(x)}_{\omega ,x} = T_x M \end{aligned}$$

and numbers $\lambda ^1(x)< \lambda ^2(x)< \dots < \lambda ^{k(x)}(x)$ (depending only on x) called Lyapunov exponents defined by

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{1}{n} \log \Vert Df_\omega ^n(x) v\Vert =\lambda ^i(x), \quad \forall v\in V^i_{\omega ,x} {\setminus } V^{i-1}_{\omega ,x}, \; \forall 1 \leqslant i \leqslant k(x). \end{aligned}$$

Moreover, $Df_\omega (x) V^i_{\omega ,x} = V^i_{\theta \omega ,f_\omega (x)}$ for every $(\omega ,x)\in \Sigma$ and the functions k(x), $\lambda ^i(x)$ and $V^i_{\omega ,x}$ vary measurably with $(\omega ,x)$, for every $1\leqslant i \leqslant k(x)$.

We will make use of the unstable manifold theorem for invariant probabilities having only positive Lyapunov exponents.

Theorem 10.5

Assume the hypothesis of Theorem 10.4 and that all Lyapunov exponents of $\eta$ are bounded below by $\lambda >0$. For any $\varepsilon >0$ small, for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$ there are $\delta _\varepsilon (\omega ,{\underline{x}})>0$ and $\gamma (\omega ,\underline{x}) >0$ and an embedded $C^1$-disk $W^u_{\text {loc}}(\omega ,\underline{x}) \subset X_\omega$, varying measurably with $(\omega ,{\underline{x}})$, so that:

• (1) For every $y \in W^u_{\text {loc}}(\omega ,{\underline{x}})$ there is a unique ${\underline{y}} \in {{\tilde{X}}}_\omega$ such that $\tilde{\pi }_0({\underline{y}})=y_0$ and $d(x_{-n},y_{-n}) \leqslant \gamma (\omega ,{\underline{x}}) \, e^{-(\lambda -\varepsilon ) n} \; \forall n \geqslant 0;$

• (2) If a point ${\underline{z}} \in {{\tilde{X}}}_\omega$ satisfies

  $$\begin{aligned} d({\underline{x}},{\underline{z}}) \leqslant \delta _\varepsilon (\omega ,{\underline{x}}) \hbox { and } d(x_{-n},z_{-n}) \leqslant \delta _\varepsilon (\omega ,{\underline{x}}) e^{-(\lambda -\varepsilon ) n} \end{aligned}$$

  for every $n \geqslant 0$ then $z_0$ belongs to $W^u_{\text {loc}}(\omega ,{\underline{x}})$;

• (3) If ${{\tilde{W}}}^u_{\text {loc}}(\omega ,{\underline{x}})$ is the set of points ${\underline{y}} \in {{\tilde{X}}}_\omega$ given by (2) then

  $$\begin{aligned} d(y_{-n},z_{-n}) \leqslant \gamma (\omega ,{\underline{x}}) \, e^{-(\lambda -\varepsilon ) n} d(y,z) \end{aligned}$$

  for every ${\underline{y}},{\underline{z}} \in \tilde{W}^u_{\text {loc}}(\omega ,{\underline{x}})$ and every $n \geqslant 0$.

Since local unstable leaves vary measurably with the point, there are compact sets of arbitrary large measure, referred to as hyperbolic blocks, restricted to which the local unstable leaves passing through those points vary continuously as follows (see e.g. [32, pp. 96–97]).

Corollary 10.6

There are countably many compact sets $({{\tilde{\Lambda }}}_i)_{i \in {\mathbb {N}}}$ whose union is a ${{\tilde{\eta }}}$-full measure set such that the following holds. For every $i \geqslant 1$ there are positive numbers $\varepsilon _i\ll 1$, $\lambda _i$, $r_i$, $\gamma _i$ and $R_i$ such that for every $(\omega ,{\underline{x}}) \in {{\tilde{\Lambda }}}_i$ there exists an embedded submanifold $W^u_{\text {loc}}(\omega ,{\underline{x}}) \subset X_\omega \subset M$ of dimension $\dim M$, and:

• (1) if $y_0 \in W^u_{\text {loc}}(\omega ,{\underline{x}})$ then there is a unique ${\underline{y}} \in \tilde{X}_\omega$ such that for every $n \geqslant 1$

  $$\begin{aligned} d(x_{-n},y_{-n}) \leqslant r_i e^{-\varepsilon _i n} \quad \text {and}\quad d(x_{-n},y_{-n}) \leqslant \gamma _i e^{-\lambda _i n}; \end{aligned}$$

• (2) for every $0<r \leqslant r_i$ the set $W^u_{\text {loc}}(\omega , {\underline{y}}) \cap B(x_0,r)$ is connected and the map

  $$\begin{aligned} B((\omega ,{\underline{x}}), \varepsilon _i r) \cap {{\tilde{\Lambda }}}_i\ni {\underline{y}} \mapsto W^u_{\text {loc}}(\omega ,{\underline{y}}) \cap B(x_0,r) \end{aligned}$$

  is continuous (in the Hausdorff topology); 

• (3) if ${\underline{y}}$ and ${\underline{z}}$ belong to $B((\omega ,{\underline{x}}), \varepsilon _i r) \cap {{\tilde{\Lambda }}}_i$ then either $W^u_{\text {loc}}(\omega ,{\underline{y}}) \cap B(x_0,r)$ and $W^u_{\text {loc}}(\omega ,{\underline{z}}) \cap B(x_0,r)$ coincide or are disjoint; in the later case, if ${\underline{y}} \in {{{\tilde{W}}}}^u(\omega , {\underline{z}})$ then $d(y_0,z_0)>2r_i$;

• (4) if $(\omega _1, {\underline{y}}) \in {{\tilde{\Lambda }}}_i \cap B((\omega ,{\underline{x}}), \varepsilon _i r)$ then $W^u_{\text {loc}}(\omega _1,{\underline{y}})$ contains the ball of radius $R_i$ around $W^u_{\text {loc}}(\omega _1,{\underline{y}}) \cap B(x_0,r)$.

1.4 Measurable generating partition

We proceed with the construction of a special partition in $\tilde{X}$, adapted to an invariant and expanding measure, that is closely related with Ledrappier’s geometric construction in [27, Proposition 3.1]. In fact this exposition is inspired by a dual argument in [32, Chapter 4, §2] on the construction of measurable partitions adapted to stable foliations.

While ${{\tilde{X}}}$ is not a manifold in general, its fibered structure makes reasonable to code inverse branches according to random orbits by $\theta ^{-1}$. Given a partition ${{\tilde{{\mathcal {Q}}}}}$ denote by ${{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})$ the element of the partition containing $(\omega ,{\underline{x}}) \in {{\tilde{X}}}$. We say that ${{\tilde{Q}}}$ is an increasing partition if $({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega , {\underline{x}}) \subset {{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})$ for ${{\tilde{\eta }}}$-almost every $(\omega , {\underline{x}})$, in which case we write ${{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}}\succ {{\tilde{{\mathcal {Q}}}}}$.

All partitions considered throughout are fibered, meaning that ${{\tilde{Q}}}$ is a refinement of the partition in fibers $\tilde{\mathcal {F}}_0:=\{\{\omega \} \times {{\tilde{X}}}_\omega {:}\, \omega \in \Omega \}$. Making this implicit requirement simplifies the notation as it avoids using an extra conditional entropy term as in [32]. Assume, for the time being, the following two instrumental results.

Proposition 10.7

Let $\eta$ be an F-invariant probability such that $\pi _*\eta ={\mathbb {P}}$ and $\eta (H)=1$. There exists an $\tilde{F}$-invariant and full ${{\tilde{\eta }}}$-measure subset ${{\tilde{S}}} \subset {{\tilde{H}}}$, and a measurable partition ${{\tilde{{\mathcal {Q}}}}}$ of ${{\tilde{S}}}$ such that:

• (1) ${{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}}\succ {{\tilde{{\mathcal {Q}}}}}$,

• (2) $\bigvee _{j=0}^{+ \infty } {{\tilde{F}}}^{-j} {{\tilde{{\mathcal {Q}}}}}$ is the partition of ${{\tilde{S}}}$ into points

• (3) The sigma-algebras ${\mathcal {M}}_n$ generated by the partitions ${{\tilde{F}}}^{-n} {{\tilde{{\mathcal {Q}}}}}$, $n\geqslant 1$, generate the $\sigma$-algebra in ${{\tilde{S}}}$,

• (4) For ${{\tilde{\eta }}}$-almost every $(\omega , {\underline{x}})$ the element ${{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})\subset {{\tilde{W}}}^u(\omega , {\underline{x}})$ contains a neighborhood of ${\underline{x}}$ in $\tilde{W}^u(\omega ,{\underline{x}}) \subset {{\tilde{X}}}_\omega$ and the projection $\pi ({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}}))$ contains a neighborhood of $x_0$ in $X_\omega$.

Due to the non-expanding nature of the dynamics we cannot ensure in Proposition 10.7 that the refined partition $\bigvee _{j=0}^{+\infty } ({{\tilde{F}}}^j_\omega )^{-1} {{\tilde{Q}}}_{\theta ^j\omega }$ is a partition into points on $\{\omega \} \times {{\tilde{X}}}_\omega$ (compare item (2) above).

Proposition 10.8

$h_{{{\tilde{\eta }}}}({{\tilde{F}}}) =H_{{{\tilde{\eta }}}}( {{\tilde{F}}}^{-1} {{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{{\mathcal {Q}}}}})$.

The previous propositions, whose proofs are postponed to the end of the appendix, justify the construction of the previous family of measurable partitions.

1.5 Rokhlin-type formula

A main step in the proof of Theorem E is to prove that every expanding equilibrium state $\eta$ for f with respect to $\phi$ satisfies a Rokhlin-like formula involving the Jacobian with respect to the conformal measure.

Let $\eta$ be an equilibrium state for f with respect to $\phi$ and assume, without loss of generality, that $\eta$ is ergodic. Let ${{\tilde{\eta }}}=({{\tilde{\eta }}}_\omega )_\omega$ be the lift of $\eta$ to $\tilde{X}$, and denote by $({{\tilde{\eta }}}_{\omega ,{\underline{x}}})_{(\omega , \underline{x})}$ be the disintegration of the measure ${{\tilde{\eta }}}_\omega$ with respect to the partition ${{\tilde{{\mathcal {Q}}}}}_\omega$ on $\{\omega \} \times \tilde{X}_\omega$. We also consider the lift of the conformal measure $\nu$ which is adapted to ${{\tilde{\eta }}}$ which is constructed as follows. Since ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$ is Oseledets typical, we define the measure ${{\tilde{\nu }}}_{\omega ,{\underline{x}}}$ as the pull back $\big ({{\tilde{\pi }}}_0 \mid _{\widetilde{W}^u_\text {loc}(\omega , {\underline{x}})}\big )^* \nu _\omega$ (recall Remark 10.3). Namely, the lift ${{\tilde{\nu }}} = (\tilde{\nu }_{\omega ,{\underline{x}}})_{\omega ,{\underline{x}}}$ is defined by

$$\begin{aligned} {{\tilde{\nu }}}({{\tilde{E}}}) = \int {{\tilde{\nu }}}_{\omega ,{\underline{x}}} ({{\tilde{E}}}) \, d{{\tilde{\eta }}} (\omega ,{\underline{x}}) \end{aligned}$$

(10.4)

for every measurable subset ${{\tilde{E}}} \subset {{\tilde{X}}}$.

Remark 10.9

By construction, ${{\tilde{\nu }}}_{\omega ,\underline{x}}({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})) =\nu _\omega \big ( \pi ({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}}))\cap W^u_{\text {loc}}(\omega ,\underline{x}) \big )$ for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$. Since ${{\tilde{\eta }}}$ is an expanding measure then $\widetilde{W}^u_{\text {loc}}(\omega ,{\underline{x}})$ forms an open neighborhood of ${\underline{x}} \in {{\tilde{X}}}_\omega$ and $\widetilde{W}^u_{\text {loc}}(\omega ,{\underline{x}}) \cap \pi ({{\tilde{{\mathcal {Q}}}}}(\omega ,\underline{x}))$ contains a neighborhood of x in $X_\omega$. Using that $\eta ({\text {supp}}\nu )=1$ we get $x\in {\text {supp}}(\nu )$ and, we hereafter conclude that $0<{{\tilde{\nu }}}_{\omega ,\underline{x}}({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}}))\leqslant 1$ for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$.

We are in a position to show that the equilibrium state satisfies a second Rokhlin-type formula.

Lemma 10.10

The measure ${{\tilde{\nu }}}_{\omega ,{\underline{x}}}$ has a Jacobian $J_{\tilde{\nu }_{\omega ,{\underline{x}}}} \, {{\tilde{F}}}_\omega = J_{\nu _\omega } f_\omega \circ (\pi \circ {{\tilde{\pi }}}_0)$ with respect to ${{\tilde{F}}}_\omega$, for ${\mathbb {P}}$-a.e. $\omega$. In addition,

$$\begin{aligned} h_{{{\tilde{\eta }}}}({{\tilde{F}}}) = \int \log J_{{{\tilde{\nu }}}_{\omega ,\underline{x}}} \, {{\tilde{F}}}_\omega \;d{{\tilde{\eta }}}(\omega ,{\underline{x}}). \end{aligned}$$

(10.5)

Furthermore, for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$ and every ${\underline{y}} \in {{\tilde{{\mathcal {Q}}}}}(\omega , {\underline{x}})$ the product

$$\begin{aligned} \Delta _\omega ({\underline{x}},{\underline{y}}) = \prod _{j=1}^{\infty } \frac{J_{{{\tilde{\nu }}}_{\theta ^{-j}\omega },{{\tilde{F}}}_\omega ^{-j}({\underline{x}})} f_{\theta ^{-j}\omega } ({{\tilde{F}}}_\omega ^{-j}({\underline{x}}))}{J_{\tilde{\nu }_{\theta ^{-j}\omega },{{\tilde{F}}}_\omega ^{-j}({\underline{x}})} f_{\theta ^{-j}\omega } ({{\tilde{F}}}_\omega ^{-j}({\underline{y}}))} \end{aligned}$$

(10.6)

is positive and finite.

Proof

By definition ${{\tilde{\nu }}}_{\theta \omega ,F_\omega {\underline{x}}}=({{\tilde{\pi }}}_0 \mid _{W^u_{\text {loc}}(\theta \omega ,F_\omega (\underline{x}))})^*(\nu _{\theta \omega }\mid _{W^u_{\text {loc}}(\theta \omega ,f_\omega (x))})$. Thus, if $E_\omega \subset X_\omega$ is measurable, $f_\omega \mid _{E_\omega }$ is injective and ${{\tilde{E}}}_\omega = {{\tilde{\pi }}}_0^{-1}(E_\omega )$ is a cylinder then

$$\begin{aligned} {{\tilde{\nu }}}_{\theta \omega , F_\omega ({\underline{x}})} ({{\tilde{F}}}_\omega ({{\tilde{E}}}))&= {{\tilde{\nu }}}_{\theta \omega , F_\omega ({\underline{x}})} ( F_\omega ({{\tilde{E}}}_\omega \cap (F_\omega ^{-1} {{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) ) \\&= \nu _{\theta \omega } ( W^u_{\text {loc}}(\theta \omega ,f_\omega (x)) \cap f_\omega (E_\omega \cap {{\tilde{\pi }}}_0( ({{\tilde{F}}}^{-1} {{\tilde{{\mathcal {Q}}}}})(\omega ,\underline{x})))) \\&= \int _{E_\omega \cap {{\tilde{\pi }}}_0( ({{\tilde{F}}}^{-1} {{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}}))) \cap f_\omega ^{-1}(W^u_{\text {loc}}(\theta \omega ,f_\omega (x)))} \, J_{\nu _\omega } f_\omega \, d\nu _\omega \\&= \int _{{{\tilde{E}}}_\omega \cap ({{\tilde{F}}}^{-1} {{\tilde{{\mathcal {Q}}}}})(\omega ,\underline{x})} \, J_{\nu _\omega } f_\omega \circ (\pi \circ {{\tilde{\pi }}}_0) \, d\tilde{\nu }_{\omega ,{\underline{x}}}. \end{aligned}$$

Since the sigma-algebra ${{\tilde{{\mathcal {B}}}}}$ is the completion of the sigma-algebra generated by the cylinders then the first statement in the proposition holds. The Rokhlin formula (10.5) follows, by simple algebraic manipulations, using that $h_{\eta }(f) = h_{{{\tilde{\eta }}}}({{\tilde{F}}})$ and

$$\begin{aligned} \pi _\phi (f,X)=\int \log \lambda _\omega \, d{{\mathbb {P}}}(\omega ) = h_\eta (f) + \int \!\!\!\int \phi _\omega (\cdot ) \, d\eta _\omega \, d{{\mathbb {P}}}(\omega ). \end{aligned}$$

Finally, the convergence of the product  (10.6) follows by the Hölder continuity of the Jacobian $J_{\tilde{\nu }_{\omega ,{\underline{x}}}} \, {{\tilde{F}}}_\omega = \lambda _\omega e^{-\phi _\omega \circ (\pi \circ {{\tilde{\pi }}}_0)}$, the fact that for ${{\mathbb {P}}}$-a.e. $\omega$ the partition ${{\tilde{{\mathcal {Q}}}}}(\omega ,\cdot )$ is subordinated to unstable leaves, and the backward distance contraction for points in the same unstable leaf. This completes the proof of the lemma. $\quad \square$

1.6 Absolutely continuous disintegration

In this subsection we complete the proof of Theorem E using the main tools provided by Propositions 10.7 and 10.8. Namely, we obtained that

$$\begin{aligned} H_{{{\tilde{\eta }}}}( {{\tilde{F}}}^{-1} {{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{{\mathcal {Q}}}}}) = \int \log J_{{{\tilde{\nu }}}_{\omega ,{\underline{x}}}} \, {{\tilde{F}}}_\omega \;d{{\tilde{\eta }}}(\omega ,{\underline{x}}). \end{aligned}$$

(10.7)

The statement in the theorem is a direct consequence of the following:

Proposition 10.11

For ${\mathbb {P}}$-a.e. $\omega$ the following properties hold:

• (1) ${{\tilde{\eta }}}_{\omega ,{\underline{x}}}$ is absolutely continuous with respect to ${{\tilde{\nu }}}_{\omega ,{\underline{x}}}$, for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$;

• (2) $\eta _\omega$ is absolutely continuous with respect to $\nu _\omega$.

Proof

By construction, the product $\Delta _\omega ({\underline{x}}, {\underline{y}})$ [recall (10.6)] is bounded away from zero and infinity for ${{\mathbb {P}}}$-a.e. $\omega$. Thus, the term $Z(\omega ,\underline{x})=\int _{{{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})} \Delta _\omega (\underline{x},{\underline{y}}) \,d{{\tilde{\nu }}}_{\omega , {\underline{x}}}({\underline{y}})$ satisfies $0<Z(\omega ,{\underline{x}})<\infty$ for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})\in {{\tilde{X}}}$.

Consider the probability ${{\tilde{\zeta }}}_{\omega ,{\underline{x}}}$ given by

$$\begin{aligned} {{\tilde{\zeta }}}_{\omega ,{\underline{x}}} (B):= \frac{1}{Z(\omega ,{\underline{x}})} \int _{{{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}}) \cap B} \Delta _\omega (\underline{x},{\underline{y}}) \;d{{\tilde{\nu }}}_{\omega ,{\underline{x}}}({\underline{y}}) \end{aligned}$$

for every measurable $B\subset {{\tilde{X}}}_\omega$, and we proceed to prove that ${{\tilde{\eta }}}_{\omega ,{\underline{x}}} =\tilde{\zeta }_{\omega ,{\underline{x}}}$. First, the property ${{\tilde{F}}}^{-1} \tilde{\mathcal {Q}}\succ {{\tilde{{\mathcal {Q}}}}}$ ensures that

$$\begin{aligned}&{{\tilde{\zeta }}}_{\omega ,{\underline{x}}}(({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) \nonumber \\&\quad = \frac{1}{Z(\omega ,{\underline{x}})} \int _{({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})} \Delta _\omega (\underline{x},{\underline{y}}) \,d{{\tilde{\nu }}}_{\omega ,{\underline{x}}}({\underline{y}}) \nonumber \\&\quad = \frac{1}{Z(\omega ,{\underline{x}})} \int _{{{\tilde{{\mathcal {Q}}}}}(\theta \omega ,F_\omega ({\underline{x}}))} J_{\nu _\omega } f_\omega \circ (\pi \circ {{\tilde{\pi }}}_0 )(\omega ,{\underline{y}}) \;\Delta _\omega (\underline{x},{\underline{y}}) \,d{{\tilde{\nu }}}_{\theta \omega ,F_\omega ({\underline{x}})}(\underline{y}) \nonumber \\&\quad = \frac{Z(\theta \omega , F_\omega ({\underline{x}}))}{Z(\omega ,{\underline{x}})} \cdot \frac{1}{\;J_{{{\tilde{\nu }}}_{\omega ,{\underline{x}}}} \tilde{F}_\omega ({\underline{x}})}. \end{aligned}$$

(10.8)

An argument identical to [32, Lemma VI.8.1] ensures that for ${{\mathbb {P}}}$-almost every $\omega$ the map ${\mathcal {Q}}(\omega ,{\underline{x}}) \ni {\underline{y}} \mapsto \Delta _\omega ({\underline{x}},{\underline{y}})$ is Hölder continuous and uniformly bounded away from zero and infinity (by a constant depending only on $\omega$). In particular

$$\begin{aligned} Z(\omega ,{\underline{x}}))= \lim _{n\rightarrow \infty } \int _{{{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})} \prod _{j=1}^{n} \frac{J_{\tilde{\nu }_{\theta ^{-j}\omega },{{\tilde{F}}}_\omega ^{-j}({\underline{x}})} f_{\theta ^{-j}\omega } ({{\tilde{F}}}_\omega ^{-j}({\underline{x}}))}{J_{\tilde{\nu }_{\theta ^{-j}\omega },{{\tilde{F}}}_\omega ^{-j}({\underline{x}})} f_{\theta ^{-j}\omega } ({{\tilde{F}}}_\omega ^{-j}({\underline{y}}))} \,d{{\tilde{\nu }}}_{\omega , \underline{x}}({\underline{y}}) \end{aligned}$$

and $\log ^+ \frac{Z(\theta \omega , F_\omega ({\underline{x}}))}{Z(\omega ,\underline{x})} \leqslant \log ^+ J_{{{\tilde{\nu }}}_{\omega ,{\underline{x}}}} {{\tilde{F}}}_\omega \in L^1({{\tilde{\eta }}})$. Together with (10.8), this ensures that

$$\begin{aligned} \int - \log {{\tilde{\zeta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) \,d{{\tilde{\eta }}}(\omega ,{\underline{x}}) = \int \log J_{{{\tilde{\nu }}}_{\omega ,{\underline{x}}}} {{\tilde{F}}}_\omega (\underline{x}) \,d{{\tilde{\eta }}}(\omega ,{\underline{x}}). \end{aligned}$$

(10.9)

Equations (10.7) and (10.9) together with

$$\begin{aligned} H_{{{\tilde{\eta }}}}({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{{\mathcal {Q}}}}}) = \int - \log {{\tilde{\eta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) \;d{{\tilde{\eta }}}(\omega ,{\underline{x}}) \end{aligned}$$

imply that

$$\begin{aligned} \int - \log {{\tilde{\zeta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) \,d{{\tilde{\eta }}}(\omega ,{\underline{x}}) = \int - \log {{\tilde{\eta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}})) \;d{{\tilde{\eta }}}(\omega ,\underline{x}). \end{aligned}$$

This ensures

$$\begin{aligned} 0=\int \log \frac{{{\tilde{\zeta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}}))}{{{\tilde{\eta }}}_{\omega ,\underline{x}}(({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}}))} \;d{{\tilde{\eta }}}(\omega ,{\underline{x}}) \leqslant \log \int \frac{\tilde{\zeta }_{\omega ,{\underline{x}}}(({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,\underline{x}))}{{{\tilde{\eta }}}_{\omega ,{\underline{x}}}((\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}})(\omega ,{\underline{x}}))} \;d{{\tilde{\eta }}}(\omega ,\underline{x}) =0 \end{aligned}$$

which together with the strict convexity of the logarithm ensures that the measures ${{\tilde{\eta }}}_{\omega ,{\underline{x}}}$ and $\tilde{\zeta }_{\omega ,{\underline{x}}}$ coincide on the sigma-algebra generated by ${{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}}$. Replacing ${{\tilde{F}}}$ by any power $\tilde{F}^n$ in the previous computations it is not difficult to check that ${{\tilde{\eta }}}_{\omega ,{\underline{x}}}$ and ${{\tilde{\zeta }}}_{\omega ,{\underline{x}}}$ coincide in the increasing family of sigma-algebras generated by $({{\tilde{F}}}^{-n}({{\tilde{{\mathcal {Q}}}}}))_{n \geqslant 1}$, proving Item (1) in the proposition.

Now, just observe that Item (1) and the definition of the lifted measures ${{\tilde{\nu }}}_{\omega ,{\underline{x}}}$ implies $({{\tilde{\pi }}}_0)_*{{\tilde{\eta }}}_{\omega , {\underline{x}}} \ll \nu$ for ${{\tilde{\eta }}}$-almost every ${\underline{x}}$. Since $({{\tilde{\eta }}}_{\omega ,{\underline{x}}})$ is a disintegration of ${{\tilde{\eta }}}$ and $({{\tilde{\pi }}}_0)_*{{\tilde{\eta }}}=\eta$ then $\eta \ll \nu$, proving Item (2). $\quad \square$

1.7 Main estimates

The present subsection is devoted to the proof of the two main propositions used in the proof of Theorem E. We combine modification of arguments in [32, pp. 96–103]. While the latter considers random positive iterations to consider partitions subordinated to stable manifolds, we need to consider inverse iterations to capture backward contraction of unstable manifolds. Moreover, we avoid the use of stationary measures.

1.7.1 Proof of Proposition 10.7

Take an F-invariant probability $\eta$ such that $\pi _*\eta ={\mathbb {P}}$ and $\eta (H)=1$ and let ${{\tilde{\eta }}}$ be the unique ${{\tilde{F}}}$-invariant probability projecting on it. We proceed to construct a ${{\tilde{\eta }}}$-almost everywhere generating partition ${{\tilde{Q}}}$.

Since ${{\tilde{\eta }}}$ is an expanding measure, Proposition 10.5 guarantees the existence of local unstable manifolds at ${{\tilde{\eta }}}$-almost every point. Take $i \geqslant 1$ such that the Pesin block ${{\tilde{\Lambda }}}_i$ satisfies ${{\tilde{\eta }}}({{\tilde{\Lambda }}}_i)>0$, and let $r_i$, $\varepsilon _i$, $\gamma _i$ and $R_i$ be given by Corollary 10.6.

Fix $0<r\leqslant r_i$ and $(\omega _0, {\underline{x}}) \in {\text {supp}}({{\tilde{\eta }}}\mid _{{{\tilde{\Lambda }}}_i})$. By construction $W^u_{\text {loc}}(\omega ,{\underline{y}}) \cap B(x_0,r)$ is connected and the map $(\omega ,{\underline{y}}) \mapsto W^u_{\text {loc}}(\omega ,\underline{y}) \cap B(x_0,r)$ is a continuous function on $B((\omega _0,\underline{x}), \varepsilon _i r) \cap {{\tilde{\Lambda }}}_i$. Consider the sets

$$\begin{aligned} {{\tilde{V}}}_\omega ({\underline{y}},r)=\{{\underline{z}} \in W^u_{\text {loc}}(\omega ,{\underline{y}}){:}\,z_0 \in B(x_0,r)\} \subset {{\tilde{X}}}_\omega \end{aligned}$$

defined for any $(\omega ,{\underline{y}}) \in B((\omega _0,{\underline{x}}), \varepsilon _i r) \cap {{\tilde{\Lambda }}}_i$, and consider the subset of $\tilde{X}$ given by

$$\begin{aligned} {{\tilde{S}}}_{\omega _0} ({\underline{x}},r) =\bigcup \Big \{\{\omega \} \times {{\tilde{V}}}_\omega ({\underline{y}}, r){:}\,(\omega ,{\underline{y}}) \in B((\omega _0,{\underline{x}}),\varepsilon _i r) \cap {{\tilde{\Lambda }}}_i\Big \}. \end{aligned}$$

Consider an initial partition defined as follows. Take the partition ${{\tilde{{\mathcal {Q}}}}}_0(\omega )$ of ${{\tilde{X}}}_\omega$ (depending on r) whose elements are the connected components ${{\tilde{V}}}_\omega ({\underline{y}},r)$ of the unstable manifolds and their complement ${{\tilde{X}}}_\omega {\setminus } {{\tilde{S}}}_\omega ({{\tilde{x}}},r)$ for every $\omega \in \Omega$ satisfying $(\{\omega \} \times {{\tilde{X}}}_\omega ) \cap {{\tilde{S}}}_{\omega _0} ({\underline{x}},r) \ne \emptyset$, and take ${{\tilde{{\mathcal {Q}}}}}_0(\omega )=\tilde{X}_\omega$ otherwise.

Now, the partition ${{\tilde{Q}}}$ is obtained by the refinements by the dynamics, exploring the ergodicity of ${{\tilde{\eta }}}$. Indeed, on the one hand Poincaré’s recurrence theorem implies that

$$\begin{aligned} {{\tilde{S}}} = {{\tilde{S}}}_r := \bigcap _{j=0}^{+\infty } \bigcup _{n=j}^{+\infty } {{\tilde{F}}}^n({{\tilde{S}}}_{\omega _0}({\underline{x}},r)) \end{aligned}$$

is a ${{\tilde{\eta }}}$-full measure set. On the other hand, taking the partition

$$\begin{aligned} {{\tilde{Q}}}(\omega ) = \bigvee _{n=0}^{+\infty } \tilde{F}_{\theta ^{-n}\omega }^n({{\tilde{Q}}}_0(\theta ^{-n}\omega )) = \bigvee _{n=0}^{+\infty } f_{\theta ^{-n}\omega }^n({{\tilde{Q}}}_0(\theta ^{-n}\omega )), \end{aligned}$$

on ${{\tilde{X}}}_\omega$ and the partition ${{\tilde{Q}}}$ formed by the previous ones on each fiber, by construction one has that $\tilde{F}^{-1}{{\tilde{{\mathcal {Q}}}}}\succ {{\tilde{{\mathcal {Q}}}}}$. Indeed,

$$\begin{aligned} {{\tilde{F}}}^{-1}(\{\omega \} \times {{\tilde{Q}}}(\omega ))&= \{\theta ^{-1}\omega \} \times f_{\theta ^{-1}\omega }^{-1} \Big ( \bigvee _{n=0}^{+\infty } f_{\theta ^{-n}\omega }^n(\tilde{Q}_0(\theta ^{-n}\omega )) \Big ) \\&= \{\theta ^{-1}\omega \} \times \bigvee _{n=-1}^{+\infty } f_{\theta ^{-n}(\theta ^{-1}\omega )}^{n}({{\tilde{Q}}}_0(\theta ^{-n}(\theta ^{-1}\omega ))) \\&\succ \{\theta ^{-1}\omega \} \times {{\tilde{Q}}}(\theta ^{-1}\omega ). \end{aligned}$$

Hence Item (1) holds. Moreover, since ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{y}})$ belongs to ${{\tilde{S}}}$, and these points have infinitely many returns to the set ${{\tilde{S}}}_{\omega _0} (\underline{x},r)$ under iteration by ${{\tilde{F}}}$, and ${{\tilde{S}}}_{\omega _0} ({\underline{x}},r)$ is formed by pieces of unstable manifolds, the backward contraction along unstable leaves guarantee not only that the diameter of the partition $\bigvee _{j=0}^{n} \tilde{F}^{-j}{{\tilde{{\mathcal {Q}}}}}$ tend to zero as $n \rightarrow \infty$, as the sigma-algebras ${\mathcal {M}}_n$ generated by the partitions ${{\tilde{F}}}^{-n} {{\tilde{{\mathcal {Q}}}}}$, $n\geqslant 1$, generate the $\sigma$-algebra in $\tilde{S}$. This proves Items (2) and (3).

In what follows we prove that, diminishing r if needed, the resulting partition ${{\tilde{Q}}}={{\tilde{Q}}}_r$ satisfies Item (4): for ${{\tilde{\eta }}}$-almost every $(\omega , {\underline{y}})$ the element ${{\tilde{{\mathcal {Q}}}}}(\omega )({\underline{y}})\subset {{\tilde{W}}}^u(\omega , {\underline{y}})$ contains a neighborhood of ${\underline{y}}$ in $\tilde{W}^u(\omega ,{\underline{y}}) \subset {{\tilde{X}}}_\omega$ and the projection $\pi ({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{y}}))$ contains a neighborhood of $y_0$ in $X_\omega$. We proceed to show that the partition ${{\tilde{{\mathcal {Q}}}}}(r)$ satisfies (4) for Lebesgue almost every parameter r. Given $0<r\leqslant r_i$ and $(\omega ,{\underline{y}}) \in {{\tilde{S}}}_r$ define

$$\begin{aligned} \beta _r(\omega ,{\underline{y}}) =\inf _{n \geqslant 0}\Big \{R_i, \frac{r}{\gamma _i}, \frac{1}{2\gamma _i} e^{\lambda _i n} d(y_{-n}, \partial B(x_0,r))\Big \}, \end{aligned}$$

that it clearly non-negative. First we observe the following:

• (a) If $(\omega ,{\underline{y}}) \in {{\tilde{S}}}_{\omega _0}(x_0,r)$, $z_0 \in W^u(\omega ,{\underline{y}}) \subset X_\omega$ and $d(y_0,z_0)<\beta _r(\omega ,{\underline{y}})$ then there exists $\underline{z} \in {{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{y}})$ such that $\pi (\underline{z})=z_0$;

• (b) There exists a full Lebesgue measure set of parameters $0<r\leqslant r_i$ such that the function $\beta _r(\cdot )$ is strictly positive almost everywhere and ${{\tilde{\eta }}}(\partial {{\tilde{{\mathcal {Q}}}}}_r)=0$.

Indeed, any point $(\omega ,{\underline{y}}) \in \tilde{S}_{\omega _0}({\underline{x}},r)$ belongs to the local unstable manifold of some element $(\omega ,{\underline{t}}) \in B((\omega _0,{\underline{x}}),\varepsilon _i r) \cap {{\tilde{\Lambda }}}_i$. If $z_0 \in W^u(\omega ,{\underline{y}})$ and $d(y_0,z_0)<\beta _r(\omega ,{\underline{y}})< R_i$ then there exists ${\underline{z}} \in {{\tilde{W}}}^u(\omega ,{\underline{y}})$ such that $\pi ({\underline{z}})=z_0$ (cf. Corollary 10.6). In particular

$$\begin{aligned} d(y_{-n},z_{-n}) \leqslant \gamma _i e^{-n \lambda _i} d(y_0,z_0), \quad \;\forall n \in {\mathbb {N}}\end{aligned}$$

ensuring that $d(y_{-n},z_{-n}) \leqslant r$ and $d(y_{-n},z_{-n}) \leqslant 1/2 \,d(y_{-n},\partial B(x_0,r))$ for every $n \in {\mathbb {N}}$. Altogether, this ensures that the iterates $\tilde{F}_\omega ^{-n}({\underline{y}})$ and ${{\tilde{F}}}_\omega ^{-n}({\underline{z}})$ belong to the same element of the partition ${{\tilde{{\mathcal {Q}}}}}_0$ for every $n \geqslant 1$, and conclude that ${\underline{z}} \in {{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{y}})$. This proves Item (a).

The argument in the proof of (b) uses the following remark from measure theory (see e.g. [32, Chapter 4, Lemma  2.1]): if $r_0>0$, $\vartheta$ is a Borel measure in $[0,r_0]$ and $0<a<1$ then Lebesgue almost every $r \in [0,r_0]$ satisfies

$$\begin{aligned} \sum _{k=0}^{\infty } \vartheta \big ( [r-a^k, r+a^k] \big )< \infty . \end{aligned}$$

(10.10)

Let ${{\tilde{\eta }}}_M$ denote the marginal of ${{\tilde{\eta }}}$ on $M^{{\mathbb {N}}}$. If $\vartheta$ is the measure on the interval $[r_i/2,r_i]$ given by

$$\begin{aligned} \vartheta ( E ) = {{\tilde{\eta }}}_M \big (\, {\underline{y}} \in M^{\mathbb {N}}{:}\,d(\pi _0({\underline{y}}), x_0) \in E \,\big ) \end{aligned}$$

the previous assertion guarantees that

$$\begin{aligned} \sum _{k=0}^{\infty } {{\tilde{\eta }}}_M \big (\, {\underline{y}} \in M^{{\mathbb {N}}}{:}\,|d(x_0,\pi _0({\underline{y}}))-r|< e^{-\lambda _i k} \,\big ) < \infty \end{aligned}$$

(10.11)

for Lebesgue almost every $r \in [r_i/2,r_i]$. On the other hand, there exists $D>0$ such that $|d(z_0,x_0)-r|<D\tau$ whenever $d(z_0,\partial B(x_0,r))<\tau$ and $0<\tau <r\leqslant r_i$. Therefore, by ${{\tilde{F}}}$-invariance of ${{\tilde{\eta }}}$ and (10.11),

$$\begin{aligned} \sum _{k=0}^{\infty } {{\tilde{\eta }}} \Big (\, (\omega ,{\underline{y}})&\in {{\tilde{X}}} {:}\, d(y_{-n}, \partial B(x_0,r))< D^{-1}e^{-\lambda _i k} \,\Big ) \\&= \sum _{k=0}^{\infty } \int _\Omega {{\tilde{\eta }}}_\omega \Big (\, {\underline{y}} \in {{\tilde{X}}}_\omega {:}\, d(y_{-n}, \partial B(x_0,r))< D^{-1}e^{-\lambda _i k} \,\Big ) \, d{\mathbb {P}}(\omega )\\&= \sum _{k=0}^{\infty } \int _\Omega (f_\omega ^n)_* {{\tilde{\eta }}}_\omega \Big (\, {\underline{y}} \in {{\tilde{X}}}_{\theta ^n\omega } {:}\, d(\pi _0({\underline{y}}), \partial B(x_0,r))< D^{-1}e^{-\lambda _i k} \,\Big ) \, d{\mathbb {P}}(\omega )\\&\leqslant \sum _{k=0}^{\infty } {{\tilde{\eta }}}_M \Big (\, {\underline{y}} \in M^{{\mathbb {N}}} {:}\, d(\pi _0({\underline{y}}), \partial B(x_0,r))< D^{-1}e^{-\lambda _i k} \,\Big )<\infty . \end{aligned}$$

Then the Borel–Cantelli lemma assures that for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{y}})\in {{\tilde{X}}}$ the condition $|d(y_{-n},\partial B(x_0,r))| < D^{-1}e^{-\lambda _i k}$ holds for at most finitely many positive integers k, proving that $\beta _r(\omega , {\underline{y}})>0$. Finally, since ${{\tilde{\eta }}}(\cup _{n\geqslant 0} {{\tilde{F}}}^n (\Omega \times \partial B(x_0,r)^{{\mathbb {N}}}))=0$ for all but a countable set of parameters $0<r \leqslant r_i$ then ${{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{y}})={{\tilde{{\mathcal {Q}}}}}_r(\omega ,{\underline{y}})$ contains a neighborhood of ${\underline{y}}$ in ${{\tilde{W}}}^u(\omega ,{\underline{y}})$ for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{y}})\in {{\tilde{X}}}$. This proves claim (b) above and finishes the proof of the proposition. $\quad \square$

1.7.2 Proof of Proposition 10.8

Given $i \geqslant 1$, let ${{\tilde{\Lambda }}}_i$ and $r_i$ be as in the proof of Proposition 10.7. The proof involves a preliminary lemma, adapted from [35], which ensures the construction of measurable and finite entropy partitions with arbitrarily small diameter.

Lemma 10.12

[32, Chapter VI, Lemma 5.2] If $\Delta {:}\,{{\tilde{X}}} \rightarrow (0,1)$ is a measurable and log-integrable function with respect to ${{\tilde{\eta }}}$ then there exists a measurable partition ${\mathcal {P}}_0$ of ${{\tilde{X}}}$ such that ${\mathcal {P}}_0\succ {{\tilde{{\mathcal {F}}}}}_0$ and ${\text {diam}}{\mathcal {P}}_0(\omega , {\underline{x}}) \leqslant \Delta (\omega ,{\underline{x}})$ for ${{\tilde{\eta }}}$-a.e. $(\omega ,{\underline{x}}) \in {{\tilde{X}}}$.

This lemma can be used to construct measurable partitions whose refinements are finer than ${{\tilde{Q}}}$. More precisely:

Lemma 10.13

For any $0<\delta <1$ there exists a measurable partition $\tilde{\mathcal {P}}$ of ${{\tilde{S}}}$ such that $H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {P}}}}})<\infty$, ${\text {diam}}({{\tilde{{\mathcal {P}}}}}(\omega ,{\underline{x}})) \leqslant \delta$ for ${{\tilde{\eta }}}$-almost every $(\omega ,{\underline{x}})$, and so that the partition

$$\begin{aligned} {{\tilde{{\mathcal {P}}}}}^{(\infty )} =\bigvee _{n=0}^{+\infty } {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}} \end{aligned}$$

is finer than ${{\tilde{Q}}}$.

Proof

The argument is identical to the proof of [32, Chapter VI, Proposition 5.1]. $\quad \square$

We are now in a position to complete the proof of Proposition 10.8. Let ${{\tilde{{\mathcal {P}}}}}$ be a measurable finite entropy partition so that ${{\tilde{{\mathcal {P}}}}}^{(\infty )} \succ {{\tilde{{\mathcal {Q}}}}}$, given by Lemma 10.13. Hence

$$\begin{aligned} h_{{{\tilde{\eta }}}}({{\tilde{F}}},{{\tilde{{\mathcal {P}}}}}) = h_{{{\tilde{\eta }}}}(\tilde{F},{{\tilde{{\mathcal {P}}}}}^{(\infty )})&= h_{{{\tilde{\eta }}}}({{\tilde{F}}},{{\tilde{{\mathcal {P}}}}}^{(\infty )} \vee {{\tilde{{\mathcal {Q}}}}}) \\&= h_{{{\tilde{\eta }}}}({{\tilde{F}}},{{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )} \vee {{\tilde{{\mathcal {Q}}}}}) \\&= H_{{{\tilde{\eta }}}}( {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )} \vee {{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}^{n+1} {{\tilde{{\mathcal {P}}}}}^{(\infty )} \vee {{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}}) \end{aligned}$$

for every $n \geqslant 1$ (here we used that $h_{{{\tilde{\eta }}}}(\tilde{F},{{\tilde{\zeta }}})=H_{{{\tilde{\eta }}}}({{\tilde{F}}}^{-1} {{\tilde{\zeta }}},\tilde{\zeta })$ whenever the partition ${{\tilde{\zeta }}}$ satisfies $\tilde{F}^{-1}{{\tilde{\zeta }}} \succ {{\tilde{\zeta }}}$). Consequently,

$$\begin{aligned} h_{{{\tilde{\eta }}}}({{\tilde{F}}}, {{\tilde{{\mathcal {P}}}}}) = H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}) + H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {P}}}}}^{(\infty )} \mid {{\tilde{F}}}^{-n} {{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}} {{\tilde{{\mathcal {P}}}}}^{(\infty )}). \end{aligned}$$

(10.12)

The second term in the right hand side above is bounded by $H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {P}}}}})$, which is finite. Then Item (3) in Proposition 10.7 implies that it tends to zero as $n\rightarrow \infty$. It remains to estimate the first term above, which is given by

$$\begin{aligned} H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}) = \int - \log {{\tilde{\eta }}}_{({{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )})(\omega ,{\underline{x}})} (\tilde{\mathcal {Q}}(\omega ,{\underline{x}}))\, d{{\tilde{\eta }}}(\omega ,{\underline{x}}), \end{aligned}$$

where the measure ${{\tilde{\eta }}}_{{{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}}$ denotes the conditional measure of $\tilde{\eta }$ with respect to the partition ${{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee \tilde{F}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}$. Notice that since the diameter of almost every element in ${{\tilde{F}}}^{-n+1}{{\tilde{{\mathcal {Q}}}}}$ tends to zero as $n \rightarrow \infty$, there exists a sequence of sets $(\tilde{\Gamma }_n)_{n\geqslant 1}$ in ${{\tilde{X}}}$ so that $\lim _n {{\tilde{\eta }}}({{\tilde{\Gamma }}}_n)=1$ and ${{\tilde{F}}}{\mathcal {Q}}(\omega ,{\underline{x}}) \subset {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}(\omega ,{\underline{x}}) \;\text {for every}\; (\omega ,{\underline{x}}) \in {{\tilde{\Gamma }}}_n$. Then

$$\begin{aligned} H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}} \vee {{\tilde{F}}}^n {{\tilde{{\mathcal {P}}}}}^{(\infty )}) = \lim _{n \rightarrow \infty } \int _{{{\tilde{\Gamma }}}_n} - \log {{\tilde{\eta }}}_{({{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}})(\omega , {\underline{x}})} (\tilde{\mathcal {Q}}(\omega ,{\underline{x}})) \, d{{\tilde{\eta }}}(\omega ,{\underline{x}}), \end{aligned}$$

where the measure ${{\tilde{\eta }}}_{{{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}}}$ is the conditional measure of ${{\tilde{\eta }}}$ with respect to the partition ${{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}}$. This proves that, taking the limit as $n\rightarrow \infty$ in (10.12),

$$\begin{aligned} h_{{{\tilde{\eta }}}}({{\tilde{F}}}, {{\tilde{{\mathcal {P}}}}}) = \lim _{n\rightarrow \infty } \int _{\tilde{\Gamma }_n} - \log {{\tilde{\eta }}}_{({{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}})(\omega , \underline{x})} ({{\tilde{{\mathcal {Q}}}}}(\omega ,{\underline{x}})) \, d{{\tilde{\eta }}}(\omega ,{\underline{x}}) = H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}{{\tilde{{\mathcal {Q}}}}}). \end{aligned}$$

We conclude that $h_{{{\tilde{\eta }}}}({{\tilde{F}}})=H_{{{\tilde{\eta }}}}({{\tilde{{\mathcal {Q}}}}} \mid {{\tilde{F}}}{\mathcal {Q}})=H_{{{\tilde{\eta }}}}({{\tilde{F}}}^{-1}{{\tilde{{\mathcal {Q}}}}} \mid {\mathcal {Q}})$, proving the proposition.