2.1 Hypothesis about the generating maps

For each $w\in X,$ let $f_{w}:M \rightarrow M$ be a local homeomorphism satisfying the following requirement: there exist $\delta _{w}>\delta >0$ and a continuous function $L_{w} :M\rightarrow \mathbb {R}_{+}$ such that for every $x \in M$ we can find a neighborhood $U_{x}$ where $f_{w}: U_{x} \rightarrow B_{\theta (w)}(f_{w}(x), \delta _{w})$ is invertible and

$$ \begin{align*}d(f_{w}^{-1}(y), f_{w}^{-1}(z)) \leq L_{w}(x)d(y, z)\quad \mbox{for all } y,z \in f_{w}(U_{x})=B_{\theta(w)}(f_{w}(x), \delta_{w}).\end{align*} $$

As $f_{w}$ is a local homeomorphism defined on a compact metric space, we have that the number of preimages $\deg (f_{w}):\#f_{w}^{-1}(x)$ is constant for all $x \in M$ . We assume that $\deg (F)= \sup _{w} \deg (f_{w}) < \infty $ .

Suppose that there are an open region $\mathcal {A}_{w}\subset M$ , constants $\sigma _{w}>1$ and $L_{w}\geq 1$ close enough to $1$ such that the following conditions hold.

1. (I) $L_{w}(x) \leq L_{w}$ for every $x \in \mathcal {A}_{w}$ and $L_{w}(x) < \sigma _{w}^{-1}$ for every $x \in \mathcal {A}_{w}^{c}=M\setminus \mathcal {A}_{w}$ .

2. (II) There exists a finite covering $\mathcal {U}_{w}$ of M, by open domains of injectivity for $f_{w}$ , such that $\mathcal {A}_{w}$ can be covered by $q_{w} < \deg (f_{w})$ elements of $\mathcal {U}_{w}$ .

3. (III) For every $\varepsilon>0$ we can find some positive integer $\tilde {n}=\tilde {n}(w,\varepsilon )$ satisfying $f_{\theta ^{j}(w)}^{\tilde {n}}(B_{\theta ^{j}(w)}(f^{j}_{w}(x),\varepsilon ))=M$ for any $j\geq 0.$

We observe that the continuous function $L_{w}(\cdot )$ associates the Lipschitz constant of the inverse branches. This property and the assumption $\delta _{w}>\delta >0$ imply the uniform openness $B_{\theta (w)}(f_{w}(x), \delta )\subset f_{w}(U_{x})$ , and thus for every $(w, x)\in X\times M$ there exists a unique continuous inverse branch of $f_{w}$ defined on $B_{\theta (w)}(f_{w}(x), \delta )$ sending $f_{w}(x)$ to x. Conditions (I) and (II) mean that expanding and contracting behavior may exist in M, but at least one preimage is required for every point in the expanding region. Condition (III) means that the skew-product F is topologically exact.

Next, we present the setting of potentials that will be considered. For $\alpha>0$ , consider the space $C^{\alpha }(M)$ of Hölder continuous functions $\varphi :M\to \mathbb {R}$ endowed with the seminorm

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

and the norm

$$ \begin{align*}\|\varphi\|_{\alpha}=\|\varphi\|_{\infty}+|\varphi|_{\alpha},\end{align*} $$

where $\|\cdot \|_{\infty }$ stands for the $\sup $ norm. Denote by $\mathbb {L}^{1}_{\mathbb {P}}(X,C^{\alpha }(M))$ the space of all measurable functions $\phi :X\times M\rightarrow \mathbb {R}$ such that for all $w\in X$ , the fiber potential $\phi _{w}:M\rightarrow \mathbb {R}$ defined by $\phi _{w}(x):=\phi (w,x)$ is Hölder continuous and $\lVert \phi \rVert _{1}=\int _{X}\lVert \phi _{w}\rVert _{\infty }\,d\mathbb {P}(w)<+\infty $ . For $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{\alpha }(M))$ we assume the existence of some positive $\varepsilon _{\phi }>0$ satisfying, for all $w\in X$ ,

(IV) $$ \begin{align} \sup \phi_{w} - \inf \phi_{w} + \varepsilon_{\phi}< \log\deg f_{w}- \log q_{w} \quad \mbox {and}\quad |e^{\phi_{w}} |_{\alpha} < \varepsilon_{\phi} e^{\inf\phi_{w}}. \end{align} $$

Notice that all potentials $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{\alpha }(M))$ in a neighborhood of zero satisfy condition (IV). In the literature this class of potentials is referred to as small variation.

Let $p_{w}:=\deg f_{w}-q_{w}.$ The choice of $\varepsilon _{\phi }$ and $L_{w}$ must satisfy, for each $w\in X$ ,

(V) $$ \begin{align} \gamma_{w}:=e^{\varepsilon_{\phi}}\bigg[\frac{ p_{w}\sigma_{w}^{-\alpha}+ q_{w} L_{w}^{\alpha} (1+ (L_{w}-1)^{\alpha} )}{\deg(f_{w})}\bigg] + \varepsilon_{\phi} L_{w}^{\alpha} \bigg[ 1+ m(\mathrm{diam}M)^{\alpha}\bigg]\leq\gamma<1. \end{align} $$

2.2 Statement of results

Consider the space $C^{0}(M)$ of real continuous functions $\psi :M\rightarrow \mathbb {R}$ endowed with the uniform convergence norm. Given $w\in X$ , let $f_{w}:M\rightarrow M$ be the dynamics and $\phi _{w}:M\rightarrow \mathbb {R}$ be the potential on the fiber. The Ruelle–Perron–Frobenius operator or simply transfer operator associated to $(f_{w}, \phi _{w})$ is the linear operator $\mathcal {L}_{w}:C^{0}(M)\rightarrow C^{0}(M)$ defined by

$$ \begin{align*}\mathcal{L}_{w}(\psi)(x)=\sum_{y\in f_{w}^{-1}(x)}e^{\phi_{w}(y)}\psi(y).\end{align*} $$

Its dual operator $\mathcal {L}_{w}^{\ast }:[C^{0}(M)]^{\ast }\to [C^{0}(M)]^{\ast } $ acts on the space of Borel measures as follows:

$$ \begin{align*}\int \psi\, d\mathcal{L}_{w}^{\ast}(\rho_{\theta(w)})=\int\mathcal{L}_{w}(\psi)\, d\rho_{\theta(w)}.\end{align*} $$

In our first result we describe the thermodynamic formalism for random non-uniformly expanding maps.

We also derive that the F-invariant family $\{\mu _{w}\}_{w\in X}$ obtained in the last theorem has an exponential decay of correlations for Hölder continuous observables.

Theorem B. There exists $0<\tau < 1$ such that for any $\varphi \in L^{1}(\mu _{\theta ^{n}(w)})$ and $\psi \in C^{\alpha }(M) $ there exists a positive constant $K(\varphi , \psi ) $ satisfying

$$ \begin{align*}\bigg|\int (\varphi \circ f^{n}_{w}) \psi \,d\mu_{w} - \int \varphi \,d\mu_{\theta^{n}(w)} \int \psi\,d\mu_{w} \bigg| \leq K(\varphi, \psi) \tau^{n},\end{align*} $$

for all $n\geq 1$ .

The weak hyperbolicity property of the generating maps allows us to prove that the F-invariant measure given by Theorem A is indeed an equilibrium state for the random dynamical system. Moreover, it is unique if we consider only the measures whose pressure is located on the expanding region. We specify the setting as follows.

Suppose that there exists $c>0$ such that for $\mathbb {P}$ -almost every $w\in X$ we can find $\tilde {L}_{w}$ close enough to $1$ and $\tilde {\sigma }_{w}>1$ satisfying for every $j\geq 0$ that

(VI) $$ \begin{align} L_{\theta^{j}(w)}\leq \tilde{L}_{w},\quad \tilde{\sigma}_{w}\leq\sigma_{\theta^{j}(w)} \quad \mbox{and}\quad \tilde{L}_{w}^{\rho}\tilde{\sigma}_{w}^{-(1-\rho)} < e^{-2c} <1, \end{align} $$

where $\rho $ is given by Lemma 4.2.

We say that a subset H of $X\times M$ is non-uniformly expanding if there exists some positive constant $c>0$ such that

(⋆) $$ \begin{align} H:=\bigg\{(w, x)\in X\times M; \limsup_{n\to+\infty} \frac{1}{n}\sum^{n-1}_{j=0}\log L_{\theta^{j}(w)}(f^{j}_{w}(x))\leqslant -2c<0\bigg\}. \end{align} $$

A probability measure $\eta $ , not necessarily invariant, is called non-uniformly expanding with exponent c if $\eta (H)=1$ . Condition (VI) above will be used to prove that the F-invariant measure given by Theorem A is non-uniformly expanding.

Our next result states uniqueness of equilibrium states for random dynamical systems among non-uniformly expanding measures.

Theorem C. Let $F:X\times M\to X\times M$ be a random dynamical system and $\phi :X\times M\to \mathbb {R}$ be a potential function satisfying conditions (I)–(VI). There exists only one F-invariant non-uniformly expanding measure $\mu _{F, \phi }\in \mathcal {M}_{\mathbb {P}}(F)$ maximizing the variational principle

where the supremum is taken in the set $\mathcal {M}_{\mathbb {P}}(F).$ Thus, $\mu _{F, \phi }$ is the unique non-uniformly expanding equilibrium state of $(F|\theta , \phi ).$

Once we have proved uniqueness of equilibrium states, we will investigate its persistence under small perturbations of the random system and the potential.

As defined above, consider the space $\mathbb {L}^{1}_{\mathbb {P}}(X,C^{\alpha }(M))$ of integrable potentials and let $\mathcal {D}\subset \Omega $ be the space of $C^{1}$ local diffeomorphisms defined on M. We shall consider the product topology on $\mathcal {D}\times \mathbb {L}^{1}_{\mathbb {P}}(X,C^{\alpha }(M))$ . We fix an invertible transformation $\theta : X\to X$ preserving an ergodic measure $\mathbb {P}$ and consider the family $\mathcal {S}$ of skew-products generated by maps of $\mathcal {D}$ ,

$$ \begin{align*}F:X\times M \to X\times M; F(w, x)=(\theta(w),f_{w}(x)),\end{align*} $$

where $(w, x)\mapsto f_{w}\in \mathcal {D}$ is measurable. Now we define the family

$$ \begin{align*}\mathcal{H}=\{(F, \phi)\in \mathcal{S}\times \mathbb{L}^{1}_{\mathbb{P}}(X,C^{\alpha}(M)); (F, \phi) \mbox{ satisfying conditions (I)--(VI)}\}.\end{align*} $$

By Theorem C, each $(F, \phi )\in \mathcal {H}$ has only one non-uniformly expanding equilibrium state. Our last main result establishes the continuity in the weak star topology of such equilibria within this family; this property is called equilibrium stability. We also prove the continuity of the random topological pressure in this setting.

We point out that we are fixing an invertible transformation $\theta :X\to X$ preserving an ergodic measure $\mathbb {P}.$ However, the proof of Theorem D remains true if we vary $\theta $ in the space of continuous functions.

3.1 Entropy and topological pressure

We start with the definition of entropy and topological pressure for random transformations. The reader can consult more results and properties in Kifer [Reference Kifer15] and Liu [Reference Liu18].

Let $\mu \in \mathcal {M}_{\mathbb {P}}(F)$ be an F-invariant measure. Given a finite measurable partition $\xi $ of M, we set

$$ \begin{align*}h_{\mu}(F|\theta; \xi):=\lim_{n\rightarrow +\infty} \frac{1}{n} \int_{X} H_{\mu_{w}}\bigg(\bigvee_{j=0}^{n-1}f_{w}^{-j}(\xi)\bigg)\,d\mathbb{P}(w)\end{align*} $$

where $H_{\nu }(\xi )=-\sum _{P\in \xi }\nu (P)\log \nu (P)$ for a finite partition $\xi $ and $\mu _{w}$ is the sample measure of $\mu .$ The entropy of $(F|_{\theta }, \mu )$ is

$$ \begin{align*}h_{\mu}(F|\theta):=\sup_{\xi}\{h_{\mu}(F|\theta;\xi)\} \end{align*} $$

where the supremum is taken over all finite measurable partitions of M.

Denote by $\mathbb {L}^{1}_{\mathbb {P}}(X,C^{0}(M))$ the space of all measurable functions $\phi :X\times M\rightarrow \mathbb {R}$ such that $\phi _{w}:M\rightarrow \mathbb {R}$ defined by $\phi _{w}(x):=\phi (w,x)$ is continuous for all $w\in X$ and $\lVert \phi \rVert _{1}=\int _{X}\lVert \phi _{w}\rVert _{\infty }\,d\mathbb {P}(w)<+\infty $ .

Fix $w\in X$ . Given $\varepsilon>0$ and an integer $n\geq 1$ , we say that a subset $F_{n}\subseteq M$ is $(w,n,\varepsilon )$ -separated if for every two distinct points $y,z\in F_{n}$ there exists some $j\in \{0,1,\ldots ,n-1\}$ such that $d(f^{j}_{w}(y),f^{j}_{w}(z))>\varepsilon $ .

For $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{0}(M))$ , $\varepsilon>0$ and $n\geq 1$ we consider

$$ \begin{align*}P_{F|\theta}(\phi)(w, n, \varepsilon)= \sup\bigg\{\sum_{y\in F_{n}}e^{ S_{n}\phi(w,y)}; F_{n} \mbox{ is a } (w,n,\varepsilon)\mbox{-separated set}\bigg\}\end{align*} $$

where $S_{n}\phi (w,y):=\sum ^{n-1}_{j=0}\phi _{\theta ^{j}(w)}( f^{j}_{w}(y))$ .

The random topological pressure of $\phi $ relative to $\theta $ is defined by

$$ \begin{align*}P_{F|\theta}(\phi)=\lim_{\varepsilon\rightarrow 0}\limsup_{n\rightarrow \infty} \frac{1}{n}\int_{X}\log P_{F|\theta}(\phi)(w, n, \varepsilon)\, d\mathbb{P}(w).\end{align*} $$

Thus, the following pressure map is well defined:

$$ \begin{align*} \begin{array}{cccc} P_{F|\theta} : & \! \mathbb{L}^{1}_{\mathbb{P}}(X,C^{0}(M)) & \! {\longrightarrow} & \!\mathbb{R}\cup\{\infty\} \\ & \! \phi & \! {\longmapsto} & \! P_{F|\theta}(\phi). \end{array} \end{align*} $$

In particular, the topological entropy of F relative to $\theta $ is $h_{\mathrm {top}}(F|_{\theta })=P_{F|\theta }(0)$ .

The topological pressure and the entropy are related by the well-known variational principle. The reader can see a proof of this result in [Reference Liu18].

Theorem 3.1. (Variational principle)

Assume that $(X, \mathbb {P})$ is a Lebesgue space. Then for any $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{0}(M))$ we have

(1) $$ \begin{align} P_{F|\theta}(\phi)=\sup_{\mu\in \mathcal{M}_{\mathbb{P}}(F)}\bigg(h_{\mu}(F|\theta)+\int\phi\,d\mu \bigg). \end{align} $$

Moreover, when $\mathbb {P}$ is ergodic, we can consider the supremum over ergodic measures.

Motivated by the variational principle, we say that an F-invariant measure $\mu \in \mathcal {M}_{\mathbb {P}}(F)$ is an equilibrium state for $(F|_{\theta }, \phi )$ relative to $\theta $ if the supremum (1) is attained by $\mu $ , that is,

$$ \begin{align*} P_{F|\theta}(\phi)=h_{\mu}(F|\theta)+\int\phi \,d\mu. \end{align*} $$

Next we define the random topological pressure via open coverings and via dynamic balls. We take as reference the deterministic case where this approach is characteristic in dimension theory. We refer the reader to [Reference Pesin22, Reference Stadlbauer, Suzuki and Varandas28] for more details.

Let $w\in X$ and denote by $\mathcal {U}$ a finite open cover of M. Denote by $\mathcal {S}_{n}(\mathcal {U}{\kern1pt})$ the set of all strings $\mathbf {U}=\{U_{i_{0}}, \ldots , U_{i_{n-1}}; U_{i_{j}}\in \mathcal {U}\}$ of length $n=n(\mathbf {U})$ and put $\mathcal {S}=\mathcal {S}(\mathcal {U}{\kern1pt})=\bigcup _{n\geq 0}\mathcal {S}_{n}(\mathcal {U}{\kern1pt})$ . Given a string $\mathbf {U}=\{U_{i_{0}}, \ldots , U_{i_{n-1}}\}\in \mathcal {S}(\mathcal {U}{\kern1pt})$ , we consider the cylinder

$$ \begin{align*}X_{w}=X_{w}(\mathbf{U}):=\{x\in M; f^{j}_{w}(x)\in U_{i_{j}} \mbox{ for } j=0, \ldots, n(\mathbf{U})-1\}.\end{align*} $$

Let $\mathcal {F}_{(N, w)}$ be the collection of all cylinders of depth at least N, that is,

$$ \begin{align*}\mathcal{F}_{(N, w)}=\mathcal{F}_{(N, w)}(\mathcal {U}{\kern1pt})=\{X_{w}(\mathbf{U}); \mathbf{U}\in \mathcal{S}_{n}(\mathcal {U}{\kern1pt}) \mbox{ for } n\geq N \}.\end{align*} $$

For $\beta \in \mathbb {R}$ and $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{0}(M))$ let

(2) $$ \begin{align} m_{\beta}(w, \phi, F|\theta, \mathcal{U}, N)=\displaystyle{\inf_{\mathcal{F}}}\bigg\{ \sum_{X_{w}\in\mathcal{F}_{(N, w)}} e^{-\beta n(\mathbf{U}) + S_{n(\mathbf{U})}\phi(X_{w})} \bigg\}, \end{align} $$

where $S_{n(\mathbf {U})}\phi (X_{w})=\sup _{y\in X_{w}}\sum _{j=0}^{n(\mathbf {U})-1}\phi _{\theta ^{j}(w)}(f^{j}_{w}(y))$ and the infimum is taken over all finite families $\mathcal {F}$ of $\mathcal {F}_{(N, w)}$ in order that (2) is measurable in w (see, for example, §9 of [Reference Stadlbauer, Suzuki and Varandas28]). As N goes to infinity we define

$$ \begin{align*} m_{\beta}(w, \phi, F|\theta, \mathcal{U})=\lim_{N\to \infty}m_{\beta}(w, \phi, F|\theta, \mathcal{U}, N). \end{align*} $$

The existence of the limit above is guaranteed by the function $m_{\beta }(w, \phi , F|\theta , \mathcal {U}, N)$ to be increasing with N. Taking the infimum over $\beta $ we call

$$ \begin{align*} P_{F|\theta}(w, \phi, \mathcal {U})=\inf\{\,\beta: m_{\beta}(w, \phi, F|\theta, \mathcal{U})=0 \}. \end{align*} $$

Let $|\mathcal {U}|=\max \{\mathrm {diam} U_{i}; U_{i}\subset \mathcal {U}\}$ be the diameter of the cover $\mathcal {U}$ and consider

$$ \begin{align*}P_{F|\theta} (w, \phi)=\lim_{|\mathcal{U}|\rightarrow 0} {P_{F|\theta} (w, \phi, \mathcal{U})}.\end{align*} $$

In [Reference Pesin22, Theorem 11.1] it was showed that this quantity is well defined and does not depend on the cover $\mathcal {U}$ . Moreover, since all quantities defined above are measurable functions of $w\in X$ (see, for example, §9 of [Reference Stadlbauer, Suzuki and Varandas28]), we can define the random topological pressure of $(F|\theta , \phi )$ as

$$ \begin{align*} P_{F|\theta} (\phi)= \int P_{F|\theta} (w, \phi)\,d\mathbb{P}(w). \end{align*} $$

In the following we present another way to define the random topological pressure. We fix $w\in X$ and $\varepsilon>0$ . For $n\in \mathbb {N}$ , $x\in M$ , let $B_{w}(x,n,\varepsilon )$ be the dynamic ball

$$ \begin{align*} B_{w}(x,n,\varepsilon):=\{y\in M: d(f_{w}^{j}(x),f_{w}^{j}(y) )< \varepsilon, \mbox{for } 0\leq j \leq n \}. \end{align*} $$

We denote by $G_{(N,w)}$ the collection of dynamic balls:

$$ \begin{align*} G_{(N,w)}:=\{ B_{w}(x,n,\varepsilon): x\in M \ \mbox{and} \ n\ge N \}. \end{align*} $$

Let $U_{w}$ be a finite or countable family of $G_{(N,w)}$ which covers M. For every $\beta \in \mathbb {R}$ and $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X,C^{0}(M))$ let

$$ \begin{align*} m_{\beta}(w, \phi, F|\theta, \varepsilon, N)=\inf_{U_{w}\subset G_{(N,w)}}\bigg\{ \sum_{B_{w}(x,n,\varepsilon)\in U_{w}} e^{-\beta n + S_{n}\phi(B_{w}(x,n,\varepsilon))} \bigg\}, \end{align*} $$

where $S_{n}\phi (B_{w}(x,n,\varepsilon ))=\sup _{y\in B_{w}(x,n,\varepsilon )}\sum _{j=0}^{n-1}\phi _{\theta ^{j}(w)}(f^{j}_{w}(y))$ . When N goes to infinity we consider

$$ \begin{align*} m_{\beta}(w, \phi, F|\theta, \varepsilon)=\lim_{N\to \infty}m_{\beta}(w, \phi, F|\theta, \varepsilon, N). \end{align*} $$

Taking the infimum over $\beta $ , we define

$$ \begin{align*} P_{F|\theta}(w, \phi, \varepsilon)=\inf\{\beta: m_{\beta}(w, \phi, F|\theta, \varepsilon)=0 \}. \end{align*} $$

Since $P_{F|\theta }(w, \phi , \varepsilon )$ is decreasing on $\varepsilon $ we can take the limit

$$ \begin{align*}P_{F|\theta} (w, \phi)=\lim_{\varepsilon\rightarrow 0} {P_{F|\theta} (w, \phi, \varepsilon)}.\end{align*} $$

Now if we consider a finite open cover $\mathcal {U}$ of M with Lebesgue number $\varepsilon (\mathcal {U}{\kern1pt})$ we have

$$ \begin{align*}B_{w}\big(x, n(\mathbf{U}), \tfrac{1}{2}\varepsilon\big)\subset X_{w}(\mathbf{U})\subset B_{w}(x, n(\mathbf{U}), 2|\mathcal {U}|)\end{align*} $$

which implies that

$$ \begin{align*}P_{F|\theta} (w, \phi)=\lim_{\varepsilon\rightarrow 0} {P_{F|\theta} (w, \phi, \varepsilon)}=\lim_{|\mathcal{U}|\rightarrow 0} {P_{F|\theta} (w, \phi, \mathcal{U})}.\end{align*} $$

Therefore, the definitions of random topological pressure via coverings and via dynamic balls coincide.

3.2 Hyperbolic times

In order to explore the non-uniform expansion of the set H we need the notion of hyperbolic times. The reader can obtain more details of this concept in [Reference Alves, Bonatti and Viana2, Reference Arbieto, Matheus and Oliveira5]. In our context, the function $L_{w}(\cdot )$ plays the role of the derivative $\|Df_{w}(\cdot )^{-1}\|.$

Definition 3.1. Let $w\in X$ and $L_{w}:X\to \mathbb {R}$ be as in §2. We say that $n\in \mathbb {N}$ is a c-hyperbolic time for $(w,x)\in X\times M$ if

(3) $$ \begin{align} \prod^{n-1}_{j=n-k}L_{\theta^{j}(w)}(f^{j}_{w}(x)) \leqslant e^{-ck} \quad \text{for every} \ 1\leqslant k\leqslant n. \end{align} $$

It is a well-known fact that if $\eta $ is a non-uniformly expanding measure with exponent c then $\eta $ -almost every point $(w,x)\in H$ has infinitely many c-hyperbolic times. A proof of this result can be found in [Reference Alves, Bonatti and Viana2].

The uniform domination required in condition (VI) will allow us to prove that the F-invariant measure $\mu $ given by Theorem A is non-uniformly expanding. Thus, we will conclude that $\mu $ -almost every point $(w, x)\in X\times M$ has infinitely many hyperbolic times.

Lemma 3.1. Given $c>0$ , there exists $\tilde {\delta }=\tilde {\delta }(c)>0$ such that, for $\mathbb {P}$ -almost every ${w\in X}$ , if n is a c-hyperbolic time of $(w,x)$ then the dynamical ball $B_{w}(x, n,\tilde {\delta })$ around x is mapped homeomorphically onto the ball $B_{\theta ^{n}(w)}(f^{n}_{w}(x), \tilde {\delta }).$ Moreover, for $z\in B_{w}(x, n, \tilde {\delta })$ and $f^{n}_{w}(z)\in B_{\theta ^{n}(w)}(f^{n}_{w}(x), \tilde {\delta })$ , we have

$$ \begin{align*} d(f^{n-k}_{w}(z), f^{n-k}_{w}(x))\leq e^{-ck/2}d(f^{n}_{w}(z),f^{n}_{w}(x)), \end{align*} $$

for each $1\leq k\leq n$ .

We point out that the proof of the lemma above is analogous to the proof of Lemma 5.5 in [Reference Arbieto, Matheus and Oliveira5] since in the definition of hyperbolic times we just replace the function $\|Df_{w}(\cdot )^{-1}\|$ by the Lipschitz function of the inverse branches $L_{w}(\cdot )$ .

Let $\mathcal {B}$ be the Borel $\sigma $ -algebra of $M.$ We say that $\xi $ is a $\mu $ -generating partition if

$$ \begin{align*}\bigvee_{j=0}^{+\infty}f_{w}^{-j}(\xi)\equiv_{\mu}\mathcal{B}\quad\mbox{for } \mathbb{P}\mbox{-almost every } w\in X.\end{align*} $$

The next result states that every non-uniformly expanding measure admits a generating partition. See a proof of this in [Reference Arbieto, Matheus and Oliveira5].

3.3 Projective metrics

To finish this section we present the definition of projective metrics associated to convex cones. This theory was introduced by Birkhoff [Reference Birkhoff10] and provides an interesting way to obtain spectral properties of the transfer operator (see, for instance, [Reference Baladi8, Reference Liverani19]).

Consider a Banach space V. We say that a subset $\mathcal {C}\subset V\setminus \{0\}$ is a cone in V if $\mathcal {C}\cap (-\mathcal {C}) = \{0\} $ and $\unicode{x3bb} \cdot v\in \mathcal {C}$ for all $v\in \mathcal {C}$ , $\unicode{x3bb}>0$ . Moreover, a cone $\mathcal {C}$ is convex if $v,w\in \mathcal {C}$ and $\unicode{x3bb} ,\eta>0$ and we have $\unicode{x3bb} \cdot v + \eta \cdot w\in \mathcal {C}.$ The closure of a cone $\mathcal {C}$ , denoted by $\bar {C}$ , is the set

We say that a cone $\mathcal {C} $ is closed if $\bar {\mathcal {C}}= \mathcal {C} \cup \{0\}$ .

Consider a closed convex cone $\mathcal {C}$ . Given $v, w \in \mathcal {C}$ , define

$$ \begin{align*} A(v,w)= \sup \{t>0 : w-tv \in \mathcal{C} \} \quad \mbox{and} \quad B(v,w)=\inf \{s>0 : sv -w \in \mathcal{C} \}, \end{align*} $$

where by convention $\sup \emptyset = 0$ and $\inf \emptyset = +\infty $ . It is straightforward to check that $A(v,w)$ is finite, $B(v,w)$ is positive and $A(v,w)\leq B(v,w)$ for all $v, w \in \mathcal {C}$ . We set

$$ \begin{align*} \Theta(v,w) = \log\bigg( \frac{B(v,w)}{A(v,w)}\bigg).\end{align*} $$

From the properties of A and B it follows that $\Theta (v,w)$ is well defined and takes values in $[0,+\infty ]$ . Notice that $\Theta (v,w)= 0$ if and only if $v= tw$ for some $t>0$ . Therefore, $\Theta $ defines a pseudo-metric in the cone $\mathcal {C}$ , and so it induces a metric on a projective quotient space of $\mathcal {C}$ . This metric is called the projective metric of $\mathcal {C}$ .

It is easy to verify that the projective metric depends monotonically on the cone: if $\mathcal {C}_{1} \subset \mathcal {C}_{2}$ are two convex cones in V, then $\Theta _{2}(v,w) \leq \Theta _{1}(v,w)$ for any $v,w \in \mathcal {C}_{1}$ , where $\Theta _{1}$ and $\Theta _{2}$ are the projective metrics in $\mathcal {C}_{1}$ and $\mathcal {C}_{2}$ , respectively.

In particular, if $V_{1},V_{2}$ are complete vector spaces and $L:{V}_{1} \to {V}_{2} $ is a linear operator such that $L(\mathcal {C}_{1}) \subset \mathcal {C}_{2}$ for $\mathcal {C}_{1}, \mathcal {C}_{2}$ convex cones in ${V}_{1}, {V}_{2}$ respectively, then $\Theta _{2}(L(v),L(w)) \leq \Theta _{1}(v,w)$ for any $v,w \in \mathcal {C}_{1}$ , where $\Theta _{1}$ and $\Theta _{2}$ are the projective metrics in $\mathcal {C}_{1}$ and $\mathcal {C}_{2}$ , respectively. The next result states that L will be a strict contraction if $L(\mathcal {C}_{1})$ has finite diameter in $\mathcal {C}_{2}$ .

Theorem 3.2. Let $\mathcal {C}_{1}$ and $\mathcal {C}_{2}$ be closed convex cones in the Banach spaces ${V}_{1}$ and ${V}_{2}$ , respectively. If $L:V_{1} \to V_{2} $ is a linear operator such that $L(\mathcal {C}_{1}) \subset \mathcal {C}_{2}$ and $\Delta = \mathrm { diam}_{\Theta _{2}}(L(\mathcal {C}_{1})) < \infty $ , then

$$ \begin{align*}\Theta_{2} (L(\varphi), L(\psi)) \leq (1-e^{-\Delta}) \cdot \Theta_{1} ( \varphi, \psi ) \quad \mbox{for all } \varphi, \psi \in \mathcal{C}_{1}.\end{align*} $$

In this work we will restrict our attention to cones of locally Hölder continuous observables. We prove that, applying the last result, the transfer operator is a contraction in this setting.

We fix $\delta>0$ as in §2 and we say that a function $\varphi :M\to \mathbb {R}$ is $(C, \alpha )$ -Hölder continuous in balls of radius $\delta $ if for some constant $C>0$ it follows that

$$ \begin{align*}|\varphi(x)-\varphi(y)|\leq Cd(x, y)^{\alpha} \quad \mbox{for all} \ y \in B(x, \delta).\end{align*} $$

Denote by $|\varphi |_{\alpha ,\delta }$ the smallest Hölder constant of $\varphi $ in balls of radius $\delta>0$ .

The next lemma states that every locally Hölder continuous function defined on a compact and connected metric space is Hölder continuous.

Proof. The compactness allows us to cover M with N balls of radius $\delta $ where N depends only on $\delta $ . Moreover, since M is connected, given $x, y\in M$ there are $z_{0}=x, z_{1}\ldots z_{N+1}=y$ satisfying $d(z_{i}, z_{i+1})\leq \delta $ and $d(z_{i}, z_{i+1})\leq d(x, y)$ for all $i=0,\ldots , N$ . Since $\varphi $ is $(C, \alpha )$ -Hölder continuous in balls of radius $\delta $ we have that

$$ \begin{align*} |\varphi(x)-\varphi(y) |\leq \sum_{i=0}^{N} | \varphi(z_{i})-\varphi(z_{i+1}) |\leq \sum_{i=0}^{N} C d(z_{i}, z_{i+1})^{\alpha}\leq C(N\!+\!1)d(x, y)^{\alpha}\end{align*} $$

which implies that $\varphi $ is $(C\cdot m, \alpha )$ -Hölder continuous for $m=N + 1$ .

Notice that the same argument used in the lemma above gives an estimate for the Hölder constant of $\varphi $ in balls of radius $(1+r)\delta $ for $0 < r \leq 1.$ Indeed, let $r\in [0, 1]$ and $x, y\in M$ with $d(x, y)<(1+r)\delta $ . Since M is connected there exists $z\in M$ such that $d(x, z)=\delta $ and $d(z, y)<rd(x, z)$ . Thus,

$$ \begin{align*} |\varphi(x)-\varphi(y) |&\leq |\varphi(x)-\varphi(z) |+ |\varphi(z)-\varphi(y) | \nonumber \\ &\leq Cd(x, z)^{\alpha}+Cd(z, y)^{\alpha}\leq C(1+r^{\alpha})d(x, y)^{\alpha}. \end{align*} $$

Therefore, we conclude that if $\varphi : M \to \mathbb {R}$ is $(C, \alpha )$ -Hölder continuous in balls of radius $\delta $ then $\varphi $ is $(C(1+r^{\alpha }), \alpha )$ -Hölder continuous in balls of radius $(1+r)\delta $ for each $0 < r \leq 1.$

For each $k>0$ we consider the convex cone of locally Hölder continuous observables defined on M by

(4) $$ \begin{align} \mathcal{C}^{k}_{\delta}= \bigg\{ \varphi:M\to \mathbb{R} : \varphi>0 \ \mbox{and} \ \frac{|\varphi|_{\alpha,\delta}}{\inf \varphi} \leq k \bigg\}. \end{align} $$

It follows by definition that $\mathcal {C}^{k_{1}}_{\delta } \subset \mathcal {C}^{k_{2},}_{\delta }$ if $k_{1} \leq k_{2}$ .

From Lemma 3.3 and from the definition of $|\varphi |_{\alpha ,\delta }$ we have that

(5) $$ \begin{align} \sup \varphi - \inf\varphi \leq |\varphi|_{\alpha,\delta}\cdot m \cdot d(x, y)^{\alpha} \leq (\inf \varphi \cdot k )\cdot m\cdot (\mathrm{diam}\,M)^{\alpha}, \end{align} $$

and thus $\sup \varphi \leq \inf \varphi \cdot (1+ m(\mathrm {diam}\,M)^{\alpha }k)$ for any $\varphi \in \mathcal {C}^{k}_{\delta }.$

In the cone $\mathcal {C}^{k}_{\delta }$ of locally Hölder continuous observables we can give a more explicit expression for the projective metric. We refer the reader to [Reference Castro and Varandas12] for its proof.

Lemma 3.4. The projective metric $\Theta _{k}$ in the cone $\mathcal {C}^{k}_{\delta }$ is given by

$$ \begin{align*}\Theta_{k}(\varphi , \psi)= \log \bigg( \frac{B_{k}(\varphi, \psi)}{ A_{k}(\varphi, \psi) }\bigg),\end{align*} $$

where

$$ \begin{align*}A_{k}(\varphi, \psi):=\displaystyle\inf_{d(x,y)< \delta, z\in M} \frac{k|x-y|^{\alpha}\psi(z) - (\psi(x)-\psi(y))}{k|x-y|^{\alpha}\varphi(z) - (\varphi(x) - \varphi(y))} \end{align*} $$

and

$$ \begin{align*}B_{k}(\varphi, \psi) := \displaystyle\sup_{d(x,y)< \delta, z\in M} \frac{k|x-y|^{\alpha}\psi(z) - (\psi(x)-\psi(y))}{k|x-y|^{\alpha}\varphi(z) - (\varphi(x) - \varphi(y))}.\end{align*} $$

In particular, we have that

$$ \begin{align*}A_{k}(\varphi, \psi)\leq \inf_{x\in M}\bigg\{\frac{\varphi(x)}{\psi(x)}\bigg\} \quad \mbox{and} \quad B_{k}(\varphi, \psi)\geq \sup_{x\in M}\bigg\{\frac{\varphi(x)}{\psi(x)} \bigg\}.\end{align*} $$

From the expression for the projective metric in the cone $\mathcal {C}^{k}_{\delta }$ one can prove that its diameter is finite for k large enough; see [Reference Castro and Varandas12].

5.1 Invariant family

For the construction of the invariant family $\{h_{w}\}_{w}$ we follow the ideas of Castro and Varandas [Reference Castro and Varandas12].

Let $\delta>0$ be as in §2 and consider for each $k>0$ the cone of locally Hölder continuous functions

(10) $$ \begin{align} \mathcal{C}^{k}_{\delta}(w)= \bigg\{ \varphi_{w}:M\to \mathbb{R} : \varphi_{w}>0 \ \mbox{and} \ \frac{|\varphi_{w}|_{\alpha,\delta}}{\inf \varphi_{w}} \leq k \bigg\}. \end{align} $$

Since the cone does not depend on w, we denote this by $ \mathcal {C}^{k}_{\delta }$ . The next proposition shows its invariance by the transfer operator.

Proposition 5.1. For every $w\in X$ , there exists $0< \gamma _{w} < 1$ such that

$$ \begin{align*} \mathcal{L}_{w}(\mathcal{C}^{k}_{\delta}) \subset \mathcal{C}^{\gamma_{w}k}_{\delta} \subset \mathcal{C}^{k}_{\delta}\end{align*} $$

for some positive constant k large enough.

Proof. For any $\varphi \in \mathcal {C}^{k}_{\delta }$ we will show that

$$ \begin{align*}\frac{| \mathcal{L}_{w}(\varphi)|_{\alpha, \delta}}{\inf \mathcal{L}_{w}(\varphi)} \leq \gamma_{w} k \quad \mbox{for some } 0<\gamma_{w} <1.\end{align*} $$

Given $y, z \in M$ satisfying $d(y, z)< \delta $ , we denote by $y_{j}, z_{j}$ , $1\leq j\leq \deg (f_{w})$ , the respective preimages under $f_{w}$ . Note that for any continuous function $\varphi $ we have

(11) $$ \begin{align} \mathcal{L}_{w}(\varphi)(y) = {\displaystyle \sum_{j=1}^{{\deg(f_{w})}}} e^{\phi_{w}(y_{j})}\varphi(y_{j}) \geq\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi. \end{align} $$

From the definition of $\mathcal {L}_{w}$ and the constant $|\mathcal {L}_{w}(\varphi )|_{\alpha , \delta }$ we obtain

$$ \begin{align*} \dfrac{|\mathcal{L}_{w}(\varphi)|_{\alpha, \delta}}{\inf \mathcal{L}_{w}(\varphi)}&= \sup_{d(y,z)<\delta}\dfrac{|\mathcal{L}_{w}(\varphi(y)) - \mathcal{L}_{w}(\varphi(z)|}{\inf \mathcal{L}_{w}(\varphi) \, d(y,z)^{\alpha}}\\ &\leq \sup_{d(y,z)<\delta}\sum_{j=1}^{{\deg(f_{w})}} \dfrac{ | e^{\phi_{w}(y_{j})}\varphi(y_{j}) -e^{\phi_{w}(z_{j})} \varphi(z_{j}) | }{\inf \mathcal{L}_{w}(\varphi) \, d(y,z)^{\alpha}}. \end{align*} $$

By equation (11) the last inequality is less than or equal to

(12) $$ \begin{align} \sup_{d(y,z)<\delta}\sum_{j=1}^{{\deg(f_{w})}}\bigg(\frac{e^{\sup\phi_{w}} |\varphi(y_{j}) - \varphi(z_{j})|}{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi \, d(y,z)^{\alpha}}+\dfrac{\sup \varphi |e^{\phi_{w}(y_{j})} - e^{\phi_{w}(z_{j})}| }{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi\, d(y,z)^{\alpha}}\bigg). \end{align} $$

Recall that, by hypothesis, we can take some $x\in M$ and neighborhood $U_{x}\subset M$ such that $y, z\in B_{\theta (w)}(f_{w}(x), \delta )$ and so the inverse branches satisfy

$$ \begin{align*}d(f_{w}^{-1}(y), f_{w}^{-1}(z)) \leq L_{w}(x)\,d(y, z).\end{align*} $$

Moreover, we are assuming that every point has $q_{w}$ preimages in the region $\mathcal {A}$ and $p_{w}$ preimages in the expanding region $M\setminus \mathcal {A}$ where $L_{w}\leq \sigma _{w}^{-1}$ . Since, by Lemma 3.3, $\varphi $ is $((1+ (L_{w}-1)^{\alpha })|\varphi |_{\alpha ,\delta }, \alpha )$ -Hölder continuous in balls of radius $L_{w}\delta $ we conclude that the sum (12) is bounded from above by

$$ \begin{align*} &\sum_{j=1}^{q_{w}}\bigg(\dfrac{e^{\sup\phi_{w}}(1+ (L_{w}-1)^{\alpha} ) |\varphi|_{\alpha,\delta}L_{w}^{\alpha} d(y,z)^{\alpha} }{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi \, d(y,z)^{\alpha}}+\dfrac{\sup \varphi |e^{\phi_{w}}|_{\alpha} L_{w}^{\alpha} d(y,z)^{\alpha}}{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi\,d(y,z)^{\alpha}}\bigg)\\ &\quad+\sum_{j=1}^{p_{w}}\bigg(\dfrac{e^{\sup\phi_{w}}|\varphi|_{\alpha,\delta}\sigma_{w}^{-\alpha} d(y,z)^{\alpha} }{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi d(y,z)^{\alpha}}+\frac{\sup \varphi |e^{\phi_{w}}|_{\alpha} L_{w}^{\alpha} d(y,z)^{\alpha} }{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi\, d(y,z)^{\alpha}}\bigg). \end{align*} $$

And this expression is equal to

$$ \begin{align*}\dfrac{e^{\sup\phi_{w}} [p_{w}\sigma_{w}^{-\alpha}+ q_{w} L_{w}^{\alpha} (1+ (L_{w}-1)^{\alpha} ) ] \ |\varphi|_{\alpha,\delta} }{\deg(f_{w})e^{\inf \phi_{w}} \inf\varphi}+ \dfrac{\sup \varphi|e^{\phi_{w}}|_{\alpha} L_{w}^{\alpha}}{e^{\inf \phi_{w}} \inf\varphi }.\end{align*} $$

Using inequality (5), the definition of cone and condition (IV), it follows that the sum above is less than or equal to

$$ \begin{align*}\bigg[e^{\varepsilon_{\phi}}\bigg[\frac{ p_{w}\sigma_{w}^{-\alpha}+ q_{w} L_{w}^{\alpha} (1+ (L_{w}-1)^{\alpha} )}{\deg(f_{w})}\bigg] + \varepsilon_{\phi} L_{w}^{\alpha} [ 1+ m (\mathrm{diam}M)^{\alpha}]\bigg]k. \end{align*} $$

By hypothesis (condition (V)), there exists some positive constant $0<\gamma _{w}< 1$ such that the previous sum is bounded from above by $\gamma _{w} k$ . This finishes the proof.

From the last proposition we have the invariance of the cone $\mathcal {C}^{k}_{\delta }.$ Since this cone has finite diameter, according to Proposition 3.1, we can apply Theorem 3.2 to conclude the next result.

Proposition 5.2. For every $w\in X$ the operator $\mathcal {L}_{w}$ is a contraction in the cone $\mathcal {C}^{k}_{\delta }$ , that is, writing $\Delta _{w} = \mathrm {diam}_{\Theta _{k}}(\mathcal {C}^{\gamma _{w} k}_{\delta })>0$ , it follows that

$$ \begin{align*}\Theta_{k} (\mathcal{L}_{w}(\varphi), \mathcal{L}_{w}(\psi) ) \leq (1-e^{-\Delta_{w}}) \cdot \Theta_{k} ( \varphi, \psi ) \quad \mbox{for all } \varphi, \psi \in \mathcal{C}^{k}_{\delta}.\end{align*} $$

Since we assume in condition (V) the existence of $\gamma \in (0,1)$ such that $\gamma _{w}\leq \gamma $ for all $w\in X$ we conclude that

$$ \begin{align*}\Theta_{k} (\mathcal{L}_{w}(\varphi), \mathcal{L}_{w}(\psi) ) \leq (1-e^{-\Delta}) \cdot \Theta_{k} ( \varphi, \psi ) \quad \mbox{for all } \varphi, \psi \in \mathcal{C}^{k}_{\delta}\ \mbox{and} \ w\in X\end{align*} $$

where $\Delta = \sup _{w} (\Delta _{w})\leq \mathrm {diam}_{\Theta _{k}}(\mathcal {C}^{\gamma k}_{\delta }).$

Let $\{\nu _{w}\}_{w}$ be the family of reference measures and $\unicode{x3bb} _{w}=\nu _{\theta (w)}(\mathcal {L}_{w}(1))$ . The contraction in the cone allows us to prove the existence of the family $\{h_{w}\}_{w}$ invariant by the transfer operator.

Proof. Consider the normalized operator $\hat {\mathcal {L}}_{w}:=\unicode{x3bb} ^{-1}_{w}\mathcal {L}_{w}$ and define the sequence $(\varphi _{n})_{n}$ by $\varphi _{n}:=\hat {\mathcal {L}}^{n}_{\theta ^{-n}(w)}(\mathbf {1})$ where

$$ \begin{align*} \hat{\mathcal{L}}^{n}_{\theta^{-n}(w)}:= \hat{\mathcal{L}}_{\theta^{-1}(w)}\circ \hat{\mathcal{L}}_{\theta^{-2}(w)}\circ \cdots \circ \hat{\mathcal{L}}_{\theta^{-(n-1)}(w)}\circ \hat{\mathcal{L}}_{\theta^{-n}(w)} \end{align*} $$

for each $n\geq 0$ . By definition of conformal measure we have

$$ \begin{align*} \int \varphi_{n}\,d\nu_{w} = \int \hat{\mathcal{L}}^{n}_{\theta^{-n}(w)}(\mathbf{1})\,d\nu_{w} =\int\mathbf{1}\,d(\hat{\mathcal{L}}^{\ast})^{n}_{\theta^{-n}(w)} \nu_{w} = \int \mathbf{1}\,d \nu_{\theta^{-n}(w)}=1. \end{align*} $$

Hence, each term $\varphi _{n}$ satisfies $\sup \varphi _{n}\geq 1$ and $\inf \varphi _{n}\leq 1$ . Since $\mathbf {1}\in \mathcal {C}^{k}_{\delta }$ and $\mathcal {C}^{k}_{\delta }$ is invariant, it follows that $\varphi _{n}\in \mathcal {C}^{k}_{\delta } $ and so, applying inequality (5), we obtain that the sequence $(\varphi _{n})_{n}$ is uniformly bounded away from zero and infinity by

$$ \begin{align*}\frac{1}{R}\leq\inf\varphi_{n} \leq 1\leq \sup \varphi_{n} \leq R. \end{align*} $$

where $R=(1 + mk \mathrm {diam}(M)^{\alpha })$ . Moreover, as $\varphi _{n}$ is C-Hölder continuous in balls of radius $\delta $ , by Lemma 3.3 we obtain that $\varphi _{n}$ is a $Cm$ -Hölder continuous function.

Next we prove that $(\varphi _{n})_{n}$ is a Cauchy sequence in the $C^{0}$ -norm. From Proposition 5.2, for every $m, l\geq n$ the projective metric satisfies

$$ \begin{align*}\Theta_{k}(\varphi_{m}, \varphi_{l})=\Theta_{k}(\hat{\mathcal{L}}^{m}_{\theta^{-m}(w)}(\mathbf{1}), \hat{\mathcal{L}}^{l}_{\theta^{-l}(w)}(\mathbf{1})) \leq \Delta \tau^{n} \quad \mbox{where } \tau:= 1 - e^{-\Delta}. \end{align*} $$

Recalling the expression for the projective metric $\Theta _{k}(\varphi _{m}, \varphi _{l})= \log ({B_{k}(\varphi _{m}, \varphi _{l})}/{A_{k}} {(\varphi _{m}, \varphi _{l})})$ , we apply Lemma 3.4 to obtain

$$ \begin{align*} e^{-\Delta \tau^{n}}\leq A_{k}(\varphi_{m}, \varphi_{l})\leq\inf \frac{\varphi_{m}}{\varphi_{l}} \leq1\leq \sup \frac{\varphi_{m}}{\varphi_{l}} \leq B_{k}(\varphi_{m}, \varphi_{l}) \leq e^{\Delta \tau^{n}}. \end{align*} $$

Thus, for all $m, l\geq n$ , we have

$$ \begin{align*} \|\varphi_{m} - \varphi_{l} \|_{\infty} \leq \|\varphi_{l} \|_{\infty} \bigg\|\frac{\varphi_{m}}{\varphi_{l}} - 1 \bigg\|_{\infty} \leq R (e^{\Delta \tau^{n}}-1) \leq \tilde{R} \tau^{n} \end{align*} $$

which proves that $(\varphi _{n})_{n}$ is a Cauchy sequence. Hence, $(\varphi _{n})_{n}$ converges uniformly to a function $h_{w}:M\to \mathbb {R}$ in the cone $\mathcal {C}^{k}_{\delta }$ satisfying $\int h_{w} \,d\nu _{w} =1$ for almost every $w\in X$ . In particular, this function is Hölder continuous and uniformly bounded away from zero and infinity. To complete the proof of the proposition, we will show that $\mathcal {L}_{w}h_{w}=\unicode{x3bb} _{w} h_{\theta (w)}.$ Consider the sequence

$$ \begin{align*}\tilde{\varphi}_{n, w}:=\frac{1}{n}\sum_{j=0}^{n-1}\varphi_{j}=\frac{1}{n}\sum_{j=0}^{n-1}\hat{\mathcal{L}}^{j}_{\theta^{-j}(w)}(\mathbf{1}).\end{align*} $$

By what we have proved above, $(\tilde {\varphi }_{n, w})$ converges uniformly to $h_{w}$ for almost every $w\in X.$ From the continuity of $\mathcal {L}_{w}$ we obtain

$$ \begin{align*} \hat{\mathcal{L}}_{w}(h_{w})=\lim_{n\to +\infty}\hat{\mathcal{L}}_{w}(\tilde{\varphi}_{n, w})&=\lim_{n\to +\infty}\frac{1}{n}\sum_{j=0}^{n-1}\hat{\mathcal{L}}_{w}(\hat{\mathcal{L}}^{j}_{\theta^{-j}(w)}(\mathbf{1}))\\ &=\lim_{n\to +\infty}\frac{1}{n}\sum_{j=0}^{n-1}\hat{\mathcal{L}}^{j}_{\theta^{-j}(\theta(w))}(\mathbf{1})+\frac{1}{n}(\hat{\mathcal{L}}^{n}_{\theta^{-n}(\theta(w))}(\mathbf{1})-\mathbf{1}). \end{align*} $$

Since $\hat {\mathcal {L}}^{n}_{\theta ^{-n}(\theta (w))}(\mathbf {1})$ is uniformly bounded we conclude that $\hat {\mathcal {L}}_{w}(h_{w})=h_{\theta (w)}.$

From the proof of the previous proposition we conclude that the family $\{h_{w}\}_{w\in X}$ is uniquely determined. Moreover, every $h_{w}$ satisfies

$$ \begin{align*}\frac{1}{R}\leq\inf h_{w} \leq 1\leq \sup h_{w} \leq R. \end{align*} $$

where $R=(1 + mk\, \mathrm {diam}(M)^{\alpha })$ .

5.2 Measurability

In the previous subsection we proved the existence of a family $\{h_{w}\}_{w}$ invariant under the action of the transfer operator. Here we prove that this family is measurable as well as the family $\{\nu _{w}\}_{w}$ . Moreover, defining the probability measure $\mu _{w}:=h_{w}\nu _{w}$ , we also prove that $\mu _{w}$ has an exponential decay of correlations and that the family $\{\mu _{w}\}_{w}$ is F-invariant.

The next proposition states an exponential approximation of functions in the cone to the invariant family $\{h_{w}\}_{w}$ . This is the main ingredient in the proof of the exponential decay of correlations.

Proposition 5.4. For almost $w\in X$ there exist constants $K>0$ and $0<\tau <1$ such that for every $\varphi \in \mathcal {C}^{k}_{\delta }$ satisfying $\int \varphi \,d\nu _{w} = 1$ we have that

$$ \begin{align*}\| \hat{\mathcal{L}}_{w}^{n}(\varphi) - \hat{\mathcal{L}}_{w}^{n}(h_{w}) \|_{\infty} \leq K\tau^{n} \quad \mbox{for all } n\geq 1,\end{align*} $$

where $\hat {\mathcal {L}}_{w}=\unicode{x3bb} _{w}^{-1}\mathcal {L}_{w}$ is the normalized operator.

Proof. Given $\varphi \in \mathcal {C}^{k}_{\delta }$ satisfying $\int \varphi \,d\nu _{w} = 1$ , we have for every $n \geq 1$ ,

$$ \begin{align*} \int \hat{\mathcal{L}}_{w}^{n}(\varphi) \,d\nu_{\theta^{n}(w)} = \int \varphi \,d(\mathcal{L}^{\ast}_{w})^{n}(\nu_{\theta^{n}(w)}) = \int \varphi\,d\nu_{w} =1 .\end{align*} $$

Since $h_{w} \in \mathcal {C}^{k}_{\delta }$ also satisfies $\int h_{w} \,d\nu _{w} = 1$ , we derive for every $n\geq 1$ that

$$ \begin{align*}\inf\frac{\hat{\mathcal{L}}_{w}^{n}(\varphi)}{\hat{\mathcal{L}}_{w}^{n}(h_{w})} \leq 1\leq \sup \frac{\hat{\mathcal{L}}_{w}^{n}(\varphi)}{\hat{\mathcal{L}}_{w}^{n}(h_{w})}.\end{align*} $$

Recalling that $\hat {\mathcal {L}}^{n}_{w}(h_{w})=h_{\theta ^{n}(w)}$ for all $n\geq 1$ , we apply the same projective metric argument used in the proof of Proposition 5.3 to obtain

$$ \begin{align*}\| \hat{\mathcal{L}}_{w}^{n}(\varphi) - \hat{\mathcal{L}}_{w}^{n}(h_{w}) \|_{\infty} \leq \|h_{\theta^{n}(w)} \|_{\infty} \bigg\| \frac{\hat{\mathcal{L}}_{w}^{n}(\varphi)}{\hat{\mathcal{L}}_{w}^{n}(h_{w})} -1 \bigg\|_{\infty} \leq R (e^{\Delta\tau^{n}} -1 ) \leq K\tau^{n}.\\[-44pt] \end{align*} $$

Let $\mu _{w}$ be the probability measure defined by $\mu _{w}:=h_{w}\nu _{w}.$ From the last proposition we derive the proof of Theorem B.

Theorem B. For almost every $w\in X,$ the probability measure $\mu _{w}$ has exponential decay of correlations for Hölder continuous observables: there exists $0<\tau < 1$ such that for any $\varphi \in L^{1}(\mu _{\theta ^{n}(w)})$ and $\psi \in C^{\alpha }(M) $ there exists a positive constant $K(\varphi , \psi ) $ satisfying for all $n\geq 1$ that

$$ \begin{align*}\bigg|\int (\varphi \circ f^{n}_{w}) \psi \,d\mu_{w} - \int \varphi \,d\mu_{\theta^{n}(w)} \int \psi\,d\mu_{w} \bigg| \leq K(\varphi, \psi) \tau^{n}.\end{align*} $$

Proof. Given $\varphi \in L^{1}(\mu _{\theta ^{n}(w)})$ and $\psi \in C^{\alpha }(M)$ , we suppose without loss of generality that $\int \psi \,d\mu _{w} = 1$ . Let $\hat {\mathcal {L}}_{w}=\unicode{x3bb} ^{-1}_{w}\mathcal {L}_{w}$ be the normalized operator. As a first case we consider $\psi \cdot h_{w}$ in the cone $\mathcal {C}^{k}_{\delta }$ for k large enough. Recalling that $\mu _{w}=h_{w}\nu _{w}$ and that $\mathcal {L}^{*}_{w}\nu _{\theta (w)}=\unicode{x3bb} _{w}\nu _{w}$ , we have

$$ \begin{align*} &\bigg|\int (\varphi \circ f^{n}_{w}) \psi \,d\mu_{w} - \int\varphi\,d\mu_{\theta^{n}(w)} \int \psi\,d\mu_{w} \bigg|\\[6pt] &\quad= \bigg|\int (\varphi \circ f^{n}_{w}) \psi\cdot h_{w} \,d\nu_{w} - \int \varphi\cdot h_{\theta^{n}(w)} \,d\nu_{\theta^{n}(w)} \bigg|\\[6pt] &\quad=\bigg| \int \varphi \cdot \hat{\mathcal{L}}_{w}^{n}( \psi \cdot h_{w} ) \,d\nu_{\theta^{n}(w)} - \int \varphi\cdot h_{\theta^{n}(w)} \,d\nu_{\theta^{n}(w)} \bigg|\\[6pt] &\quad\leq \|\varphi\|_{1}\| \hat{\mathcal{L}}_{w}^{n}( \psi \cdot h_{w} ) -\hat{\mathcal{L}}_{w}^{n}(h_{w} ) \|_{\infty}. \end{align*} $$

Since $\psi \cdot h_{w}\in \mathcal {C}^{k}_{\delta }$ and $\int \psi \cdot h_{w} \,d\nu _{w}=\int \psi \,d\mu _{w} = 1$ we can apply Proposition 5.4 to conclude the existence of constants $K>0$ and $0<\tau <1$ such that

$$ \begin{align*}\| \hat{\mathcal{L}}_{w}^{n}( \psi \cdot h_{w} ) -\hat{\mathcal{L}}_{w}^{n}(h_{w} ) \|_{\infty}\leq K \tau^ n\quad\mbox{for every } n\geq 1.\end{align*} $$

For the general case we write $\psi \cdot h_{w}= g$ where

$$ \begin{align*}g=g^{+}-g^{-}; g^{\pm}=\tfrac{1}{2}(|g|\pm\psi)+C\quad \mbox{and}\quad C=k^{-1}|\psi \cdot h_{w}|_{\alpha, \delta}.\end{align*} $$

Therefore, $g^{\pm }\in \mathcal {C}^{k}_{\delta }.$ From the previous estimates on $g^{\pm }$ and by linearity the proposition holds.

Now we state the measurability of the families $\{\nu _{w}\}_{w}$ and $\{h_{w}\}_{w}$ . We start by observing that for almost every $w\in X$ and every continuous function $g\in C^{0}(M)$ we have

(13) $$ \begin{align} \frac{1}{h_{\theta^{n}(w)}}\,\hat{\mathcal{L}}^{n}_{w}(g\cdot h_{w})\longrightarrow \int g\cdot h_{w}\, d\nu_{w}\quad \mbox{when } n\to\infty, \end{align} $$

where $\hat {\mathcal {L}}$ is the normalized operator $\hat {\mathcal {L}}_{w} = \unicode{x3bb} ^{-1}_{w} \mathcal {L}_{w}.$ Indeed, we can suppose that the function $g_{w}$ is Hölder continuous because any continuous function is approximated by such functions. Moreover, following the proof of Theorem 5.2, we can just consider the case $ g \cdot h_{w} \in \mathcal {C}^{k}_{\delta }$ for k large enough. We have that

$$ \begin{align*} \bigg\|\frac{1}{h_{\theta^{n}(w)}}\,\hat{\mathcal{L}}^{n}_{w}(g h_{w}) - \int g h_{w}\, d\nu_{w} \bigg\|_{\infty} &\leq \bigg\|\dfrac{1}{h_{\theta^{n}(w)} }\bigg\| \bigg\| \hat{\mathcal{L}}^{n}_{w}(g h_{w}) - \int g h_{w}\, d\nu_{w}\cdot h_{\theta^{n}(w)} \bigg\| \\ &\leq \displaystyle R \|g h_{w}\|_{\infty} \bigg\| \hat{\mathcal{L}}^{n}_{w}\bigg(\frac{gh_{w}}{\int g h_{w}\, d\nu_{w}}\bigg) - \hat{\mathcal{L}}^{n}_{w}(h_{w}) \bigg\|_{\infty}. \end{align*} $$

By Proposition 5.4 the convergence in (13) follows. We use this on the next result.

Lemma 5.1. Let $\unicode{x3bb} _{w}=\nu _{\theta (w)}(\mathcal {L}_{w}(1)).$ The family $\{\nu _{w}\}$ is uniquely determined by

Moreover, the map $w\mapsto \nu _{w}(g_{w})$ is measurable for any $g\in \mathbb {L}^{1}_{\mathbb {P}}(X, C^{0}(M))$ .

Proof. Fix $w\in X$ and let $(x_{n})$ be a sequence of points in M. Define the probability

$$ \begin{align*}\nu_{w, n}=\frac{(\mathcal{L}^{n}_{w})^{\ast}\delta_{x_{n}}}{\mathcal{L}^{n}_{w}1(x_{n})}.\end{align*} $$

Since $\nu _{w}$ satisfies the condition $\mathcal {L}_{w}^{\ast }\nu _{\theta (w)}=\unicode{x3bb} _{w}\nu _{w} $ we can apply the convergence (13) to conclude that for almost every $w\in X$ and any continuous function $g_{w}$ we have

$$ \begin{align*}\lim_{n\to\infty}{\nu_{w, n}(g_{w})}=\lim_{n\to\infty}\frac{(\mathcal{L}^{n}_{w})^{\ast}\delta_{x_{n}}(g_{w})}{\mathcal{L}^{n}_{w}1(x_{n})}&=\lim_{n\to\infty}\frac{\mathcal{L}^{n}_{w}g_{w}(x_{n})}{\mathcal{L}^{n}_{w}1(x_{n})}\\[6pt] &=\displaystyle\lim_{n\to\infty}\frac{\mathcal{L}^{n}_{w}\bigg(\displaystyle\frac{g_{w}}{h_{w}}\cdot h_{w}\bigg)(x_{n})}{\mathcal{L}^{n}_{w}\bigg(\displaystyle\frac{1}{h_{w}}\cdot h_{w}\bigg)(x_{n})}=\nu_{w}(g_{w}). \end{align*} $$

The convergence of $\nu _{w,n}\stackrel {w*}\longrightarrow \nu _{w}$ therefore follows. Since the sequence $(x_{n})$ was arbitrary the uniqueness of the family $\{\nu _{w}\}$ is proved. Moreover, the equality

$$ \begin{align*}\lim_{n\to\infty}\frac{\|\mathcal{L}^{n}_{w}g_{w}\|_{\infty}}{\|\mathcal{L}^{n}_{w}1\|_{\infty}}=\nu_{w}(g_{w})\end{align*} $$

implies the measurability of $w\mapsto \nu _{w}(g_{w})$ since the transfer operator is measurable.

The lemma above enables us to define the probability measure $\nu $ on the Borel sets of $X\times M$ by

$$ \begin{align*}\nu(g)=\int_{X}\, \int_{M}\, g_{w}\, d\nu_{w}\,d\mathbb{P}(w).\end{align*} $$

Let $c>0$ be given by condition (VI) and let $H\subset X\times M$ be the non-uniformly expanding set defined in (⋆). As in §4, for $w\in X$ consider

$$ \begin{align*}H_{w}:=\bigg\{x\in M; \limsup_{n\to+\infty} \frac{1}{n}\sum^{n-1}_{j=0}\log L_{\theta^{j}(w)}(f^{j}_{w}(x))\leqslant -2c<0\bigg\}.\end{align*} $$

By Proposition 4.2 for almost every $w\in X$ we have $\nu _{w}(H_{w})=1$ . Thus, we conclude that $\nu (H)=1$ , that is, $\nu $ is a non-uniformly expanding measure.

Notice that since $\unicode{x3bb} _{w}=\nu _{\theta (w)}(\mathcal {L}_{w}(1))$ we have that the map $w\mapsto \unicode{x3bb} _{w}\in \mathbb {R}$ is measurable. Recalling that for almost every $w\in X$ the function $h_{w}$ is given by

$$ \begin{align*}h_{w}=\lim_{n\to\infty}{\hat{\mathcal{L}}^{n}_{\theta^{-n}(w)}1},\end{align*} $$

we deduce the measurability of the map $(w, y)\mapsto h_{w}(y)$ from the measurability of $\unicode{x3bb} _{w}$ and the transfer operator. From this result the probability measure $\mu _{F, \phi }\in \mathcal {M}_{\mathbb {P}}(X\times M)$ is well defined by the formula

$$ \begin{align*}\mu_{F, \phi}(g)=\int_{X}\, \int_{M} \, g_{w}\cdot h_{w}\, d\nu_{w}\, d\mathbb{P}(w).\end{align*} $$

In order to prove the F-invariance of $\mu _{F, \phi }$ we first observe that

$$ \begin{align*} \mu_{w} (g_{\theta(w)}\circ f_{w}) &= \int\! g_{\theta(w)}\circ f_{w}\cdot h_{w}\,d\nu_{w}\\ &=\int \hat{\mathcal{L}}_{w} (g_{\theta(w)}\circ f_{w} \cdot h_{w}) \,d\nu_{\theta(w)}\\ &=\int\dfrac{g_{\theta(w)}\cdot \hat{\mathcal{L}}_{w} (h_{w})}{h_{\theta(w)}} \,d\mu_{\theta(w)} = \mu_{\theta(w)}(g_{\theta(w)}). \end{align*} $$

Therefore, for every integrable function we get

$$ \begin{align*} \int\!\! g \circ F \,d\mu_{F, \phi} = \int\!\!\! \int\!\! g_{\theta(w)}\circ f_{w}(x) \,d\mu_{w} d\mathbb{P}(w) = \int\!\!\! \int\!\!g_{\theta(w)} \,d\mu_{\theta(w)} \,d\mathbb{P}(w) = \int\!\! g\,d\mu_{F, \phi}. \end{align*} $$

This finishes the proof of Theorem A.

6.1 Existence

Consider the family $\{\nu _{w}\}_{w\in X}$ of reference measures and $\{h_{w}\}_{w\in X}$ as in Theorem A. For each $w\in X$ let $\mu _{w}$ be the probability measure defined by $\mu _{w}=h_{w}\nu _{w}$ . Recalling that the Jacobian of $\nu _{w}$ is $J_{\nu _{w}}f_{w}=\unicode{x3bb} _{w}e^{-\phi _{w}}$ , it is easy to verify that the Jacobian of $\mu _{w}$ relative to $f_{w}$ is given by

(14)

Consider the probability measure $\mu _{F, \phi }$ whose disintegration is $\{\mu _{w}\}_{w}$ , that is,

$$ \begin{align*}\mu_{F, \phi}(g)=\int_{X}\, \int_{M} \, g_{w}\cdot h_{w}\, d\nu_{w}\, d\mathbb{P}(w),\end{align*} $$

for every continuous function $g:X\times M\to \mathbb {R}$ . As we saw in §5.2, $\mu _{F,\phi }$ is F-invariant. In the next proposition we show its ergodicity.

Proof. Given a F-invariant set $A\subset X\times M$ , for each $w\in X$ denote by $A_{w}$ the set $A_{w}=\{z\in M; (w, z)\in A\}.$ The F-invariance of A implies that $f_{w}^{-1}(A_{\theta (w)})=A_{w}.$ Consider $X_{0}=\{w\in X; \mu _{w}(A_{w})>0\}$ . It is straightforward to check that $X_{0}$ is a $\theta $ -invariant subset of X. Since $\theta $ is ergodic with respect to $\mathbb {P}$ , we will obtain the ergodicity of $\mu _{F, \phi }$ by showing that for almost every $w\in X_{0}$ we have $\mu _{w}(A_{w})=1$ , if $\mathbb {P}(X_{0})>0.$

Let $\varphi _{w}$ be the characteristic function of $A_{w}$ , that is, $\varphi _{w}=\mathbf {1}_{A_{w}}$ . Notice that $\varphi _{\theta ^{n}(w)}\circ f_{w}^{n}=\varphi _{w}$ holds $\mathbb {P}$ -almost everywhere. Given $\psi _{w}\in L^{1}(\mu _{w})$ such that $\int \psi _{w}\, d\mu _{w}=0$ , it follows from the decay correlation property of $\mu _{w}$ (Theorem B) that

$$ \begin{align*}\mu_{w}((\varphi_{\theta^{n}(w)}\circ f_{w}^{n})\cdot \psi_{w})\rightarrow 0\quad \mbox{when } n\to+\infty.\end{align*} $$

And thus $\int _{A_{w}} \psi _{w}\, d\mu _{w}=0$ for any $\psi _{w}\in L^{1}(\mu _{w})$ satisfying $\int \psi _{w}\, d\mu _{w}=0$ . This proves that $\mu _{w}(A_{w})=1$ for $\mathbb {P}$ -almost every $w\in X_{0}$ which finishes the proof.

In §5.2 we observe that the measure $\nu $ defined by

$$ \begin{align*}\nu(g)=\int_{X}\, \int_{M}\, g_{w}\, d\nu_{w}\,d\mathbb{P}(w)\end{align*} $$

is non-uniformly expanding. Since $\mu _{F, \phi }$ is absolutely continuous with respect to it, we have that $\mu _{F, \phi }$ is also a non-uniformly expanding measure. In particular, by Lemma 3.2, it admits a generating partition. Thus, we can use the random Rokhlin formula to express the entropy of $\mu _{F, \phi }$ in terms of its Jacobian.

Theorem 6.1. (Random Rokhlin formula)

Let $\mu \in \mathcal {M}_{\mathbb {P}}(F)$ be an ergodic measure which admits a $\mu $ -generating partition. Then

$$ \begin{align*} h_{\mu}(F|\theta)= \int \log J_{\mu} (F) \,d\mu = \int_{X}\bigg( \int_{M}\log J_{\mu_{w}}f_{w}(y) \,d\mu_{w}(y)\bigg) \,d\mathbb{P}(w), \end{align*} $$

where $J_{\mu _{w}}f_{w}$ denotes the Jacobian of $f_{w}$ relative to $\mu _{w}$ .

The reader can consult [Reference Przytycki and Urbanski23, Theorem 1.9.7] for a proof of the last result. Now we are ready to prove that $\mu _{F, \phi }$ is an equilibrium state for $(F|\theta , \phi )$ . We have

From the F-invariance of $\mu _{F, \phi }$ we derive that

$$ \begin{align*}\int_{X} \int_{M} \bigg(\log h_{\theta(w)}\circ f_{w}(y) - \log h_{w}(y)\bigg)\, d\mu_{w} (y) \,d\mathbb{P}(w)=0.\end{align*} $$

Thus, we can write

Applying the variational principle (1) and Proposition 6.1, it follows that

which implies, in particular, that $P_{ F|\theta }(\phi )=\int _{X}\log \unicode{x3bb} _{w}\, d\mathbb {P}(w)$ and so $\mu _{F, \phi }$ is an equilibrium state.

6.2 Uniqueness

So far we have proved the existence of an equilibrium state for $(F|_{\theta }, \phi )$ . Here we prove uniqueness in the set of non-uniformly expanding measures.

Let $\eta $ be an ergodic non-uniformly expanding equilibrium state for $(F|_{\theta }, \phi )$ . We prove that the disintegration of $\eta $ is absolutely continuous to the reference measure. For this we use the following remark from the basic calculus.

Proof. We begin by proving that for almost every $w\in X$ the Jacobian of $\eta _{w}$ is given by

Indeed, since $\eta $ is an ergodic non-uniformly expanding equilibrium state we can apply the Rokhlin formula to obtain that

$$ \begin{align*}h_{\eta}(F|\theta)= \int_{X}\bigg( \int_{M}\log J_{\eta_{w}}f_{w}(y)\, d\eta_{w}(y)\bigg)\, d\mathbb{P}(w)=\int_{X\times M}\log J_{\eta_{w}}f_{w}(y)\, d\eta(w, y).\end{align*} $$

Recalling that $h_{w}$ is a bounded function and that the Jacobian of $\mu _{w}$ is given by $J_{{\mu }_{w}}{f_{w}}\;=\unicode{x3bb} _{w} e^{-{\phi _{w}}}\cdot {h_{\theta (w)}}\circ {f_{w}}/{h_{w}}$ , we have

From the definition of Jacobian we can write

(15) $$ \begin{align} \int{\sum_{z=f_{w}^{-1}(y)}{J_{\eta_{w}}f_{w}^{-1}(z)}\log\frac{J_{\eta_{w}}f_{w}}{J_{\mu_{w}}f_{w}}(z)}\,d\eta(w, y)=\int{\log\frac{J_{\eta_{w}}f_{w}}{J_{\mu_{w}}f_{w}}(y)}\,d\eta(w, y)\geq 0. \end{align} $$

Take $p_{i}=J_{\eta _{w}}f_{w}^{-1}(z_{i})$ and $q_{i}=J_{\eta _{w}}f_{w}(z_{i})/J_{\mu _{w}}f_{w}(z_{i})$ where the $z_{i}$ are the preimages of y. Since $(f_{w})_{\ast }\eta _{w}=\eta _{\theta (w)}$ we have $\sum _{i=1}^{\deg (f_{w})}{p_{i}}=\sum _{z=f_{w}^{-1}(y)}J_{\eta _{w}}f_{w}^{-1}(z)=1$ for $\eta _{\theta (w)}$ -almost every $y\in M.$ Therefore, we can apply Remark 6.1 to conclude that

for $\eta _{\theta (w)}$ -almost every $y\in M$ . Recalling the inequality (15), we obtain that

$$ \begin{align*}0\leq\int{\log\frac{J_{\eta_{w}}f_{w}}{J_{\mu_{w}}f_{w}}(y)}\,d\eta(w, y)=\int{\sum_{z=f_{w}^{-1}(y)}{J_{\eta_{w}}f_{w}^{-1}(z)}\log\frac{J_{\eta_{w}}f_{w}}{J_{\mu_{w}}f_{w}}(z)}\,d\eta(w, y)=0.\end{align*} $$

Thus, from the second part of Remark 6.1, the values $q_{i}=J_{\eta _{w}}f_{w}(z_{i})/J_{\mu _{w}}f_{w}(z_{i})$ must be the same for all $z_{i}\in f_{w}^{-1}(y)$ in a full $\eta _{\theta (w)}$ -measure set. In other words, for every $y\in M$ on the preimage of a full $\eta _{\theta (w)}$ -measure set, we have

To finish the proof of the proposition we observe that ${1}/{h_{w}}\cdot \eta _{w}$ is a reference measure associated to $\unicode{x3bb} _{w}$ for the dual transfer operator:

From the uniqueness given by Theorem A we conclude that ${1}/{h_{w}}\cdot \eta _{w}$ is equivalent to $\nu _{w}$ and thus $\eta _{w}$ is absolutely continuous to the latter.

Finally, we prove the uniqueness of the equilibrium state associated to $(F|_{\theta }, \phi ).$ Suppose that there exist two ergodic equilibrium states $\mu $ and $\eta $ . Let $(\mu _{w})_{w\in X}$ and $(\eta _{w})_{w\in X}$ be the disintegration of $\mu $ and $\eta $ , respectively.

By the proposition above we have that $\mu _{w}$ and $\eta _{w}$ are equivalent measures. From the Radon–Nikodym theorem we know that there exists a mensurable function $q_{w}: M\rightarrow \mathbb {R}$ such that $\mu _{w}=q_{w}\eta _{w}$ for every $w\in X$ . Consider $q: X\times M \rightarrow \mathbb {R}$ , defined by $q(w,x)=q_{w}(x)$ . Given a measurable set $E\subset X\times M$ , consider $E_{w}\subset M$ the intersection $E_{w}=E\cap M$ . Then we have that

$$ \begin{align*} \mu(E)=\int_{X}\mu_{w}(E_{w})\,d\mathbb{P}(w)=\int_{X}\int_{E_{w}}q_{w}\,d\eta_{w}\,d\mathbb{P}(w)=\int_{E}q \,d\eta. \end{align*} $$

Moreover, it follows from the F-invariance of $\mu $ and $\eta $ that

$$ \begin{align*} \mu(E)=F_{\ast}\mu(E)=(q\circ F)F_{\ast}\eta(E)=(q\circ F)\eta(E). \end{align*} $$

Since the Radon–Nikodym derivative is essentially unique, we conclude that $q=q\circ F$ at $\eta $ -almost every point. By ergodicity we have that q is constant everywhere and thus $\mu =\eta .$

6.3 Positive Lyapunov exponents

The main tool in the proof of Proposition 6.3 is the existence of a generating partition for the equilibrium state. In the context of random dynamical systems generated by non-uniformly expanding maps, the existence of generating partitions for ergodic measures with Lyapunov exponents bounded away from zero was proved by Bilbao and Oliveira in [Reference Bilbao and Oliveira9]. Therefore, in this setting we can also apply our Proposition 6.3 to obtain uniqueness of equilibrium states. This is our goal now.

Consider a compact and connected Riemann manifold $M^{d}$ of dimension d. Let $F:X\times M\to X\times M$ be the skew-product $(\theta (w), f_{w}(x))$ generated by $C^{1}$ local diffeomorphisms $f_{w}: M\to M$ satisfying conditions (I)–(III). For $1\leq k\leq d-1$ , define

$$ \begin{align*} C_{k}(w,x)=\limsup_{n\rightarrow +\infty} \frac{1}{n}\log \|\Lambda^{k}Df_{w}^{n}(x)\| \quad \mbox{and} \quad C_{k}(w,F)=\max_{x \in M}C_{k}(w,x),\end{align*} $$

where $\Lambda ^{k}$ is the kth exterior product. We suppose that, for some $\varepsilon>0$ ,

$$ \begin{align*} \beta(F):=(1-\varepsilon)\int \log \deg(f_{w}) d \mathbb{P}(w) - \max_{1\leqslant k\leqslant d -1}\int_{X} C_{k}(w,F) \,d\mathbb{P}(w)>0. \end{align*} $$

For potentials $\phi \in \mathbb {L}^{1}_{\mathbb {P}}(X, C^{\alpha }(M))$ satisfying conditions (IV) and (V) such that for almost every $w\in X$ the inequality $\sup \phi _{w}-\inf \phi _{w}<\varepsilon \int \log \deg (f_{w})\,d \mathbb {P}(w)$ holds, we obtain the following result.

Proof. Let $\eta \in \mathcal {M}_{\mathbb {P}}(F|\theta )$ be an ergodic equilibrium state for $(F|_{\theta }, \phi )$ . Denote by $\unicode{x3bb} _{1}(w,x)\leq \cdots \leq \unicode{x3bb} _{d}(w,x)$ the Lyapunov exponents of $\eta $ at $(w,x)$ . We claim that they are bigger than $\beta (F)>0$ . If not, by applying the random version of the Marquis–Ruelle inequality [Reference Liu18, Theorem 2.4] we have

Thus, for potentials such that $\sup \phi _{w}-\inf \phi _{w}<\varepsilon \int \log \deg (f_{w})\,d \mathbb {P}(w)$ it follows that

$$ \begin{align*} h_{\eta}(F|\theta) + \int \phi \,d\eta &\leq (1-\varepsilon)\int \log \deg(f_{w}) \,d \mathbb{P}(w) + \int\sup\phi_{w} \,d \eta\\ &<\int \log \deg(f_{w}) \,d \mathbb{P}(w)+ \int \inf \phi_{w} \,d\eta \leq P_{F|\theta}(\phi). \end{align*} $$

which is a contradiction. Therefore, the Lyapunov exponents of $\eta $ are bigger than $\beta (F)$ , and so $\eta $ admits generating partitions with small diameter. Applying the proof of Proposition 6.3, we have that $\eta $ is absolutely continuous with respect to $\nu $ , and thus the uniqueness is proved.