Equilibrium states for non-uniformly expanding skew products

Abstract

We study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.

1 Introduction and main results

The study of equilibrium states for uniformly hyperbolic dynamical systems via thermodynamic formalism dates back to Ruelle, Sinai, and Bowen in the mid-1970s. They proved the existence and uniqueness of an equilibrium state for Hölder continuous potentials by (semi)conjugating with two-sided subshifts of finite type (SFTs) via Markov partitions. The Ruelle–Perron–Frobenius (RPF) operator has been a key tool in proving the existence of equilibrium states. In particular, it is well known that given a Hölder potential on an expanding dynamical system $(X,f)$ , the eigendata of the RPF operator

$$ \begin{align*}\mathcal{L}_\varphi\psi(x)=\sum_{\overline{x}\in f^{-1}(x)}e^{\varphi(\overline{x})}\psi(\overline{x}),\end{align*} $$

where $\psi \colon X\to X$ is a continuous function, uniquely determines the equilibrium state.

The purpose of this paper is to understand the construction of equilibrium states for higher-dimensional spaces. Let X and Y be compact connected manifold. We will consider equilibrium states for certain non-uniformly expanding skew products $F(x,y)=(fx,g_xy)$ on $X\times Y$ . In the 1950s, Rohklin [10] proved that every measure can be disintegrated along measurable partitions into a unique family of conditional measures. Denote by $Y_x$ the vertical fiber above $x\in X$ . We will show that the conditional measures given by Rohklin can be obtained via a family of fiberwise transfer operators $\mathcal {L}_x\colon C(Y_x)\to C(Y_{fx})$ defined such that for any $\psi \in C(Y_x)$ ,

In the setting of random dynamics, Kifer [6] used these operators to prove the existence and uniqueness of equilibrium states that satisfy a fiberwise scaling property on almost every fiber if the system exhibits expansion on average. Denker and Gordin were able to strengthen this to results on every fiber in expanding systems. In their 1999 paper [4], they used these fiberwise transfer operators to show that if a fibred system, a class of systems including skew products, is uniformly expanding and topologically exact along fibers, then given a Hölder potential , there is a unique equilibrium state on $X\times Y$ that has conditionals defined by a fiberwise Gibbs property and whose transverse measure on X is a Gibbs measure for a certain Hölder potential on X.

Varandas and Viana [12] and Castro and Varandas [2] studied equilibrium states for non-uniformly expanding maps. In particular, [2] proved that for a certain class of non-uniformly expanding maps on compact, connected manifolds equipped with almost constant Hölder potentials, eigendata for the Ruelle operator acting on the space of Hölder potentials can be used to construct a unique equilibrium state $\mu $ on $X\times Y$ . This gives us existence and uniqueness of an equilibrium state on $X\times Y$ . Pollicott and Kempton [9] and Piraino [8] give conditions for when Gibbsianness is preserved for factors of SFTs. Given a Gibbs measure for a Hölder (or Walters) potential and a factor of the SFT, it has been shown that the potential

(1)

exists independent of probability measure $\sigma $ on Y, is Hölder (or Walters, respectively) and that the pushforward of the Gibbs measure is $\Phi $ -Gibbs.

In §2 we recall some background on skew products, describe our non-uniform expansion assumptions, and state our main result (see Theorem A). In §3, we use arguments similar to Piraino [8] to prove the existence and regularity of $\Phi $ using the Hilbert metric and the Birkhoff contraction theorem on convex cones. This requires new arguments since the non-uniform expansion in the fiberwise maps $\{g_x\}$ makes estimates on distances between preimages more complex. The proofs in this paper are similar to those in Stadlbauer, Suzuki and Varandas [11] and Hafouta [5]. In [11], Stadlbauer et al showed that hyperbolic potentials for random non-uniformly expanding dynamical systems admit an equilibrium state whose disintegration satisfies a weak-Gibbs property almost everywhere along the base. In §3.4 we describe a process of coding the orbits of points in $X\times Y$ that are similar to the ideas of hyperbolic time in [11]. We also show the existence of a family of Gibbs measures in the sense of [4] (see Theorem 3.6). In §4 we complete the proof of Theorem A by establishing that $\hat {\mu }$ is the equilibrium state for $\Phi $ and the existence of fiberwise measures that form the unique family of conditional measures for the equilibrium state $\mu $ .

We remark that the conditional measures constructed here should satisfy a weak Gibbs property similar to that in Stadlebauer, Suzuki and Varandas [11]. However, they assume that almost every fiber is uniformly topologically exact, which is key to their proofs. We do not assume this but note that a similar property should follow from topological exactness in the product. The author also expects that it can be shown that the family of conditionals measures we construct in §4 uniquely achieves the relative pressure of the system. The proofs in [11] rely on hyperbolic times so the proofs of a relative variational principle would need to be adapted in the setting of the current paper.

2 Non-uniformly expanding skew products

Let X and Y be compact, connected Riemannian manifolds. Consider the product space $X\times Y$ . We will refer to X as the base and $\{Y_x=\{x\}\times Y\}_{ x\in X}$ as the fibers of the product since

$$ \begin{align*}X\times Y=\bigcup_{x\in X}\{x\}\times Y.\end{align*} $$

Note that each fiber $Y_x$ can be identified with Y. We will make the necessary distinctions as needed. Denote by d the $L_1$ distance on $X\times Y$ and by $\pi _X$ and $\pi _Y$ the natural projection maps from $X\times Y$ onto X and Y, respectively.

2.1 Dynamics of skew products

Let F be a continuous skew product on $X\times Y$ ; that is, there are continuous maps $f\colon X\to X$ and $\{g_x\colon Y\to Y|\ x\in X\}$ such that

$$ \begin{align*}F(x,y)=(f(x),g_x(y)). \end{align*} $$

To understand the dynamics of F on $X\times Y$ , define for any $n\geq 0$ and $x\in X$ , the map

$$ \begin{align*}g_x^n:=\ g_{f^{n-1}x}\circ\dots\circ g_x\colon Y_x\to Y_{f^nx}.\end{align*} $$

Then for any $(x,y)\in X\times Y$ , the behavior of the system can be investigated through the sequence $F^n(x,y)=(f^n(x),g_x^n(y)).$

For each $n\geq 0$ , define the nth Bowen metric as

$$ \begin{align*}d_n((x,y),(x',y'))=\max_{0\leq i\leq n}\{d(F^i(x,y),F^i(x',y'))\}.\end{align*} $$

Also denote the nth Bowen ball centered at $(x,y)$ of radius $\delta>0$ by

$$ \begin{align*}B_n((x,y),\delta)=\{(x',y')\colon d_n((x,y),(x',y'))<\delta\}.\end{align*} $$

2.2 Non-uniform expansion along fibers

The following describes our assumptions of non-uniform expansion along the fibers on $X\times Y$ . We shall assume that F is a topologically exact local homeomorphism and that there is a continuous function $(x,y)\mapsto L(x,y)$ such that the following statements hold.

1. (A1) There exists open $U\ni (x,y)$ such that $F|_U$ is invertible and

   $$ \begin{align*}d(F^{-1}(u_1,u_2),F^{-1}(v_1,v_2))\leq L(x,y)d((u_1,u_2),(v_1,v_2))\end{align*} $$
   for all $(u_1,u_2),(v_1,v_2)\in F(U)$ .

2. (A2) There exist constants $\gamma>1$ and $L\geq 1$ , and an open region $\mathcal {A}\subset X\times Y$ such that $L(x,y)\leq L$ for every $(x,y)\in \mathcal {A}$ and $L(x,y)<\gamma ^{-1}$ for all $(x,y)\not \in \mathcal {A}$ , and L is close enough to 1 so that inequality (6) below is satisfied.

Assumption (A1) gives us control of preimages in the image of small balls for which the product map is invertible. Assumption (A2) says that F is uniformly expanding outside of some region $\mathcal {A}$ and not too contracting in $\mathcal {A}$ . Thus, if $\mathcal {A}$ is empty, then everything is reduced to the uniformly expanding case.

We say that an open cover $\mathcal {U}$ of $X\times Y$ separates curves if, given a distance-minimizing geodesic segment c on $X\times Y$ , each element of $\mathcal {U}$ can intersect at most one curve in $F^{-1}(c)$ . Denote by the collection of open covers $\mathcal {U}$ of $X\times Y$ that separates curves and each $U\in \mathcal {U}$ satisfies assumption (A1). We further assume the following.

1. (A3) There exists a finite covering such that $\mathcal {A}$ can without loss of generality be covered by the first $q<\deg (F)$ elements of $\mathcal {U}$ .

2. (A4) For every $x\in X$ , there exists a finite covering $\mathcal {U}_x$ of $Y_x$ which separates curves by sets in and $\mathcal {A}\cap Y_x$ can be covered by the first $q<d$ elements of $\mathcal {U}_x$ .

Assumption (A3) ensures that every point has at least one preimage in the expanding region. Note that (A3) is a strengthened version of assumption H2 from Castro and Varandas [2]. Assumption (A4) guarantees that every point in a single fiber has at least preimage in the expanding region. This will be crucial to the arguments in §3 as discussed below.

A map $f\colon X\to X$ is uniformly expanding if there exist $C, \delta _f>0$ and $\gamma>1$ such that

$$ \begin{align*}d(f^n(x),f^n(x'))\geq C\gamma^n d(x,x')\end{align*} $$

whenever $d_n(x,x')\leq \delta _f$ . One can assume without loss of generality that $C=1$ by passing to an adapted metric. This reduces the expanding property to

$$ \begin{align*}d(f(x),f(x'))\geq\gamma d(x,x')\end{align*} $$

whenever $d(x,x')\leq \delta _f$ .

Lemma 2.1. If $F\colon X\times Y\to X\times Y$ is a skew product that satisfies assumptions (A1)–(A4), then f is uniformly expanding.

Proof. Fix $(x,y)\in X\times Y$ . Choose $x'\in X$ such that $(f(x'),g_x(y))\in F(U{(x,y)})$ . Since F is a local homeomorphism, there exists $y'\in Y$ such that $(x',y')\in U{(x,y)}$ and $F(x',y')=(f(x'),g_x(y))$ . Thus,

$$ \begin{align*} d(x,x')&\leq d((x,y),(x',y'))\\ &\leq L(x,y)d(F(x,y),F(x',y'))\\ &\leq L(x,y)d((f(x),g_x(y)),(f(x'),g_x(y)))\\ &\leq L(x,y)d(fx,fx'). \end{align*} $$

This finishes the proof since (A4) implies $\inf _{y\in Y}L(x,y)<1$ .

We remark that it would be interesting to see how the proofs in this paper would need to be changed if we removed assumption (A4). In such a setting, the base map could be non-uniformly expanding. It is known that the geometric potential for the Manneville–Pomeau map is Hölder continuous and has two equilibrium states. Uniqueness of the equilibrium state on the base is crucial to the proofs in §4 about the conditional measures of the equilibrium state $\mu $ . It is well known that Hölder potentials associated to expanding maps admit a unique equilibrium states. The uniform expansion in the base given by Lemma 2.1 is essential to proving that the potential $\Phi $ defined in equation (1) is Hölder continuous.

As a consequence of Lemma 2.1, we see that the number of preimages along fibers must be fixed. Indeed, since F is a local homeomorphism on a compact connected manifold, F is a covering map. Thus, we have that $\deg (F)$ is constant. Similarly, since f is expanding on a compact connected manifold onto itself, f is a covering map. Thus, $\hat {d}:={\deg (f)}$ is constant in x. Thus, $d:={\deg (g_x)}$ is constant for all $x\in X$ and $y\in Y_x$ and $\deg (F)=\hat {d}d$ .

The following example shows that there is a robust class of systems that satisfies the given assumptions.

Example 2.2. The Manneville–Pomeau map on is a classic example of a system that displays non-uniform expansion. Define a map $F\colon X\times Y\to X\times Y$ by taking the base map f to be the doubling map on and Manneville–Pomeau maps in the fibers where $p(x)>0$ varies continuously in the base point. Each of these maps has two branches so $d=2$ . Note that $g_x'(y)>g_x'(0)=1$ for all $y\not =0$ . Let $\mathcal {A}$ be any small neighborhood around . Then on $\mathcal {A}^c$ the product map F does not decrease distances; that is, if $(x,y),(x',y')\in \mathcal {A}^c$ , then

$$ \begin{align*}d((f(x),g_x(y)),(f(x'),g_{x'}(y')))\geq \gamma d((x,y),(x',y')).\end{align*} $$

So $q=1$ . Then $F(x,y)=(f(x),g_x(y))$ satisfies assumptions (A1) and (A2) and thus Theorem A holds for this example.

Lemma 2.3. If F satisfies (A1) and (A2), then for any $x,x'\in X$ and $y,y'\in Y$ , we can pair off the preimages of $g_x^{-1}(y)=\{y_1,\ldots ,y_{d}\}$ and $g_{x'}^{-1}(y')=\{y_1',\ldots ,y_{d}'\}$ where for any $k=1,2,\ldots ,q$ ,

$$ \begin{align*}d((x,y_k),(x,y_k'))\leq Ld((fx,y),(fx',y'))\end{align*} $$

while for any $k=q+1,\ldots ,d$ ,

$$ \begin{align*}d((x,y_k),(x',y_k'))\leq \gamma^{-1}d((fx,y),(fx',y')).\end{align*} $$

Proof. Let $(x,y),(x',y')\in X\times Y$ and c be a distance-minimizing geodesic segment between these points. Let $g_x^{-1}(y)=\{y_1,\ldots ,y_d\}$ . Since F is a covering map, we can uniquely lift c to curves $c_1,\dots ,c_d$ such that each $c_k$ starts at $y_i$ and $F(c_k)=c$ for all k. Then, letting $y_k'$ be the other endpoint of $c_k$ , we get a collection of preimages $g_{x'}^{-1}(y')=\{y^{\prime }_1,\ldots ,y^{\prime }_d\}$ . Cover each $c_k$ by domains of injectivity as in (A2). Then at most q of these balls can intersect $\mathcal {A}$ and each one intersects at most one of the curves $c_k$ . Thus there are at most q curves $c_k$ that intersect $\mathcal {A}$ . Without loss of generality, we can assume that these are the first q preimages. Applying (A1) gives the desired result.

2.3 Existence and uniqueness of equilibrium states

We say is $\alpha $ -Hölder continuous for some $\alpha>0$ if

$$ \begin{align*}|\varphi|_\alpha:=\sup_{(x,y)\not=(x',y')}\frac{|\varphi(x,y)-\varphi(x',y')|}{d((x,y),(x',y'))^\alpha}<\infty.\end{align*} $$

We denote by $C^\alpha (X\times Y)$ the Banach space of $\alpha $ - $\mathrm {H}\ddot{\mathrm{o}}\mathrm{lder}$ continuous functions on $X\times Y$ . The nth Birkhoff sum is defined as $S_n\varphi (x,y)=\sum _{k=0}^n\varphi \circ F^k(x,y)$ .

We denote by $\mathcal {M}(X\times Y)$ the space of Borel probability measures on $X\times Y$ and $\mathcal {M}(X\times Y,F)$ those that are F-invariant. Given a continuous map $F\colon X\times Y\to X\times Y$ and a potential , the variational principle asserts that

(2) $$ \begin{align} P(\varphi)=\sup\!\bigg\{h_\nu(F)+\int\varphi\, d\nu \colon \nu\in\mathcal{M}(X\times Y,F) \bigg\} \end{align} $$

where $P(\varphi )$ denotes the topological pressure of F with respect to $\varphi $ and $h_\mu (F)$ denotes the metric entropy of F. An equilibrium state for F with respect to $\varphi $ is an invariant measure that achieves the supremum in the right-hand side of equation (2). For uniformly expanding maps, every equilibrium state $\mu $ satisfies the Gibbs property: for any $\varepsilon>0$ , there exists a $C>0$ such that

$$ \begin{align*}C^{-1}\leq\dfrac{\mu(B_n((x,y),\varepsilon))}{e^{-nP(\varphi)+S_n\varphi(x,y)}}\leq C\end{align*} $$

for any $(x,y)\in X\times Y$ and .

For our purposes in this paper, we fix a Holder potential $\varphi \in {C^\alpha (X\times Y)}$ satisfying

(P) $$ \begin{align} \sup\varphi-\inf\varphi<\varepsilon_\varphi\quad \text{and}\quad |e^\varphi|_\alpha<\varepsilon_\varphi e^{\inf\varphi} \end{align} $$

for some $\varepsilon _\varphi>0$ satisfying inequality (3) and equation (4) below (see §3). Almost constant potentials satisfy (P) so Theorem A applies to an open set of potentials. In particular, Theorem A holds for measures of maximal entropy. We assume that L is close enough to $1$ and $0<\varepsilon _\varphi <\log d -\log q$ so that

(3) $$ \begin{align} e^{\varepsilon_\varphi}\cdot\bigg(\dfrac{(d-q)\gamma^{-\alpha}+qL^\alpha}{d}\bigg)<1. \end{align} $$

Under these assumptions, it is known that there is a unique equilibrium state $\mu $ for $\varphi $ on $X\times Y$ .

Lemma 2.4. If F is topologically exact and satisfies (A1), (A2), and $\varphi $ satisfies $\sup \varphi -\inf \varphi <\log \deg (F)-\log q$ , then there exists an expanding conformal measure such that $\mathcal {L}_\varphi ^*\nu =\unicode{x3bb} \nu $ and $\mathrm {supp}(\nu )=\overline {X\times Y}$ , where the spectral radius of $\mathcal {L}_\varphi $ , $\unicode{x3bb} :=r(\mathcal {L}_\varphi )\geq \deg (F) e^{\inf \varphi }$ . Moreover, $\nu $ is a non-lacunary Gibbs measure and has a Jacobian with respect to F given by $J_\nu F=\unicode{x3bb} e^{-\varphi }$ .

Proof. See Theorem 4.1 in Varandas and Viana [12].

We will not use the non-lacunary property of $\nu $ or $J_\nu F$ . For more details, see [12].

Theorem 2.5. Let $F\colon X\times Y\to X\times Y$ be a local homeomorphism with Lipschitz continuous inverse and be a Hölder continuous potential satisfying (A1), (A2), and (P). Then the RPF operator has a spectral gap property in the space of Hölder continuous observables, there exists a unique equilibrium state $\mu $ for F with respect to $\varphi $ , and the density $d\mu /d\nu $ is Hölder continuous.

Proof. See Theorem A in Castro and Varandas [2].

Denote by $\hat {\mu }=\mu \circ \pi _X^{-1}$ the pushforward of the equilibrium state $\mu $ onto the base X. Throughout this paper, we shall refer to this measure as the transverse measure for our skew product.

Our main result is the following theorem.

Theorem A. Let X and Y be compact connected Riemannian manifolds and $(X\times Y,F)$ be a Lipschitz skew product that satisfies assumptions (A1)–(A4). Let $\varphi $ be a Hölder continuous potential on $X\times Y$ satisfying assumption (P) and $\mu $ be its corresponding equilibrium state. Then the following assertions are true.

1. (1) The potential $\Phi $ in equation (1) is independent of $\sigma $ , is Hölder continuous, and satisfies $P(\varphi )=P(\Phi )$ .

2. (2) $\hat {\mu }=\mu \circ \pi _X^{-1}$ is the unique equilibrium state for $\Phi $ .

3. (3) There is a unique family of measures $\{\nu _x\colon x\in X\}$ such that $\nu _x(Y_x)=1$ and

   $$ \begin{align*}\mathcal{L}_x^*\nu_{fx}=e^{\Phi(x)}\nu_x.\end{align*} $$

4. (4) $x\mapsto \nu _x$ is weak $^*$ -continuous.

5. (5) Let $\hat {h}$ and $\hat {\nu }$ be the eigendata of $\mathcal {L}_\Phi $ , that is, $\mathcal {L}_\Phi ^*\hat {\nu }=e^{P(\Phi )}\hat {\nu }$ , $\mathcal {L}_\Phi \hat {h}=e^{P(\Phi )}\hat {h}$ , and $\int \hat {h}d\hat {\nu }=1$ . Then the measures $\mu _x=({h(x,\cdot )}/{\hat {h}(x)})\nu _x$ are probability measures on $Y_x$ such that

   $$ \begin{align*}\mu=\int_X\mu_x\ d\hat{\mu}(x).\end{align*} $$

We remark that the existence in item $(1)$ follows closely proofs from Piraino [8] on SFTs. However, the proof of the Hölder continuity of $\Phi $ took new ideas on compact, connected manifolds (see §3.4). It is worth noting that Stadlbauer, Varandas and Zhang proved a similar result to item $(3)$ for conformal measure of Ruelle expanding iterated function systems.

2.4 Fiberwise transfer operators for skew products

As is common in the literature, we will utilize Ruelle operators to study the equilibrium state on $(X\times Y,F)$ . Define the transfer operator $\mathcal {L}_\varphi $ acting on $ C(X\times Y)$ by sending $\psi \in C(X\times Y)$ to

Note that under the skew product representation of F, we may write

$$ \begin{align*}\sum_{(\overline{x},\overline{y})\in F^{-1}(x,y)}e^{\varphi(\overline{x},\overline{y})}\psi(\overline{x},\overline{y})=\sum_{\overline{x}\in f^{-1}x}\sum_{\overline{y}\in g_{\overline{x}}^{-1}y}e^{\varphi(\overline{x},\overline{y})}\psi(\overline{x},\overline{y}).\end{align*} $$

This gives rise to a fiberwise transfer operator on the fibers of $X\times Y$ .

We disintegrate $\varphi $ and get the family of fiberwise potentials $\{\varphi _x(\cdot )=\varphi (x,\cdot )\}_{x\in X}$ . For every $x\in X$ , let $\mathcal {L}_x\colon C(Y_x)\to C(Y_{fx})$ be defined by

$$ \begin{align*} \mathcal{L}_x\psi_x(y)=\sum_{\overline{y}\in g_x^{-1}y}e^{\varphi_x(\overline{y})}\psi_x(\overline{y}) \end{align*} $$

for any $\psi \in C(X\times Y)$ . We shall iterate the transfer operator by letting

$$ \begin{align*}\mathcal{L}_x^n=\mathcal{L}_{f^{n-1}x}\circ\cdots\circ \mathcal{L}_x\colon C(Y_x)\to C(Y_{f^nx}).\end{align*} $$

Along with each of these fiberwise operators, we define its dual $\mathcal {L}_x^*$ by sending a probability measure $\eta \in \mathcal {M}(Y_{fx})$ to the measure $\mathcal {L}_x^*\eta \in \mathcal {M}(Y_x)$ such that for any $\psi \in C(X\times Y)$ ,

$$ \begin{align*}\int \psi\,d(\mathcal{L}_x^*\eta)=\int \mathcal{L}_x\psi\,d\eta.\end{align*} $$

3 A potential for the transverse measure

Piraino [8] shows that for some factors of mixing SFTs, a Gibbs measure $\mu $ is pushed onto a Gibbs measure $\hat {\mu }=\mu \circ \pi _X^{-1}$ for the potential

where $\sigma $ is any probability measure supported on Y. We will show in Theorem 3.5 that this potential exists in our setting. Furthermore, in Theorem 3.15 we show that $\Phi $ is Hölder continuous.

3.1 Birkhoff contraction theorem

It is not hard to check that the Ruelle operator preserves the Banach space of Hölder continuous potentials $ C^\alpha (X\times Y),\ 0<\alpha <1$ . A subset $\Lambda \subset {C^\alpha (X\kern1.3pt{\times}\kern1.3pt Y)}$ is called a cone if $a\Lambda \kern1.3pt{=}\kern1.3pt\Lambda $ for all $a\kern1.3pt{>}\kern1.3pt0$ . A cone $\Lambda $ is convex if $\psi +\zeta \in \Lambda $ for all $\psi ,\ \zeta \in \Lambda $ . We say that $\Lambda $ is a closed cone if $\Lambda \cup \{0\}$ is closed with respect to the Hölder norm. We assume our cones are closed, convex, and $\Lambda \cap (-\Lambda )=\emptyset $ . For any probability measure $\eta $ and Hölder potential $\psi $ , let $\langle \psi ,\eta \rangle \kern1.2pt{=}\kern0.2pt\int \psi\, d\eta $ . Given a closed cone $\Lambda {\kern-1pt}\subset{\kern-1pt} {C^\alpha (X{\kern-1pt}\times{\kern-1pt} Y)}$ , we can define the dual cone $\Lambda ^*{\kern-1pt}={\kern-1pt}\{\eta {\kern-1pt}\in{\kern-1pt} ({C^\alpha (X{\kern-1pt}\times{\kern-1pt} Y)})^*\colon \langle \psi ,\eta \rangle \geq 0 \text {for\ all\ } \psi {\kern-1pt}\in{\kern-1pt} \Lambda \}$ . For more on cones, see §4 in [7] or the appendices in [8].

Define a partial ordering $\preceq $ on ${C^\alpha (X\times Y)}$ by saying that $\phi \preceq \psi $ if and only if $\psi -\phi \in \Lambda \cup \{0\}$ for any $\phi ,\psi \in {C^\alpha (X\times Y)}$ . Let

$$ \begin{align*}A=A(\phi,\psi)=\sup\{t>0\colon t\phi\preceq\psi\}\quad \text{and}\quad B=B(\phi,\psi)=\inf\{t>0\colon \psi\preceq t\phi\}.\end{align*} $$

The Hilbert projective metric with respect to a closed cone $\Lambda $ is defined as

$$ \begin{align*}\Theta(\phi,\psi)=\log\frac{B}{A}.\end{align*} $$

The following lemma is useful when calculating distances in the Hilbert metric. For a proof, see §4 in [7].

Lemma 3.1. Let $\Lambda $ be a closed cone and $\Lambda ^*$ its dual. For any $\phi ,\psi \in \Lambda $ ,

$$ \begin{align*}\Theta(\phi,\psi)=\log\!\bigg(\!\!\sup\!\bigg\{\dfrac{\langle\phi,\sigma\rangle\langle\psi,\eta\rangle}{\langle\psi,\sigma\rangle\langle\phi,\eta\rangle}\colon \sigma,\eta\in \Lambda^*\ \text{and\ } \langle\psi,\sigma\rangle\langle\phi,\eta\rangle\neq 0\bigg\}\!\bigg).\end{align*} $$

The main idea of the proof of Theorem A is to find a cone on which the fiberwise transfer operator is a contraction. To accomplish this, we will need the Birkhoff contraction theorem.

Theorem 3.2. (Birkhoff [1])

Let $\Lambda _1, \Lambda _2$ be closed cones and $\mathcal {L}\colon \Lambda _1\to \Lambda _2$ a linear map such that $\mathcal {L} \Lambda _1\subset \Lambda _2$ . Then for all $\phi ,\psi \in \Lambda _1$ .

$$ \begin{align*}\Theta_{\Lambda_2}(\mathcal{L}\phi,\mathcal{L}\psi)\leq \tanh\bigg(\frac{\operatorname{\mathrm{diam}}_{\Lambda_2}(\mathcal{L} \Lambda_1)}{4}\bigg)\Theta_{\Lambda_1}(\phi,\psi)\end{align*} $$

where $\operatorname {\mathrm {diam}}_{\Lambda _2}(\mathcal {L} \Lambda _1)=\sup \{\Theta _{\Lambda _2}(\mathcal {L}\phi ,\mathcal {L}\psi )\colon \phi ,\psi \in \Lambda _1\}$ and $\tanh \infty =1$ .

3.2 Existence of $\Phi $

We will use cones of the form

$$ \begin{align*}\Lambda_K={\Lambda_K(\alpha)}=\{\psi\in {C^\alpha(X\times Y)}\colon\psi>0\ \text{and\ } |\psi|_\alpha\leq K\inf\psi\}\cup\{0\}.\end{align*} $$

It can be shown that $\Lambda _K$ is a closed cone in ${C^\alpha (X\times Y)}$ . For these cones, we get an alternate way of calculating distances in the Hilbert metric.

Lemma 3.3. For any $\phi ,\psi \in \Lambda _K$ ,

$$ \begin{align*}A(\phi,\psi)=\inf_{z_1,z_2,z_3\in X\times Y}\dfrac{Kd(z_1,z_2)^\alpha\psi(z_3)-(\psi(z_1)-\psi(z_2))}{Kd(z_1,z_2)^\alpha\phi(z_3)-(\phi(z_1)-\phi(z_2))}\end{align*} $$

and

$$ \begin{align*}B(\phi,\psi)=\sup_{z_1,z_2,z_3\in X\times Y}\dfrac{Kd(z_1,z_2)^\alpha\psi(z_3)-(\psi(z_1)-\psi(z_2))}{Kd(z_1,z_2)^\alpha\phi(z_3)-(\phi(z_1)-\phi(z_2))}.\end{align*} $$

Proof. See Lemma 4.2 in Castro and Varandas [2].

Denote by

$$ \begin{align*}\Lambda_K^x=\{\psi\in C^\alpha(X\times Y)\colon\psi_x(\cdot)>0\ \text{and\ } |\psi|_\alpha\leq K\inf\psi\}\cup\{0\}\end{align*} $$

the cross-section of $\Lambda _K$ that lives on $Y_x$ .

Let

$$ \begin{align*}s:=e^{\varepsilon_\varphi}\cdot\bigg(\dfrac{(d-q)\gamma^{-\alpha}+qL^\alpha}{d}\bigg)<1\end{align*} $$

as in inequality (3). We assume that $\varepsilon _\varphi>0$ is small enough that

(4) $$ \begin{align} \zeta:=s+2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha <1. \end{align} $$

Then we have the following lemma based on similar arguments from Castro and Varandas [2] and Stadlbauer, Suzuki and Varandas [11].

Lemma 3.4. With $\zeta $ as in (4), for all K sufficiently large, we have $\mathcal {L}_x(\Lambda _{K}^x)\subset \Lambda _{\zeta K}^{fx}$ for all $x\in X$ . Moreover, there is a constant $M=M(K)>0$ such that for all $x\in X$ , $\operatorname {\mathrm {diam}}(\mathcal {L}_x\Lambda ^x_K)\leq M<\infty $ with respect to the Hilbert projective metric on $\Lambda _K^{fx}$ .

Proof. Fix $x\in X$ and $K>0$ . Denote by $\{y_k\}$ and $\{y_k'\}$ the preimages of y and $y'$ in $Y_x$ , respectively, as given by Lemma 2.3. Now fix $\psi \in \Lambda _K$ . Since $\inf \mathcal {L}_x\psi \geq de^{\inf \varphi }\inf \psi $ and

$$ \begin{align*}&\mathcal{L}_{x}\psi_{fx}(y)-\mathcal{L}_{x}\psi_{fx}(y')\\ &\quad= \sum_{k=1}^d(e^{{\varphi}(x,y_k)}(\psi(x,y_k)-\psi(x,y_k'))+(e^{{\varphi}(x,{y_k})}-e^{{\varphi}(x,y_k')})\psi(x,y_k')),\end{align*} $$

we have

$$ \begin{align*} \dfrac{|\mathcal{L}_x\psi(fx,y)-\mathcal{L}_x\psi(fx,y')|}{\inf\mathcal{L}_x\psi}&\leq d^{-1}\sum_{k=1}^d e^{\varphi(x,y_k)-\inf\varphi}|\psi(x,y_k)-\psi(x,y_k')|(\inf\psi)^{-1}\\&\quad+d^{-1}\sum_{k=1}^d(\sup\psi/\inf\psi)e^{-\inf\varphi}|e^{\varphi(x,y_k)}-e^{\varphi(x,y_k')}|\\&=:I_1+I_2. \end{align*} $$

Note that $|\psi (x,{\kern-1pt}y_k){\kern-1pt}-{\kern-1pt}\psi (x,{\kern-1pt}y_k')|{\kern-1pt}\leq{\kern-1pt} |\psi |_\alpha d(y_k,{\kern-1pt}y^{\prime }_k)^\alpha {\kern-1pt}\leq{\kern-1pt} K\inf \psi\, d(y_k,{\kern-1pt}y^{\prime }_k)^\alpha $ . By Lemma 2.3, $d(y_k,y_k')\leq Ld(y,y')$ for any $1\leq k\leq q $ and $d(y_k,y_k')\leq \gamma ^{-1}d(y,y')$ for $q<k\leq d$ , so

$$ \begin{align*} I_1&\leq d^{-1}\sum_{k=1}^d e^{\varphi(x,y_k)-\inf\varphi}Kd(y_k,y^{\prime}_k)^\alpha\\ &\leq d^{-1}e^{\varepsilon_\varphi}K\sum_{k=1}^d d(y_k,y^{\prime}_k)^\alpha\\ &\leq K e^{\varepsilon_\varphi}d^{-1}(L^\alpha q+(d-q)\gamma^{-\alpha})d(y,y')^\alpha\\ &\leq sKd(y,y')^\alpha \end{align*} $$

where the second inequality holds by (P).

To estimate $I_2$ , note that $|e^{\varphi (x,y_k)}-e^{\varphi (x,y_k')}|\leq |e^{\varphi _x}|_\alpha d(x,y_k),(x,y_k'))^\alpha $ and

$$ \begin{align*}\sup\psi\leq\inf\psi+|\psi|_\alpha\operatorname{\mathrm{diam}}(Y)^\alpha\leq(1+K\operatorname{\mathrm{diam}}(Y)^\alpha)\inf\psi\end{align*} $$

implies that

$$ \begin{align*}\sup\psi/\inf\psi\leq 1+K\operatorname{\mathrm{diam}}(Y)^\alpha\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha\end{align*} $$

provided that K is sufficiently large. Then (P) implies that

$$ \begin{align*} I_2&\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha e^{-\inf\varphi}d^{-1}\sum_{k=1}^d|e^{\varphi_x}|_\alpha d(x,y_k),(x,y_k'))^\alpha\\ &\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha\varepsilon_\varphi d^{-1}\sum_{k=1}^d (L^\alpha q+(d-q)\gamma^{-\alpha})d(y,y')^\alpha\\ &\leq 2K\operatorname{\mathrm{diam}}(Y)^\alpha s\varepsilon_\varphi d(y,y')^\alpha\\ &\leq 2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha K d(y,y')^\alpha. \end{align*} $$

Therefore, if we let $\zeta :=s+2s\varepsilon _\varphi \operatorname {\mathrm {diam}}(Y)^\alpha $ , we have that

$$ \begin{align*}|\mathcal{L}_x\psi|_\alpha\leq (s+2s\varepsilon_\varphi\operatorname{\mathrm{diam}}(Y)^\alpha)K\inf\mathcal{L}_x\psi\leq \zeta K\inf\mathcal{L}_x\psi\end{align*} $$

so $\mathcal {L}_x\psi \in \Lambda ^{fx}_{\zeta K}$ .

Note that $\sup \mathcal {L}_x\psi \leq (1+\zeta K(\operatorname {\mathrm {diam}} Y)^\alpha )\inf \mathcal {L}_x\psi $ . Let $y_1,y_2,y_3\in Y$ . Then since $|\mathcal {L}_x\psi |_\alpha \leq \zeta K\inf \mathcal {L}_x\psi $ , we have

$$ \begin{align*} &\dfrac{Kd(y_1,y_2)^\alpha\mathcal{L}_x\psi(y_3)-(\mathcal{L}_x\psi(y_1)-\mathcal{L}_x\psi(y_2))}{Kd(y_1,y_2)^\alpha\mathcal{L}_x\phi(y_3)-(\mathcal{L}_x\phi(y_1)-\mathcal{L}_x\phi(y_2))}\\&\quad\leq\dfrac{(K\sup\mathcal{L}_x\psi+\zeta K\inf\mathcal{L}_x\psi)d(y_1,y_2)^\alpha}{(K\inf\mathcal{L}_x\phi-\zeta K\inf\mathcal{L}_x\phi)d(y_1,y_2)^\alpha}. \end{align*} $$

Thus, $B(\mathcal {L}_x\psi ,\mathcal {L}_x\phi )\leq ({K\sup \mathcal {L}_x\psi +\zeta K\inf \mathcal {L}_x\psi })/({K\inf \mathcal {L}_x\phi -\zeta K\inf \mathcal {L}_x\phi })$ . A similar calculation gives a lower bound on $A(\mathcal {L}_x\psi ,\mathcal {L}_x\phi )$ . So by Lemma 3.3, we have

$$ \begin{align*} \Theta(\mathcal{L}_x\psi,\mathcal{L}_x\phi) &\leq \log\!\bigg(\dfrac{K\sup\mathcal{L}_x\phi+\zeta K\inf\mathcal{L}_x\phi}{K\inf\mathcal{L}_x\phi-\zeta K\inf\mathcal{L}_x\phi}\cdot\dfrac{K\sup\mathcal{L}_x\psi+\zeta K\inf\mathcal{L}_x\psi}{K\inf\mathcal{L}_x\psi-\zeta K\inf\mathcal{L}_x\psi}\bigg)\\ &\leq \log\!\bigg(\dfrac{K(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)(1+\zeta)\inf\mathcal{L}_x\phi}{K(1-\zeta)\inf\mathcal{L}_x\phi}\bigg)\\&\quad+\log\!\bigg(\dfrac{K(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)(1+\zeta)\inf\mathcal{L}_x\psi}{K(1-\zeta)\inf\mathcal{L}_x\psi}\bigg)\\ &\leq 2\log\!\bigg(\dfrac{1+\zeta}{1-\zeta}\bigg)+2\log(1+\zeta K\operatorname{\mathrm{diam}}(Y)^\alpha)<\infty. \end{align*} $$

This proves the existence of M.

Theorem 3.5. Let There exist $0<\tau <1$ and $C_1>0$ such that for all , and any probability measures $\sigma _n$ on $Y_{f^nx}$ and $\sigma _m$ on $Y_{f^mx}$ , we have

$$ \begin{align*}|\Phi_n^{\sigma_n}(x)-\Phi_m^{\sigma_m}(x)|\leq C_1\ \tau^k.\end{align*} $$

Thus, $\Phi (x)=\lim _{n\to \infty }\Phi _n^{\sigma _n}(x)$ exists and $|\Phi _n^{\sigma _n}(x)-\Phi (x)|\leq C_1\tau ^n$ .

Proof. Fix $x\in X$ . Suppose $n,m\geq k \geq 1$ . Then

where $\sigma _{fx,n}=(\mathcal {L}_{f^{k+1}x})^*\cdots (\mathcal {L}_{f^nx})^*\sigma _n$ . By Lemma 3.1, we see that

Clearly,

for any $K>0$ . Then

by Lemma 3.4. Fix K large and M as in Lemma 3.4. Set $\tau =\tanh {(M/4)}$ . By Theorem 3.2, we have

Let $C_1=M/\tau $ . Hence, the sequence $\{\Phi _n\}_{n\geq 0}$ is Cauchy and the limit exists at every $x\in X$ .

This proves the existence of $\Phi $ .

3.3 Fiber measures

To completely understand the equilibrium state $\mu $ on $(X\times Y,F)$ , we need to understand how it gives weight to the fibers $\{Y_x\}_{x\in X}$ . The first step is the following non-stationary RPF theorem adapted from [3], whose proof we include here for completeness (see Hafouta [5] for a similar result when the base is invertible).

Theorem 3.6. Let $F\colon X\times Y\to X\times Y$ satisfy (A1) and (A2). For any Hölder satisfying (P) and its associated family of fiberwise transfer operators $\{\mathcal {L}_{x}\}_{x\in X}$ , there exists a unique family of probability measures $\nu _x\in \mathcal {M}(Y_x)$ such that for all $x\in X$ ,

Theorem 3.6 is a consequence of the following two propositions.

Proposition 3.7. Let $C_1>0$ and $0<\tau <1$ be as in Theorem 3.5. Given any $x\in X$ , , and $\sigma _k\in \mathcal {M}(Y_{f^kx})$ , define $\nu _{x,k}\in \mathcal {M}(Y_{x})$ by . If $m,n\geq k$ and $\psi \in \Lambda _K$ , we have

$$ \begin{align*}\bigg|\kern-2pt\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_{x,m}\bigg|\leq C_1\|\psi\|\tau^k.\end{align*} $$

In particular, $\langle \psi ,\nu _x\rangle :=\lim _{n\to \infty }\langle \psi ,\nu _{x,n}\rangle $ exists and defines a probability measure $\nu _x$ on $Y_x$ with

$$ \begin{align*}\bigg|\kern-2pt\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_x\bigg|\leq C_1\|\psi\|\tau^n.\end{align*} $$

Proof. Let

and

. Note that

. So

A similar computation shows that $b_k\leq \langle \psi ,\nu _{x,n}\rangle $ . Then $b_k\leq \langle \psi ,\nu _{x,n}\rangle \leq c_k$ for all $n\geq k$ . Therefore, $|\langle \psi ,\nu _{x,n}\rangle -\langle \psi ,\nu _{x,m}\rangle |\leq c_k-b_k$ for all $n,m\geq k$ . Lemma 3.4 implies that

. So $1\leq {c_k}/{b_k}\leq e^{M\tau ^{k-1}}$ . Thus, $b_k\leq c_k\leq b_ke^{M\tau ^{k-1}}$ which implies that $c_k-b_k\leq b_k(e^{M\tau ^{k-1}}-1)$ . Moreover, for all $y\in Y$ , we have

So $b_k\leq \|\psi \|$ . Hence,

$$ \begin{align*}|\langle\psi,\nu_{x,n}\rangle-\langle\psi,\nu_{x,m}\rangle|\leq c_k-b_k\leq\|\psi\|(e^{M\tau^{k-1}}-1).\end{align*} $$

Thus, $\{\nu _{x,n}\}$ is a Cauchy sequence. Then there is a constant $C_1>0$ such that

$$ \begin{align*}|\langle\psi,\nu_{x,n}\rangle-\langle\psi,\nu_{x}\rangle|\leq C_1\|\psi\|\tau^n\end{align*} $$

for all $n\geq 0$ .

Proposition 3.8. Let $\{\nu _x\}$ be as in Proposition 3.7. Then $\mathcal {L}_x^*\nu _{fx}=e^{\Phi (x)}\nu _x$ .

Proof. For all $\psi \in C(Y_x)$ , we have

Observe that

This completes the proof of Theorem 3.7.

As in Denker and Gordin [4], we call a system of $\{\nu _x\colon x\in X\}$ of conditional probabilities for $(X\times Y,F)$ a family of Gibbs measures for a continuous function if there exists a positive measurable function with the following property: for all $x\in X$ , the Jacobian of $\mu _x$ with respect to the map F is given by

$$ \begin{align*}\dfrac{d\nu_{f(x)}\circ g_x}{d\nu_x}=A(x)e^{-\varphi_x}.\end{align*} $$

Corollary 3.9. The measures $\{\nu _x\colon x\in X\}$ form a family of Gibbs measures for $\varphi $ in the sense of Denker and Gordin [4] with $A(x)=\unicode{x3bb} _x$ .

Proof. Choose $A\in Y_x$ such that $g_x|_A$ is invertible. This implies

Therefore,

. Since this holds for $x\in X$ , $\{\nu _x\colon x\in X\}$ forms a family of Gibbs measures.

3.4 Regularity of $\Phi $

Now we will show that $\Phi $ is Hölder continuous. A direct consequence of Lemma 3.4 is the following lemma, which we will need to prove the Hölder continuity of $\Phi $ . For convenience, we write

Lemma 3.10. Let M be as in Lemma 3.4. Then for all and $(x,y)\in X\times Y$ .

Proof. Let $\overline {\varphi }_x=\varphi _x-\log \unicode{x3bb} _x$ and write

. Theorem 3.7 gives $(\overline {\mathcal {L}}_{x})^*\nu _{fx}=\nu _x$ for all $x\in X$ . Inductively, we get that $(\overline {\mathcal {L}}^{n}_x)^*\nu _{f^{n}x}=\nu _{x}$ . Then for any $k,\ell $ ,

Let $\Lambda ^+$ be the cone of strictly positive continuous functions on $X\times Y$ . Since $\Lambda _K\subset \Lambda ^+$ , the projective metrics of the two cones satisfy $\Theta ^+(\phi ,\psi )\leq \Theta (\phi ,\psi )$ . Write

. Then

, so $1\leq {\sup \psi _k}/{\inf \psi _k}\leq e^M$ . We know that

. This implies that $e^{-M}\leq \psi _k\leq e^M$ for all

. Thus,

Let and $x,x'\in X$ , and $y\in Y$ . Let $\mathcal {W}_n=\{1,\ldots ,d\}^n$ . By Lemma 2.3, we can write

$$ \begin{align*}g^{-1}_{f^{n-1}x}(y)=\{y_1,\ldots,y_d\}\quad \text{and}\quad g^{-1}_{f^{n-1}x'}(y)=\{y^{\prime}_1,\ldots,y^{\prime}_d\}\end{align*} $$

such that

$$ \begin{align*}d((f^{n-1}x,y_k),(f^{n-1}x',y^{\prime}_k))\leq L_kd_X(f^nx,f^nx')\end{align*} $$

where $L_k=L$ if $1\leq k\leq q$ and $L_k=\gamma ^{-1}$ if $q< k\leq d$ . Continuing in this way, we get that

$$ \begin{align*}g^{-n}_{x}(y)=\{y_w\in Y_x\colon w\in\mathcal{W}_n\}\quad \text{and}\quad g^{-n}_{x'}(y)=\{y^{\prime}_w\in Y_{x^{\prime}_w}\colon w\in \mathcal{W}_n\}\end{align*} $$

such that for all $0\leq k\leq n$ ,

$$ \begin{align*}d(F^{k}(x,y_w),F^{k}(x',y^{\prime}_w))\leq L_{w_{k+1}}\cdots L_{w_{n}}d_X(f^nx,f^nx').\end{align*} $$

Let and $0<\iota <1$ . A pair of inverse branches for F of length n starting from $(f^nx,y)$ and $(f^nx',y)$ and labeled by $w\in \mathcal {W}_n$ is good if for all such that $jm\leq n$ , we have

$$ \begin{align*}\#\{n-jm< i\leq n\colon w_i\leq q\}\leq\iota jm.\end{align*} $$

This means that the last $jm$ iterates of an orbit segment of length n will be in the contraction region at most $\iota jm$ times. We will denote the collection of words corresponding to good trajectories by

$$ \begin{align*}\mathcal{W}^{\mathcal{G}}_n=\mathcal{W}^{\mathcal{G}}_n(m)=\{w\in\mathcal{W}_n\colon\ \text{for all}\ j\leq n/m, \#\{n-jm< i\leq n\colon w_i\leq q\}\leq\iota jm\}\end{align*} $$

and the collection of words for bad trajectories by

$$ \begin{align*}\mathcal{W}^{\mathcal{B}}_n&=\mathcal{W}^{\mathcal{B}}_n(m)\\&=\{w\in\mathcal{W}_n\colon \text{there exists}\ j\leq n/m \ni \#\{n-jm< i\leq n\colon w_i\leq q\}\geq\iota jm\}.\end{align*} $$

The following lemma due to Varandas and Viana gives us a way to count the number of words that code bad trajectories of a given length. Let $I(\iota ,n)=\{w\in \mathcal {W}_n\colon \#\{1\leq k\leq n\colon w_i\leq q\}\geq \iota n\}$ .

Lemma 3.11. Given $\varepsilon>0$ , there exists a $\iota _0\in (0,1)$ such that

$$ \begin{align*} \limsup_{n\to\infty}\frac{1}{n}\log\#I(\iota,n) <\log q+\varepsilon \end{align*} $$

for all $\iota \in (\iota _0,1)$ . Therefore, there exists a $C>0$ such that $\#I(\iota ,n) \leq Cq^ne^{\varepsilon n}$ for all n.

Proof. See Lemma 3.1 in Varandas and Viana [12].

Since it is assumed that $\varepsilon _\varphi <\log d-\log q$ , it follows that ${qe^{\varepsilon _\varphi }}/{d}<1$ . Choose $\varepsilon>0$ such that

(5) $$ \begin{align}\theta:=\frac{qe^\varepsilon e^{\varepsilon_\varphi}}{d}<1.\end{align} $$

Let $\iota =\iota (\varepsilon ,d,q)\in (0,1)$ be given by Lemma 3.11. We now further our assumption on the constant L by assuming that it is close enough to $1$ so that there is a $c>0$ satisfying

(6) $$ \begin{align} 0<\gamma^{-(1-\iota)}L^\iota<e^{-2c}<1. \end{align} $$

Lemma 3.12. There is a $Q>0$ such that for all , if $(x,\overline {y})$ and $(x',\overline {y}')$ are preimages coded by a word in $\mathcal {W}_n^{\mathcal {G}}{(m)}$ , then

$$ \begin{align*}d(F^k(x,\overline{y}),F^k(x',\overline{y}'))\leq Q^me^{-2c(n-k)}d(f^nx,f^nx')\end{align*} $$

for all $0\leq k<n$ .

Proof. Fix . Write $n-k=jm+i$ for $0\leq i<m$ . Since our preimage branches are assumed to be good, we get

$$ \begin{align*} d(F^k(x,\overline{y}),F^k(x',\overline{y}'))&\leq L_{w_{k+1}}\cdots L_{w_{n}}d(F^{k+i}(x,\overline{y}),F^{k+i}(x',\overline{y}'))\\ &\leq L_{w_{k+1}}\cdots L_{w_{k+i}}(L^{\iota jm}\gamma^{-(1-\iota ){jm}})d(f^nx,f^nx').\end{align*} $$

Recall from (6) that we can choose $c>0$ so that $0<\gamma ^{-(1-\iota )}L^\iota <e^{-2c}<1$ . Thus,

$$ \begin{align*} d(F^k(x,\overline{y}),F^k(x',\overline{y}'))&\leq L^m e^{-2cjm}d(f^nx,f^nx')\\ &\leq (Le^{2c})^{m}e^{-2c(n-k)}d(f^nx,f^nx').\\[-3.1pc] \end{align*} $$

In what follows it will be convenient to write $a\colon \mathcal {W}_n\to Y$ so that $a(w)=y_w$ and $a'\colon \mathcal {W}_n\to Y$ so that $a'(w)=y^{\prime }_w$ . Lemma 2.3 gives us bijections $b_j\colon \mathcal {W}_{jm}\to g^{-jm}_{f^{n-jm}x}(y)$ such that $b_j(v)=g_{x}^{n-jm}(a(uv))$ and $c_v\colon \mathcal {W}_{n-jm}\to g^{-(n-jm)}_{x}(b_j(v))$ such that $c_v(u)=a(uv)$ as well as their associated maps $b_j'$ and $c_v'$ . See Figure 1 for reference.

Figure 1 The maps a, $b_j$ , and $c_v$ on words of corresponding lengths.

Lemma 3.13. Let $\theta $ as in (5) above. There exists $C_2>0$ such that

$$ \begin{align*}\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}(m)}}e^{S_n\varphi_x(a(w))}\leq C_2\theta^m\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}(m)}}e^{S_n\varphi_x(a(w))}\end{align*} $$

for all , $x,x'\in X$ , and $y\in Y$ .

Proof. Let $x\in X$ and $y\in Y$ . For any $w\in \mathcal {W}_n^{\mathcal {B}}$ , there is $1\leq j\leq n/m$ such that $w=uv$ for some $u\in \mathcal {W}_{n-jm}$ and $v\in I(\iota ,jm)$ . Thus,

by Lemma 3.10. Note that for any $j\leq {n}/{m}$ ,

Lemma 3.11 implies that $\#I(\iota ,n)\leq Cq^{n}e^{\varepsilon n}$ for all $n\geq 0$ . Then since $\#\mathcal {W}_n$ is finite,

where the last inequality holds by Lemma 3.11 and (P). Let $\theta ={qe^\varepsilon e^{\varepsilon _\varphi }}/{d}<1$ . Then

$$ \begin{align*}\dfrac{\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}}{\sum_{w\in\mathcal{W}_n}e^{S_n\varphi_x(a(w))}}\leq e^{2M}\sum_{j=1}^{\infty}\bigg(\dfrac{qe^{\varepsilon}e^{\varepsilon_\varphi}}{d}\bigg)^{jm}= e^{2M}\bigg(\dfrac{\theta^m}{1-\theta^m}\bigg).\end{align*} $$

But

$$ \begin{align*}\sum_{w\in\mathcal{W}_n}e^{S_n\varphi_x(a(w))}=\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}+\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}}}e^{S_n\varphi_x(a(w))}.\end{align*} $$

Choose m such that $1-\theta ^m\geq \tfrac 12$ . Then

$$ \begin{align*}\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}\leq 2e^{2M}C\theta^m\bigg(\sum_{w\in{\mathcal{W}_n^{\mathcal{B}}}}e^{S_n\varphi_x(a(w))}+\sum_{w\in{\mathcal{W}_n^{\mathcal{G}}}}e^{S_n\varphi_x(a(w))}\bigg).\end{align*} $$

So if we increase m so that $2e^{2M}C\theta ^m<\tfrac 12$ , then

$$ \begin{align*}\sum_{w\in\mathcal{W}_n^{\mathcal{B}}}e^{S_n\varphi_x(a(w))}\leq 4e^{2M}C\theta^m\sum_{w\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_x(a(w))}.\end{align*} $$

This achieves the desired result.

Now we apply the above with various values of m to prove the Hölder continuity of $\Phi $ . First, a bound on $\Phi _n$ .

Lemma 3.14. There exist $C_3>0$ and $\beta>0$ such that

$$ \begin{align*}|\Phi_n(x)-\Phi_n(x')|\leq C_3d(f^nx,f^nx')^{\alpha\beta}\end{align*} $$

for all $x,x'\in X$ and .

Proof. First note that along good orbit pairs, we have by Lemma 3.12 that

$$ \begin{align*} |S_n\varphi(x,\overline{y})-S_n\varphi(x',\overline{y}')|&\leq \sum_{k=0}^{n-1}|\varphi|_\alpha d(F^k(x,\overline{y}),F^k(x',\overline{y}'))^\alpha\nonumber\\ &\leq \sum_{k=0}^{n-1}|\varphi|_\alpha Q^{\alpha m}e^{-2c\alpha (n-k)}d(f^nx,f^nx')^\alpha\nonumber\\ &\leq Q^{\alpha m}d(f^nx,f^nx')^\alpha\cdot\sum_{k=0}^{\infty}|\varphi|_\alpha e^{-2c\alpha (n-k)}.\nonumber \end{align*} $$

Let $V=\sum _{k=0}^{\infty }|\varphi |_\alpha e^{-2c\alpha (n-k)}$ . Then

(7) $$ \begin{align} |S_n\varphi(x,\overline{y})-S_n\varphi(x',\overline{y}')|\leq VQ^{\alpha m}d(f^nx,f^nx')^\alpha. \end{align} $$

For convenience, we write $\Sigma _{\mathcal {G}}=\sum _{w\in \mathcal {W}_n^{\mathcal {G}}}e^{S_n\varphi _x(a(w))}$ and $\Sigma _{\mathcal {B}}=\sum _{w\in \mathcal {W}_n^{\mathcal {B}}}e^{S_n\varphi _x(a(w))}$ as well as $\Sigma _{\mathcal {G}}'$ and $\Sigma _{\mathcal {B}}'$ for the sums of the preimages associated to $a'(w)$ . By Lemma 3.13, we get that

(8)

Note that by (7),

$$ \begin{align*} \dfrac{\Sigma_{\mathcal{G}}}{\Sigma_{\mathcal{G}}'}&=\dfrac{\sum_{w\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_x(a(w))}}{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_{x'}(a'(w))}}\\ &\leq \dfrac{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{VQ^{\alpha m}d(f^nx,f^nx')^\alpha}e^{S_n\varphi_{x'}(a'(w))}}{\sum_{w'\in\mathcal{W}_n^{\mathcal{G}}}e^{S_n\varphi_{x'}(a'(w))}}\\ &\leq e^{VQ^{\alpha m}d(f^nx,f^nx')^\alpha}. \end{align*} $$

Let $\Phi _n$ be as in Theorem 3.5 for the delta measure on Y, $\delta _y\ (y\in Y)$ . Then

where the second inequality holds due to (8).

Let $\rho _1={\theta }/{Q^\alpha }$ and note that $\rho _1<1$ . Then there is a such that

(9) $$ \begin{align} \rho_1^{k+1}\leq d(f^nx,f^nx')^\alpha\leq\rho_1^k. \end{align} $$

Now set $m=k$ . Let $\beta =({\log \theta }/{\log \rho _1})$ and note that

$$ \begin{align*}\theta^m=e^{m\log\theta}=e^{\beta m\log\rho_1}=\rho_1^{\beta m}\leq\rho_1^{-\beta}d(f^nx,f^nx')^{\alpha\beta}.\end{align*} $$

Thus, $Q^{\alpha m}d(f^nx,{\kern-1pt} f^nx')^\alpha {\kern-1pt}\leq{\kern-1pt} \theta ^m{\kern-1pt}\leq{\kern-1pt} \rho _1^{-\beta }d(f^nx,{\kern-1pt}f^nx')^{\alpha \beta }$ . Hence, letting $C_3{\kern-1pt}={\kern-1pt}2(V{\kern-1pt}+{\kern-1pt}C)\rho _1^{-\beta }$ yields

$$ \begin{align*}|\Phi_n(x)-\Phi_n(x')|\leq C_3d(f^nx,f^nx')^{\alpha\beta}.\\[-34pt]\end{align*} $$

Theorem 3.15. The potential $\Phi $ constructed in Theorem 3.5 is Hölder continuous.

Proof. Let $C=\max \{C_1,C_3\}$ . For any $n\geq 0$ ,

$$ \begin{align*} |\Phi(x)-\Phi(x')|&\leq |\Phi(x)-\Phi_n(x)|+|\Phi_n(x)-\Phi_n(x')|+|\Phi_n(x')-\Phi(x')|\\ &\leq 2C\tau^n+Cd(f^nx,f^nx')^{\alpha\beta}\\ &\leq 2C\tau^n+C\Gamma^{\alpha\beta n}d(x,x')^{\alpha\beta}. \end{align*} $$

where the second inequality follows from Theorem 3.5 and Lemma 3.14 and $\Gamma $ is the inherited Lipschitz constant for f.

Similarly to the argument in the proof of Lemma 3.14, we need to adjust the Hölder exponent to establish our bound. Let $\rho _2={\tau }/{\Gamma ^{\alpha \beta }}$ . Then there is a k such that $\rho _2^{k+1}\leq d(x,x')^{\alpha \beta }\leq \rho _2^k$ . Let $n=k$ and $\eta ={\log \tau }/{\log \rho _2}$ . Then

$$ \begin{align*}\tau^n=e^{n\log\tau}=e^{\eta n\log\rho_2}=\rho_2^{\eta n}\leq\rho_2^{-\eta}d(x,x')^{\alpha\beta\eta}.\end{align*} $$

So $\Gamma ^{\alpha \beta }d(x,x')^{\alpha \beta }\leq \tau ^n\leq \rho _2^{-\eta }d(x,x')^{\alpha \beta \eta }$ . Therefore,

$$ \begin{align*}|\Phi(x)-\Phi(x')|\leq 2C\rho_2^{-\eta}d(x,x')^{\alpha\beta\eta}.\\[-34pt]\end{align*} $$

4 Conditional measures of equilibrium states

Let $\mathcal {L}_\Phi \colon C(X)\to C(X)$ be defined by

for any $\xi \in C(X)$ . Since f is uniformly expanding on X, Theorem 3.15 implies that there is a unique equilibrium state that can be obtained via $\mathcal {L}_\Phi $ . See Theorems 6 and 8 in [13] for details.

Theorem 4.1. Let X be a compact, connected manifold and $f\colon X\to X$ is uniformly expanding. For any Hölder , the following assertions hold.

1. (1) There is a unique probability measure $\hat {\nu }\in \mathcal {M}(X)$ with the property that $\mathcal {L}_\Phi ^*\hat {\nu }$ is a scalar multiple of $\hat {\nu }$ .

2. (2) There is a unique positive continuous function $\hat {h}\in C(X)$ with the property that $\mathcal {L}_\Phi \hat {h}$ is a scalar multiple of $\hat {h}$ and $\int _X\hat {h}(x)d\hat {\nu }(x)=1$ .

3. (3) The eigenvalues associated to $\hat {\nu }$ and $\hat {h}$ are the same.

4. (4) The unique equilibrium state for $\Phi $ is .

We shall show that and construct the family of measures $\{\mu _x\}_{x\in X}$ . To do this, we first prove the following lemmas. Place on $C(Y)$ the sup norm $\|\cdot \|_\infty $ ; that is, if $\psi \in C(Y)$ , then $\|\psi \|_\infty =\sup \{\psi (y)\colon y\in Y\}$ .

Lemma 4.2. For any $\psi \in C(X\times Y)$ , the map $x\mapsto \mathcal {L}^x_{\varphi }\psi _x$ is continuous with respect to the topology induced on $C(Y)$ by $\|\cdot \|_\infty $ .

Proof. Let $\psi \in C(X\times Y)$ and $y\in Y$ . For any $x,x'\in X$ ,

$$ \begin{align*}|{\mathcal{L}_{x}}\psi_x(y)-{\mathcal{L}_{x'}}\psi_{x'}(y)| \leq\!\! \sum_{\overline{y}\in g_x^{-1}y}\!\!(e^{\varphi(x,\overline{y})}|\psi(x,\overline{y})-\psi(x',\overline{y}')|+\|\psi\|_\infty\ |e^{\varphi(x,\overline{y})}-e^{\varphi(x',\overline{y}')}|) \end{align*} $$

where $\overline {y}'$ are the preimages of y under $g_{x'}^{-1}$ given by Lemma 2.3. Fix $\epsilon>0$ . Let $M_1=\sup _{(x,y)\in X\times Y}\{\mathcal {L}_x\psi (y)\}$ . Since $\psi $ is continuous, there exists a $\delta _1>0$ such that $|\psi (u,v)-\psi (u',v')|<{\epsilon }/{2M_1}$ whenever $d((u,v),(u'v'))<\delta _1$ . Similarly, there is a $\delta _2>0$ such that $|e^{\varphi (u,v)}-e^{\varphi (u',v')}|<{\epsilon }/{2d\|\psi \|_\infty }$ whenever $d((u,v),(u'v'))<\delta _2$ .

Let $\delta _3={\min \{\delta _1,\delta _2\}}/{L}$ . By the continuity of f, there exists $\delta>0$ such that if $d(x,x')<\delta $ , then $d((fx,y),(fx',y))<\delta _3$ . By Lemma 2.3, for all $(x,\overline {y})\in F^{-1}(fx,y)\cap Y_x$ , there exists $(x',\overline {y}')\in F^{-1}(fx',y)\cap Y_{x'}$ such that if $(x,\overline {y})\in \mathcal {A}$ , then

$$ \begin{align*}d((x,\overline{y}),(x',\overline{y}'))\leq Ld(fx,fx')<L\delta_3<\min\{\delta_1,\delta_2\}\end{align*} $$

and if $(x,\overline {y})\not \in \mathcal {A}$ , then

$$ \begin{align*}d((x,\overline{y}),(x',\overline{y}'))\leq \gamma^{-1}d(fx,fx')<\delta_3.\end{align*} $$

Thus, $d((x,\overline {y}),(x',\overline {y}'))<\min \{\delta _1,\delta _2\}$ . Hence,

$$ \begin{align*} |{\mathcal{L}_{x}}\psi_x(y)-{\mathcal{L}_{x'}}\psi_{x'}(y)| &\leq \dfrac{\epsilon}{2M_1}\sum_{\overline{y}\in g_x^{-1}y} e^{\varphi(x,\overline{y})} + \|\psi\|_\infty\ \sum_{\overline{y}\in g_x^{-1}y} \dfrac{\epsilon}{2d\|\psi\|_\infty}\\&<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon. \end{align*} $$

Since this is independent of y, we have $\|{\mathcal {L}_{x}}\psi _x-{\mathcal {L}_{x'}}\psi _{x'}\|_\infty <\epsilon $ .

Remark 1. This proof can be extended to hold for all iterates of the transfer operator.

Lemma 4.3. For every continuous , the map $x\mapsto \nu _x(\psi _x)$ is continuous with respect to the usual topology.

Proof. Fix $y\in Y$ and let

be as in Proposition 3.7. So as shown there, $\nu _{x,n}\xrightarrow [\phantom {.}]{{\text {weak}^*}}\nu _x$ . For any $x,x'\in X$ ,

$$ \begin{align*} \bigg|\int\psi\, d\nu_{x}-\int\psi\, d\nu_{x'}\bigg|&\leq\bigg|\int\psi\, d\nu_{x}-\int\psi\, d\nu_{x,n}\bigg|+\bigg|\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_{x',n}\bigg|\\ &\quad+\bigg|\int\psi\, d\nu_{x',n}-\int\psi\, d\nu_{x'}\bigg|\\ &\leq 2C_1\|\psi\|\tau^n + \bigg|\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_{x',n}\bigg| \end{align*} $$

where the last inequality holds by Proposition 3.7. Note that

is continuous in x by Lemma 4.2. Given $\epsilon>0$ , choose n sufficiently large so that $2C_1\|\psi \|\tau ^n<\epsilon /2$ and $\delta>0$ such that $d(x,x')<\delta $ implies $|\kern-1.5pt\int \psi\, d\nu _{x,n}-\int \psi\, d\nu _{x',n}|<\epsilon /2$ . Then

$$ \begin{align*}\bigg|\int\psi\, d\nu_{x}-\int\psi\, d\nu_{x'}\bigg|\leq 2C_1\|\psi\|\tau^n + \bigg|\int\psi\, d\nu_{x,n}-\int\psi\, d\nu_{x',n}\bigg|<\epsilon.\end{align*} $$

This proves continuity of $x\mapsto \nu _x(\psi _x)$ .

Define $I\colon C(X\times Y)\to C(X)$ by $(I\psi )(x)=\int _{Y_x}\psi (x,y)\,d\nu _x({y})$ . Observe that for any $\eta \in \mathcal {M}(X)$ , we have

$$ \begin{align*}\langle\psi,I^*\eta\rangle=\int_X(I\psi)(x)\,d\eta(x)=\int_X\int_Y\psi(x,y)\,d\nu_x(y)\,d\eta(x).\end{align*} $$

So $\langle I\psi ,\eta \rangle \kern1.3pt{=}\kern1.3pt\langle \psi ,I^*\eta \rangle $ where $I^*\colon \mathcal {M}(X)\kern1.3pt{\to}\kern1.3pt \mathcal {M}(X\times Y)$ is defined by $I^*\eta =\int _X\nu _x\,d\eta (x)$ .

Theorem 4.4. The operators I and $I^*$ satisfy $I\circ \mathcal {L}_\varphi =\mathcal {L}_\Phi \circ I$ and $I^*\circ \mathcal {L}_\Phi ^*={\mathcal {L}_\varphi ^*}\circ I^*$ . That is, they make their respective diagrams below commute:

Proof. Given $\psi \in C(X\times Y)$ , we have

$$ \begin{align*} I\circ\mathcal{L}_\varphi\psi(x,y)&=\int_{X\times Y}\mathcal{L}_\varphi\psi(x,y)\, d\nu_x(y)\\&=\int_{X\times Y}\sum_{\overline{x}\in f^{-1}x}\sum_{\overline{y}\in g_{\overline{x}}^{-1}y}e^{\varphi(\overline{x},\overline{y})}\psi(\overline{x},\overline{y})\,d\nu_x(y)\\&=\int_{X\times Y}\sum_{\overline{x}\in f^{-1}x}(\mathcal{L}_{\overline{x}}\psi)(x,y)\,d\nu_x(y)\\&=\sum_{\overline{x}\in f^{-1}x}\langle\mathcal{L}_{\overline{x}}\psi,\nu_x\rangle\\ &=\sum_{\overline{x}\in f^{-1}x}\langle\psi,(\mathcal{L}_{\overline{x}})^*\nu_x\rangle\\ &=\sum_{\overline{x}\in f^{-1}x}e^{\Phi(\overline{x})}\langle\psi,\nu_{\overline{x}}\rangle=\sum_{\overline{x}\in f^{-1}x}e^{\Phi(\overline{x})}(I\psi)(\overline{x})\\ &=(\mathcal{L}_\Phi\circ I)\psi(x). \end{align*} $$

Duality gives $I^*\circ \mathcal {L}_\Phi ^*=\mathcal {L}_\varphi ^*\circ I^*$ .

Corollary 4.5. $P(\Phi )=P(\varphi )$ . Moreover, $\nu ,\hat {\nu }$ and $h,\hat {h}$ satisfy $\nu =I^*\hat {\nu }$ and $\hat {h}=Ih$ .

Proof. By Theorem 4.4, we have

$$ \begin{align*}\mathcal{L}^*_{\varphi}(I^*\hat{\nu})=I^*\mathcal{L}_\Phi^* \hat{\nu}=I^*e^{P(\Phi)}\hat{\nu}=e^{P(\Phi)}({I^*\hat{\nu}}).\end{align*} $$

Then items (1) and (3) of Theorem 4.1 implies $I^*\hat {\nu }=\nu $ and $P(\varphi )=P(\Phi )$ , respectively. Now we will show that $Ih$ is the eigenfunction for $\mathcal {L}_\Phi $ . We have

$$ \begin{align*}\mathcal{L}_\Phi(Ih)=I\mathcal{L}_\varphi h=Ie^{P(\varphi)}h=e^{P(\Phi)}(Ih)\end{align*} $$

where the first equality is by Theorem 4.4 and the last is by the paragraph above. Moreover, the paragraph above implies that $\int (Ih)d\hat {\nu }=\int h\, d\nu =1$ where the last equality holds by item (2) of Theorem 4.1 for the potential . Thus, item (2) of Theorem 4.1 for implies $Ih=\hat {h}$ .

Thus, given any $\psi \in C(X\times Y)$ , we have

where $\mu _x$ is defined by ${d\mu _x}/{d\nu _x}={h(x,y)}/{\hat {h}(x)}$ and $\overline {\mu }=\hat {h}\hat {\nu }$ . Note that by Corollary 4.5

$$ \begin{align*}\mu_x(Y_x)=\int_{Y_x} \frac{h(x,y)}{\hat{h}(x)}\,d\nu_x(y)=\frac{(Ih)(x)}{\hat{h}(x)}=\frac{\hat{h}(x)}{\hat{h}(x)}=1.\end{align*} $$

Therefore,

and $\{\mu _x\}_{x\in X}$ is the unique family of conditional measures for $\mu $ .