Uniqueness of minimizer for countable Markov shifts and equidistribution of periodic points

TL;DR

This paper establishes conditions under which the equilibrium state for a countable Markov shift is unique and coincides with the minimizer of a large deviations rate function, leading to equidistribution of periodic points.

Contribution

It provides a new criterion on the pressure function ensuring uniqueness of the Gibbs state and its coincidence with the large deviations minimizer for countable Markov shifts.

Findings

Unique equilibrium state under specified pressure conditions

Periodic points equidistribute with respect to the Gibbs state

Applications to Gauss map and Fuchsian groups

Abstract

For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowen's Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that all periodic points weighted with the potential equidistribute with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group with parabolic elements.

Equations

$$-\underset{{\mathcal{B}}^o}{\inf }I\le \underset{n\to \infty }{\underline{\lim }}\frac{1}{n}\log {\mu }_n({\mathcal{B}}^o)\le \underset{n\to \infty }{\overline{\lim }}\frac{1}{n}\log {\mu }_n(\overline{\mathcal{B}})\le -\underset{\overline{\mathcal{B}}}{\inf }I,$$

$$S^{\mathbb{N}}=\{x=(x_0,x_1,\dots ):x_i\in S\forall i\in \mathbb{N}\}.$$

$$X=\{x\in S^{\mathbb{N}}:t_{x_i{x}_{i+1}}=1\forall i\in \mathbb{N}\}.$$

$${\mathcal{M}}_{\varphi }(\sigma {|}_X)=\left\{\mu \in \mathcal{M}(\sigma {|}_X):\int \varphi d\mu >-\infty \right\}.$$

$$[\omega ]=[{\omega }_0,\dots ,{\omega }_{n-1}]=\{x\in X:x_i={\omega }_{i}0\le \forall i\le n-1\}.$$

$$Z_n(\varphi )=\sum _{\omega \in E^n}\underset{[\omega ]}{\sup }\exp S_n\varphi ,$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log Z_n(\varphi ).$$

$$P(\varphi )=\sup \left\{h(\mu )+\int \varphi d\mu :\mu \in {\mathcal{M}}_{\varphi }(\sigma {|}_X)\right\}.$$

$$c^{-1}\le \frac{{\mu }_{\varphi }[x_0,\dots ,x_{n-1}]}{\exp \left(-Pn+S_n\varphi (x)\right)}\le c.$$

$$Z_1(\varphi )<\infty .$$

$${\beta }_{\infty }(\varphi )=\inf \{\beta \in \mathbb{R}:\text{βϕ is summable}\}.$$

$$\sum _{n=1}^{\infty }\underset{\omega \in E^n}{\sup }\underset{x,y\in [\omega ]}{\sup }\varphi (x)-\varphi (y)<\infty .$$

$${\left(\sum _{x\in {\mathrm{Per}}_n(\sigma {|}_X)}\exp S_n\varphi (x)\right)}^{-1}\sum _{x\in {\mathrm{Per}}_n(\sigma {|}_X)}\exp S_n\varphi (x){\delta }_x^n$$

$$T:x\in (0,1]\mapsto \frac{1}{x}-\lfloor \frac{1}{x}\rfloor \in [0,1).$$

$${\left(\sum _{x\in {\mathrm{Per}}_n(T)}|(T^n{)}^{\prime }x{|}^{-1}\right)}^{-1}\sum _{x\in {\mathrm{Per}}_n(T)}|(T^n{)}^{\prime }x{|}^{-1}{\delta }_x^n(T)$$

$$J=\bigcap _{n=0}^{\infty }{f}^{-n}\Delta .$$

$${\dim }_{H}J=\inf \{\beta \ge 0:P(\beta \varphi )<0\}.$$

$${\beta }_{\infty }(\varphi )\le {\dim }_{H}J.$$

$$G_0=\left\{g_s:s\text{side of }R\right\}\subset G$$

$${\Gamma }_0=\{{\gamma }_0^{\pm 1}\}\text{ and }{H}_0=G_0\setminus {\Gamma }_0.$$

$${\Delta }_g={\mathbb{S}}^1\cap \left\{z\in {\mathbb{R}}^2:|g^{\prime }(z)|\ge 1\right\},$$

$$f{|}_{{\Delta }_g}=g{|}_{{\Delta }_g},g\in G_0.$$

$$\tilde{f}x=f^{|B_1(x)|}x,$$

$$F=\bigcup _{n=1}^{\infty }\left\{{\gamma }^{n}g:\gamma \in {\Gamma }_0,g\in G_0\setminus \left\{{\gamma }^{\pm 1}\right\}\right\}\cup \left\{hg:h\in H_0,g\in G_0\setminus \left\{h^{-1}\right\}\right\}.$$

$${\left(\sum _{x\in {\mathrm{Per}}_n(\tilde{f})}\exp \left(-\sum _{i=1}^n|B_i(x)|\right)\right)}^{-1}\sum _{x\in {\mathrm{Per}}_n(\tilde{f})}\exp \left(-\sum _{i=1}^n|B_i(x)|\right){\delta }_x^n(\tilde{f})$$

$$\begin{array}{r}F(\nu )=\{\begin{array}{ll}-P(\varphi )+h(\nu )+\int \varphi d\nu & \text{ if ν∈Mϕ(σ∣X)};\\ -\infty & \text{ otherwise.}\end{array}\end{array}$$

$$I(\mu )=-\underset{\mathcal{G}\ni \mu }{\inf }\underset{\mathcal{G}}{\sup }F,$$

$${\eta }_n={\left(\sum _{x\in {\mathrm{Per}}_n(\sigma {|}_X)}\exp S_n\varphi (x)\right)}^{-1}\sum _{x\in {\mathrm{Per}}_n(\sigma {|}_X)}\exp S_n\varphi (x){\delta }_{{\delta }_x^n},$$

$$\underset{n\to \infty }{\overline{\lim }}\frac{1}{n}\log {\mu }_{\varphi }\{x\in X:{\delta }_x^n\notin \mathcal{G}\}\le -\underset{\mathcal{M}\setminus \mathcal{G}}{\inf }I<0.$$

$$\underset{n\to \infty }{\overline{\lim }}\frac{1}{n}\log \frac{{\sum }_{\stackrel{x\in {\mathrm{Per}}_n(\sigma {|}_X)}{{\delta }_x^n\notin \mathcal{G}}}\exp S_n\varphi (x)}{{\sum }_{x\in {\mathrm{Per}}_n(\sigma {|}_X)}\exp S_n\varphi (x)}\le -\underset{\mathcal{M}\setminus \mathcal{G}}{\inf }I<0.$$

Full text

Keywords: Large Deviation Principle; countable Markov shift; Gibbs-equilibrium state; minimizer

1. Introduction

The theory of large deviations aims to characterize limit behaviors of measures in terms of rate functions. A sequence $\{\mu_{n}\}_{n=1}^{\infty}$ of Borel probability measures on a topological space $\mathcal{X}$ satisfies the Large Deviation Principle (LDP) if there exists a lower semi-continuous function $I\colon\mathcal{X}\to[0,\infty]$ such that for every Borel subset $\mathcal{B}$ of $\mathcal{X}$ the following holds:

[TABLE]

where $\log 0=-\infty$, $\inf\emptyset=\infty$, ${\mathcal{B}}^{o}$ and $\overline{\mathcal{B}}$ denote the interior and the closure of $\mathcal{B}$ respectively. The function $I$ is called a rate function, and is called a good rate function if the level set $\{x\in\mathcal{X}\colon I(x)\leq\alpha\}$ is compact for every $\alpha\in(0,\infty)$.

If $\mathcal{X}$ is a metric space, then the rate function is unique. We call $x\in\mathcal{X}$ a minimizer if $I(x)=0$. For a closed set $\mathcal{B}$ of $\mathcal{X}$ which is disjoint from the set of minimizers, the LDP ensures that $\mu_{n}(\mathcal{B})$ decays exponentially as $n\to\infty$. If moreover $I$ is a good rate function, the support of any accumulation point of $\{\mu_{n}\}_{n=1}^{\infty}$ is contained in the set of minimizers. Hence, it is important to determine the set of minimizers. The non-uniqueness of minimizer is referred to as a phase transition. The uniqueness of minimizer implies several strong conclusions.

Donsker and Varadhan have identified three levels of the LDP, see [8, Chapter 1] for details. This paper is concerned with the level-2 LDP in the thermodynamic formalism. For finite topological Markov shifts and Hölder continuous potentials, unique Gibbs-equilibrium states have been constructed by Bowen [2] and Ruelle [24]. Extensions of this construction to countable Markov shifts have been done by Mauldin and Urbański [20], Sarig [25, 26] after the works of Walters [30], Gurevic and Savchenko [11]. The level-2 LDP for finite Markov shifts has been established by Takahashi [27], Kifer [17, 18], and in some particular cases by Orey and Pelikan [22].

The level-2 LDP for countable Markov shifts has been established in [28] under the assumption of the existence of a Gibbs state and a strong connectivity of transition matrices. The assumption on potentials in [28] is rather general, and not much is known about minimizers in such a generality. As the first result of this paper, in Theorem 1.1 we provide conditions which ensure the uniqueness of minimizer.

The uniqueness of minimizer implies several strong conclusions. e.g., an exponential decay, Erdös-Rényi’s law [5] for the level-1 LDP, the differentiability of pressure [8]. The second result of this paper is Theorem 1.3 which asserts that the uniqueness of minimizer in Theorem 1.1 implies a weighted equidistribution of periodic points. For dynamical systems of compact spaces, several large deviations approaches to distributions of periodic points are known, see e.g., [18, 23]. A natural problem to consider is an extension of these results to dynamical systems of non-compact spaces, notably to countable Markov shifts. One existing result closely related to Theorem 1.3 is that of Fiebig et al. [10] to which we shall come back later on.

1.1. Thermodynamic formalism for countable Markov shifts

Let $S$ be a countable set and $\mathbb{N}$ the set of non-negative integers. Let $S^{\mathbb{N}}$ denote the set of one-sided infinite sequences over $S$ endowed with the product topology of the discrete topology on $S$, namely

[TABLE]

This topology is metrizable with a metric $d(x,y)=\exp\left({-\inf\{i\in\mathbb{N}\colon x_{i}\neq y_{i}\}}\right)$ with the convention $\exp(-\infty)=0$. The left shift $\sigma$ acts continuously on $S^{\mathbb{N}}$ by $(\sigma x)_{i}=x_{i+1}$ $(\forall i\in\mathbb{N})$. Let $T=(t_{ij})_{S\times S}$ be a matrix of zeros and ones with no column or row which is all made of zeros. A (one-sided) topological Markov shift $X$ generated by the transition matrix $T$ is given by

[TABLE]

If $\#S=\infty$ (resp. $\#S<\infty$), $X$ is called a countable (resp. finite) Markov shift. If all entries of the matrix are $1$, it is called the full shift. The restriction of the left shift to $X$ is denoted by $\sigma|_{X}$.

For two strings $\omega=\omega_{0}\cdots\omega_{m-1}$, $\alpha=\alpha_{0}\cdots\alpha_{n-1}$ of elements of $S$, let $\omega\alpha$ denote the concatenated string $\omega_{0}\cdots\omega_{m-1}\alpha_{0}\cdots\alpha_{n-1}$. An $n$-string $\omega_{0}\cdots\omega_{n-1}$ is admissible if $n=1$, or else $n\geq 2$ and $t_{\omega_{i}\omega_{i+1}}=1$ holds for every $0\leq i\leq n-1$. Let $E^{n}$ denote the set of $n$-admissible strings and put $E^{*}=\bigcup_{n=1}^{\infty}E^{n}$. A countable Markov shift $X$ is finitely irreducible if there exists a finite set $\Lambda\subset E^{*}$ such that for all $\omega,\omega^{\prime}\in E^{*}$ there exists $\lambda\in\Lambda$ for which $\omega\lambda\omega^{\prime}\in E^{*}$. If $X$ is finitely irreducible and the finite set $\Lambda$ consists of strings of the same length $N$, then $X$ is called finitely primitive. Notice that the set $\Lambda$ associated either with a finitely irreducible or primitive $X$ can be taken to be empty for the full shift (in which case $N=0$).

Let $\mathcal{M}$ denote the space of Borel probability measures on $X$ endowed with the weak*-topology, and $\mathcal{M}(\sigma|_{X})$ the set of shift-invariant elements of $\mathcal{M}$. The space $\mathcal{M}$ is metrizable with the bounded Lipschitz metric. The Kolmogorov-Sinaĭ entropy of each measure $\mu\in\mathcal{M}(\sigma|_{X})$ with respect to $\sigma|_{X}$ is denoted by $h(\mu)$. For a measurable function $\phi\colon X\to\mathbb{R}$ with $\sup\phi<\infty$ define

[TABLE]

For each $n$-string $\omega=\omega_{0}\cdots\omega_{n-1}\in E^{n}$ define an $n$-cylinder

[TABLE]

Put

[TABLE]

where $S_{n}\phi=\sum_{i=0}^{n-1}\phi\circ\sigma^{i}$. Since $n\mapsto\log Z_{n}(\phi)$ is sub-additive, the limit

[TABLE]

in (1.1) exists, and in fact never $-\infty$. Define

[TABLE]

We say the variational principle holds if the limit in (1.1) is equal to $P(\phi)$. If the variational principle holds and $P(\phi)<\infty$, then measures in $\mathcal{M}_{\phi}(\sigma|_{X})$ which attain the supremum in (1.2) are called equilibrium states for the potential $\phi$.

A Borel probability measure $\mu_{\phi}$ on $X$ is a Gibbs state (in the sense of Bowen) for the potential $\phi$ (cf. [2, 20, 24, 25]) if there exist constants $c\geq 1$ and $P\in\mathbb{R}$ such that for every $n\geq 1$ and every $x\in X$,

[TABLE]

If $\mu_{\phi}$ is shift-invariant, then it is called a shift-invariant Gibbs state.

1.2. Uniqueness of minimizer

Let $X$ be a countable Markov shift. A function $\phi\colon X\to\mathbb{R}$ is summable if

[TABLE]

The summability of $\phi$ implies $\sup\phi<\infty$ and $\inf\phi=-\infty$. For a summable function $\phi$, define

[TABLE]

If $\beta_{0}\geq 0$ and $\beta_{0}\phi$ is summable then so is $\beta\phi$ for every $\beta>\beta_{0}$. Hence, $0\leq\beta_{\infty}(\phi)\leq 1$ holds. A function $\phi\colon X\to\mathbb{R}$ has summable variations if

[TABLE]

If $X$ is finitely irreducible and $\phi$ is summable with summable variations, then the variational principle holds, and there exists a unique shift-invariant Gibbs state $\mu_{\phi}$ for the potential $\phi$. If moreover $\int\phi d\mu_{\phi}>-\infty$, then $\mu_{\phi}$ is the unique equilibrium state for the potential $\phi$ (see [20]). One can verify this integrability provided $\beta_{\infty}(\phi)<1$. In fact, this condition also allows us to show the uniqueness of minimizer. We obtain the following result.

Theorem 1.1**.**

Let $X$ be a finitely irreducible countable Markov shift and $\phi\colon X\to\mathbb{R}$ a summable function with summable variations satisfying $\beta_{\infty}(\phi)<1.$ Then, the minimizer of the rate function is unique. It coincides with the unique shift-invariant Gibbs state, and the unique equilibrium state for the potential $\phi$.

Theorem 1.1 has its analogue in statistical mechanics. Dobrushin-Lanford-Ruelle’s variational principle [6, 7, 19] states that the set of shift-invariant Gibbs states defined by the DLR-equation, that of equilibrium states (measures minimizing the free energy) and that of minimizers (of the relative entropy density) coincide for shift-invariant absolutely summable interactions.

Remark 1.2*.*

The following example inspired by [1, 20, 26] shows that the assumption $\beta_{\infty}(\phi)<1$ in Theorem 1.1 is not removable. Consider the potential $\phi\colon\mathbb{N}^{\mathbb{N}}\to\mathbb{R}$ given by $\phi(x)=\log p_{x_{0}}$ where $\{p_{k}\}_{k\in\mathbb{N}}$ is a sequence with $p_{k}\in(0,1)$, $\sum_{k\in\mathbb{N}}p_{k}=1$ and $\sum_{k\in\mathbb{N}}p_{k}\log p_{k}=-\infty$, e.g., $p_{k}\propto 1/(k(\log k)^{2})$. The Gibbs state $\mu_{\phi}$ for the potential $\phi$ is the Bernoulli measure associated with the infinite probability vector $(p_{k})_{k\in\mathbb{N}}$. It is ergodic and hence a minimizer. As $\int\phi d\mu_{\phi}=-\infty$, it is not an equilibrium state for the potential $\phi$.

1.3. Weighted equidistribution

We now proceed to distributions of periodic points. For each $n\geq 1$ and $x\in X$ let $\delta_{x}^{n}$ denote the uniform probability distribution on $\{x,\sigma x,\ldots,\sigma^{n-1}x\}$. Write ${\rm Per}_{n}(\sigma|_{X})=\{x\in X\colon\sigma^{n}x=x\}.$

Theorem 1.3**.**

Let $X$ be a finitely primitive countable Markov shift and $\phi\colon X\to\mathbb{R}$ a summable function with summable variations satisfying $\beta_{\infty}(\phi)<1$. As $n\to\infty$, the Borel probability measure on $X$ defined by

[TABLE]

converges in the weak-topology to the equilibrium state for the potential $\phi$.*

For transitive countable Markov shifts, Fiebig et al. [10] introduced the notion of $Z$-recurrence in terms of a pressure computed from weighted periodic points contained in one fixed $1$-cylinder. Under the $Z$-recurrence and some additional assumptions, they showed that the sequence of measures constructed from these periodic points are tight, and all its weak*-accumulation points are equilibrium states for the potential [10, Theorem 4.2]. Theorem 1.3 does not follow from [10, Theorem 4.2] because it concerns all periodic points in $X$ and there are infinitely many $1$-cylinders. The tightness of the sequence of measures in Theorem 1.3 is a consequence of the exponential tightness [4] of the sequence $\{\eta_{n}\}_{n=1}^{\infty}$ of measures on $\mathcal{M}$ in Theorem 1.6.

We apply Theorem 1.3 to the Gauss map

[TABLE]

Following the orbits of $T$ over the infinite Markov partition $\{(\frac{1}{k+1},\frac{1}{k}]\}_{k\geq 1}$, one can model $T$ with the countable full shift. The unbounded function $-\log|T^{\prime}|$ induces a potential $\phi$ on the full shift which is summable with summable variations (see [10, Section 7]). The Gauss measure $\frac{1}{\log 2}\frac{dx}{1+x}$ corresponds to the unique equilibrium state for the potential $\phi$ (see [31, Theorem 22]), and we have $\beta_{\infty}(\phi)=1/2$ (see [21]). By Theorem 1.1, the equilibrium state for the potential $\phi$ is the unique minimizer of the corresponding rate function.

For each integer $n\geq 1$ and $x\in(0,1)\setminus\mathbb{Q}$, let $\delta_{x}^{n}(T)$ denote the uniform probability distribution on $\{x,Tx,\ldots,T^{n-1}x\}$. Write ${\rm Per}_{n}(T)=\{x\in(0,1)\setminus\mathbb{Q}\colon T^{n}x=x\}.$ The exponential instability of each point $x\in{\rm Per}_{n}(T)$ under the iteration of the Gauss map is quantified by its Lyapunov exponent $(1/n)\log|(T^{n})^{\prime}x|$. It is well-known [12, 16] that numbers in $\bigcup_{n=1}^{\infty}{\rm Per}_{n}(T)$ correspond to solutions of quadratic equations with integer coefficients, and their Lyapunov exponents are related to the rate of Diophantine approximation by the regular continued fraction expansion.

Proposition 1.4**.**

As $n\to\infty$, the measure

[TABLE]

converges in the weak-topology to the Gauss measure.*

1.4. Finitely primitive uniformly expanding Markov maps

Examples satisfying the key condition $\beta_{\infty}(\phi)<1$ in Theorem 1.1 and Theorem 1.3 are in abundance. We show this by linking this condition to another on dimension of invariant sets of Markov interval maps.

Let $f\colon\varDelta\to[0,1]$ satisfy the following:

• •

there exists a family $\{\varDelta_{a}\}_{a\in\mathbb{N}}$ of intervals with disjoint interiors such that $\varDelta=\bigcup_{a\in\mathbb{N}}\varDelta_{a}$ and $f|_{\varDelta_{a}}=f_{a}$ for each $a\in\mathbb{N}$.

• •

for each $a\in\mathbb{N}$, $f_{a}$ is a $C^{1}$ diffeomorphism onto its image with appropriate one-sided derivatives.

• •

if $a,b\in\mathbb{N}$ and $f\varDelta_{a}\cap\varDelta_{b}$ has non-empty interior, then $f\varDelta_{a}\supset\varDelta_{b}$.

The maximal invariant set of $f$ is defined by

[TABLE]

Let $\dim_{H}J\in[0,1]$ denote the Hausdorff dimension of $J$. We assume (some iterate of) $f$ is uniformly expanding, i.e., there exists $\kappa>1$ such that $|f^{\prime}|>\kappa$ on $\varDelta$. Then, $f$ is modeled by the countable Markov shift whose transition matrix is determined by the Markov partition $\{\varDelta_{a}\}_{a\in\mathbb{N}}$. We moreover assume the Markov shift is finitely primitive, and $-\log|f^{\prime}|$ induces a summable function $\phi$ with summable variations. By [20, Theorem 4.2.13] we have

[TABLE]

From this and [20, Proposition 2.1.9] we obtain

[TABLE]

Therefore, the condition $\dim_{H}J<1$ implies $\beta_{\infty}(\phi)<1$. In particular, all periodic points of $f$ weighted with their Lyapunov exponents equidistribute as the periods tend to infinity as in Proposition 1.4.

1.5. Bowen-Series maps for free Fuchsian groups

We give another application of Theorem 1.3 to cuspidal windings for geodesic flows on hyperbolic surfaces. Our presentation below is essentially a compressed version of [14, Section 2]. For more details, see there and the references therein.

Let $\mathbb{D}\subset\mathbb{R}^{2}$ denote the open unit disk around the origin and $\mathbb{S}^{1}$ the boundary of $\mathbb{D}$. A Fuchsian group is a discrete subgroup of Möbius transformations preserving $\mathbb{D}$. Let $G$ be a finitely generated free Fuchsian group with parabolic elements. We denote by $\Lambda(G)\subset\mathbb{S}^{1}$ the limit set of $G$. Fix a Dirichlet fundamental domain $R\subset\mathbb{D}$ for $G$. Each side $s$ of $R$ gives rise to a side-pairing transformation $g_{s}\in G$, and the set

[TABLE]

is a symmetric set of generators of $G$. Since $G$ is free, each vertex of $R$ is in $\mathbb{S}^{1}$ and it is a parabolic fixed point of some element of $G$. For simplicity we assume the vertex of $R$ is unique, denoted by $p$, and there exists $\gamma_{0}\in G_{0}$ such that $\gamma_{0}(p)=p$. We define

[TABLE]

Each side $s$ of $R$ is contained in the isometric circle of $g_{s}$. For $g\in G_{0}$ define

[TABLE]

and put $\varDelta=\bigcup_{g\in G_{0}}\varDelta_{g}$. Following [3] we introduce the Bowen-Series map $f:\varDelta\rightarrow\mathbb{S}^{1}$ by

[TABLE]

The map $f$ determines a one-to-one correspondence between $\Lambda(G)$ and a subset of the set $\{(\omega_{n})_{n=0}^{\infty}\in G_{0}^{\mathbb{N}}\colon\omega_{n}\omega_{n+1}\neq 1\in G\ \forall n\geq 0\}$ of symbolic sequences. Let $\Lambda_{c}(G)$ denote the conical limit set of $G$. For $x\in\Lambda_{c}(G)$, its symbolic sequence $\omega$ is decomposed into a sequence of blocks $(B_{i}(x))_{i\geq 1}$ as in [13]. Each element of $H_{0}$ in $\omega$ forms a block of length one. For elements of $\Gamma_{0}$ in $\omega$ we build maximal blocks of consecutive appearances of the same element. Either $B_{i}(x)=h$ for some $h\in H_{0}$, or $B_{i}(x)=\gamma_{0}^{n}$ holds for some $n\in\mathbb{Z}\setminus\{0\}$.

This decomposition has the following geometric interpretation. For a fixed initial value $o\in\mathbb{D}/G$, the set of closed geodesics $\gamma:[0,\infty)\rightarrow\mathbb{D}/G$ satisfying $\gamma(0)=o$ can be identified with (not necessarily prime) periodic orbits of $f$ in $\Lambda_{c}(G)$. The symbolic sequence $\omega$ records the sides of $R$ crossed by $\gamma(t)$ consecutively as $t\rightarrow\infty$. A block $B_{i}(x)=\gamma_{0}^{\pm n}$ of length $n\geq 2$ means that the geodesic $\gamma$ spirals $n-1$ times around the cusp $\gamma_{0}$.

For each reduced word $\omega=\omega_{0}\cdots\omega_{n-1}\in G_{0}^{n}$ of length $n\geq 2$, let $\varDelta_{\omega}$ denote the domain of the diffeomorphism $f|_{\varDelta_{\omega_{n-1}}}\circ\cdots\circ f|_{\varDelta_{\omega_{0}}}$. We now define an induced map $\tilde{f}:\bigcup_{\omega\in F}\varDelta_{\omega}\rightarrow\mathbb{S}^{1}$ by

[TABLE]

where $|B_{1}(x)|$ denotes the word length of the block $B_{1}(x)$, and $F$ is the countable set of reduced words in $\bigcup_{n=2}^{\infty}G_{0}^{n}$ given by

[TABLE]

The maximal invariant set of $\tilde{f}$ is $\Lambda_{c}(G)$, and $B_{i}(x)=B_{1}(\tilde{f}^{i-1}x)$ holds for all $x\in\Lambda_{c}(G)$ and $i\geq 1$. The $\tilde{f}$ determines a transition matrix and an associated countable Markov shift $X$ with symbols in $F$. Since $\tilde{f}$ is uniformly expanding, there is a conjugacy $\pi\colon X\to\Lambda_{c}(G)$ with $\tilde{f}\circ\pi=\pi\circ\sigma|_{X}$. Moreover, $\tilde{f}$-invariant Borel probability measures and shift-invariant ones are in one-to-one correspondence. For each $n\geq 1$ and $x\in\Lambda_{c}(G)$ let $\delta_{x}^{n}(\tilde{f})$ denote the uniform probability distribution on $\{x,\tilde{f}x,\ldots,\tilde{f}^{n-1}x\}$. Write ${\rm Per}_{n}(\tilde{f})=\{x\in\Lambda_{c}(G)\colon B_{i}(x)=B_{i+n}(x)\ \forall i\geq 1\}.$ For each prime periodic point $x\in{\rm Per}_{n}(\tilde{f})$ of $f$, the total number of cuspidal windings of the corresponding closed geodesic is $-n+\sum_{i=1}^{n}|B_{i}(x)|$.

Proposition 1.5**.**

Assume $f$ is topologically mixing. As $n\to\infty$, the measure

[TABLE]

converges in the weak-topology to the Borel probability measure on $\Lambda_{c}(G)$ corresponding to the equilibrium state for the potential $-|B_{1}\circ\pi|$.*

1.6. LDPs for countable Markov shifts

We now give a precise formulation of the LDP in our context. Let $X$ be topological Markov shift and $\phi\colon X\to\mathbb{R}$ a measurable function such that $\sup\phi<\infty$ and $P(\phi)<\infty$. The (minus of the) free energy $F\colon\mathcal{M}\to[-\infty,0]$ is defined by

[TABLE]

In the case $X$ is a topologically mixing finite Markov shift and $\phi$ is a Hölder continuous potential, the LDP for empirical means was established in [17, 27], and for weighted periodic points in [18]. In these settings the rate function is $-F$, and therefore the minimizer coincides with the Gibbs-equilibrium state for the potential $\phi$. The same characterization of the rate function is no longer true for countable Markov shifts, as we explain below.

Let $X$ be a finitely irreducible countable Markov shift and $\phi\colon X\to\mathbb{R}$ a summable function with summable variations. Several level-2 LDPs have been established in [28]. Since the entropy is not upper semi-continuous [15, p.774], $-F$ cannot be the rate function. The rate function $I\colon\mathcal{M}\to[0,\infty]$ in [28] is given by

[TABLE]

where the infimum is taken over all open subsets $\mathcal{G}$ of $\mathcal{M}$ containing $\mu$.

Let $\mu_{\phi}$ denote the shift-invariant Gibbs state for the potential $\phi$. We introduce two sequences $\{\xi_{n}\}_{n=1}^{\infty}$, $\{\eta_{n}\}_{n=1}^{\infty}$ of Borel probability measures on $\mathcal{M}$:

• 1. (Empirical means) for each $n\geq 1$ and $x\in X$ let $\delta_{x}^{n}$ denote the uniform probability distribution on $\{x,\sigma x,\ldots,\sigma^{n-1}x\}$. Then $\xi_{n}$ is the distribution of the random variable $x\mapsto\delta_{x}^{n}$ on the probability space $(X,\mu_{\phi})$;
• 2. (Weighted periodic points) for each integer $n\geq 1$ define

[TABLE]

where $\delta_{\delta_{x}^{n}}$ denotes the unit point mass at $\delta_{x}^{n}$.

Theorem 1.6**.**

([28, Theorem A])* Let $X$ be a finitely irreducible countable Markov shift and $\phi\colon X\to\mathbb{R}$ a summable function with summable variations. Then $\{\xi_{n}\}_{n=1}^{\infty}$ satisfies the LDP with the good rate function $I$. If $X$ is finitely primitive, then $\{\eta_{n}\}_{n=1}^{\infty}$ satisfies the LDP with the same good rate function.*

From Theorem 1.1 and Theorem 1.6 we obtain an exponential decay.

Corollary 1.7**.**

Let $X$, $\phi$ be as in Theorem 1.1. Then, for any weak-open set $\mathcal{G}$ containing $\mu_{\phi}$, one has*

[TABLE]

If moreover $X$ is finitely primitive, then

[TABLE]

2. Proofs of the main results

A proof of Theorem 1.1 amounts to showing the next theorem which identifies minimizers as equilibrium states.

Theorem 2.1**.**

Let $X$ be a finitely irreducible countable Markov shift and $\phi\colon X\to\mathbb{R}$ a uniformly continuous summable function satisfying $\beta_{\infty}(\phi)<1$. Assume there exists a Gibbs state for the potential $\phi$. Then, any minimizer of the rate function is an equilibrium state for the potential $\phi$.

We finish the proof of Theorem 1.1 and Theorem 1.3 assuming Theorem 2.1.

Proof of Theorem 1.1

Let $X$ be finitely irreducible and $\phi\colon X\to\mathbb{R}$ be summable with summable variations satisfying $\beta_{\infty}(\phi)<1$. By [20, Theorem 1.5] the variational principle holds, and by [20, Theorem 2.2.4] there exists a unique shift-invariant Gibbs state for the potential $\phi$ which we denote by $\mu_{\phi}$.

Fix $\delta\in(0,1-\beta_{\infty}(\phi))$. Since $\phi$ has summable variations, there exists a constant $C(\phi,\delta)>0$ such that $\sup_{[k]}|\phi|\leq C(\phi,\delta)\sup_{[k]}\exp(-\delta\phi)$ for every $k\in\mathbb{N}$. Using this and (1.3) we obtain

[TABLE]

and thus $\int\phi d\mu_{\phi}>-\infty$. By [20, Theorem 2.2.9], $\mu_{\phi}$ is the unique equilibrium state for the potential $\phi$. From this and Theorem 2.1, the minimizer is unique and it is $\mu_{\phi}$. ∎

Proof of Theorem 1.3

Let $X$ be finitely primitive and $\phi\colon X\to\mathbb{R}$ be summable with summable variations satisfying $\beta_{\infty}(\phi)<1$. Let $p_{n}$ denote the Borel probability measure on $X$ in the statement of Theorem 1.3. For an arbitrary bounded continuous function $\varphi\colon X\to\mathbb{R}$, a direct calculation gives

[TABLE]

By Theorem 1.6, the sequences $\{\eta_{n}\}_{n=1}^{\infty}$ satisfy the LDP with the good rate function $I$. Since the rate function is good and the minimizer is unique, $\{\eta_{n}\}_{n=1}^{\infty}$ converges in the weak*-topology to the unit point mass at the unique minimizer $\mu_{\phi}$. Since the functional $\mu\in\mathcal{M}\mapsto\int\varphi d\mu$ is bounded continuous, $\lim_{n\to\infty}\int\left(\int\varphi d\mu\right)d\eta_{n}(\mu)=\int\varphi d\mu_{\phi}$ holds, and thus $\lim_{n\to\infty}\int\varphi dp_{n}=\int\varphi d\mu_{\phi}$ as required. ∎

2.1. Preliminaries

The rest of this paper is dedicated to a proof of Theorem 2.1. If not stated otherwise, $X$ denotes a general countable Markov shift.

Lemma 2.2**.**

Let $\phi\colon X\to\mathbb{R}$ be a summable function. For any $\delta>0$ there exists a constant $K(\delta)\in\mathbb{R}$ such that if $\mu\in\mathcal{M}_{\phi}(\sigma|_{X})$ satisfies $-h(\mu)/\int\phi d\mu>\beta_{\infty}(\phi)+\delta$ then $\int\phi d\mu\geq K(\delta)$.

Proof.

For $\delta>0$ put $\beta_{0}=\beta_{\infty}(\phi)+\delta/2$. Then

[TABLE]

The second inequality follows from [20, Theorem 2.1.7], and the last one follows from the summability gives of $\beta_{0}\phi$. Put $K(\delta)=\frac{P(\beta_{0}\phi)}{\beta_{0}-\beta_{\infty}(\phi)-\delta}$. The first inequality and the assumption on $\mu$ give

[TABLE]

and therefore $\int\phi d\mu\geq K(\delta)$ as required. ∎

Lemma 2.3**.**

*Let $\phi\colon X\to\mathbb{R}$ be a summable function satisfying $\beta_{\infty}(\phi)<1.$ Let $\{\mu_{k}\}_{k=1}^{\infty}$ be a sequence in $\mathcal{M}_{\phi}(\sigma|_{X})$ such that $\{F(\mu_{k})\}_{k=1}^{\infty}$ converges to a finite number as $k\to\infty$. Then $\inf_{k\geq 1}\int\phi d\mu_{k}>-\infty.$ *

Proof.

If the infimum is $-\infty$, then it is possible to take a subsequence $\{\mu_{k_{i}}\}_{i=1}^{\infty}$ of $\{\mu_{k}\}_{k=1}^{\infty}$ such that $\int\phi d\mu_{k_{i}}\to-\infty$ and $-h(\mu_{k_{i}})/\int\phi d\mu_{k_{i}}\to 1$ as $i\to\infty$. Fix $\delta\in(0,1-\beta_{\infty}(\phi))$. Since $-h(\mu_{k_{i}})/\int\phi d\mu_{k_{i}}>\beta_{\infty}(\phi)+\delta$ holds for sufficiently large $i$, we obtain a contradiction to Lemma 2.2. ∎

To prove Theorem 2.1, we take a sequence of measures converging to a minimizer, and evaluate the free energies along this sequence. The main difficulty is the lack of upper semi-continuity of entropy. To overcome this we show a limited form of upper semi-continuity in the following form.

Theorem 2.4**.**

Let $\phi\colon X\to\mathbb{R}$ be a uniformly continuous summable function satisfying $\beta_{\infty}(\phi)<1$. Let $\{\mu_{j}\}_{j=1}^{\infty}$ be a sequence in $\mathcal{M}_{\phi}(\sigma|_{X})$ which converges to $\mu_{0}\in\mathcal{M}_{\phi}(\sigma|_{X})$ in the weak-topology as $j\to\infty$. Assume $P(\phi)-\int\phi d\mu_{0}>0$. If*

[TABLE]

then

[TABLE]

Remark 2.5*.*

As continuous functions on $X$ bounded from above are integrated upper semi-continuously, the assumption $P(\phi)-\int\phi d\mu_{0}>0$ in Theorem 2.4 implies $P(\phi)-\int\phi d\mu_{j}>0$ for sufficiently large $j$.

A main inspiration for Theorem 2.4 is from [9, Lemma 6.5], which is used for refined descriptions of multifractal spectra of fully branched Markov interval maps. Below we finish the proof of Theorem 2.1 assuming Theorem 2.4.

Proof of Theorem 2.1

Let $\mu_{*}\in\mathcal{M}(\sigma|_{X})$ be a minimizer. Let $\{\mu_{k}\}_{k=1}^{\infty}$ be a sequence in $\mathcal{M}_{\phi}(\sigma|_{X})$ which converges in the weak*-topology to $\mu_{*}$ with $\lim_{k\to\infty}F(\mu_{k})=0$. Lemma 2.3 gives $\inf_{k\geq 1}\int\phi d\mu_{k}>-\infty$; As $\phi$ is continuous bounded from above, it is integrated upper semi-continuously [15, Lemma 1]. In particular, $\int\phi d\mu_{*}>-\infty$ holds. If $\varliminf_{k\to\infty}h(\mu_{k})=0$, then for a subsequence $\{\mu_{k_{j}}\}_{j=1}^{\infty}$ with $\lim_{j\to\infty}h(\mu_{k_{j}})=0$ the formula (1.4) gives

[TABLE]

namely $\mu_{*}$ is an equilibrium state for the potential $\phi$. If $\varliminf_{k\to\infty}h(\mu_{k})>0$, then $\varliminf_{k\to\infty}(-\int\phi d\mu_{k})>0$ and

[TABLE]

It follows that

[TABLE]

We have $-\int\phi d\mu_{*}\geq h(\mu_{*})\geq 0$. If $-\int\phi d\mu_{*}=0$, then clearly $\mu_{*}$ is an equilibrium state for the potential $\phi$. If $-\int\phi d\mu_{*}>0$, then Theorem 2.4 gives

[TABLE]

namely $\mu_{*}$ is an equilibrium state for the potential $\phi$. ∎

Proof of Theorem 2.4

We shall view measures in $\mathcal{M}(\sigma|_{X})$ as measures on the full shift $\mathbb{N}^{\mathbb{N}}$, and project them to the canonical finite subsystems $\Sigma_{p}$ $(p\in\mathbb{N})$, show for each $p\in\mathbb{N}$ a $p$-th approximation of the inequality (2.2), and finally let $p\to\infty$ to obtain (2.2). The condition $\beta_{\infty}(\phi)<1$ will be used to control tails arising in approximations of entropy and integrals of the potential.

Considering $\phi-P(\phi)$ instead of $\phi$, we may assume $P(\phi)=0$. Put $\beta_{\infty}:=\beta_{\infty}(\phi)<1$. Let $\{\mu_{j}\}_{j=1}^{\infty}$ be a sequence in $\mathcal{M}_{\phi}(\sigma|_{X})$ which converges to $\mu_{0}\in\mathcal{M}_{\phi}(\sigma|_{X})$ and assume $-\int\phi d\mu_{0}>0$. Since $\beta_{\infty}\geq 0$ and $\varliminf_{j\to\infty}(-\int\phi d\mu_{j})>0$ as in Remark 2.5, (2.1) implies

[TABLE]

It is convenient to split the rest of the proof into two cases.

Case 1: $\phi$ is constant on each $1$-cylinder of $X$. We view each measure $\mu\in\mathcal{M}(\sigma|_{X})$ as an element of $\mathcal{M}(\sigma)$ by setting $\mu(A)=\mu(A\cap X)$ for any Borel subset $A$ of $\mathbb{N}^{\mathbb{N}}$. This extension preserves entropy. For each $k\in\mathbb{N}$ let $\langle k\rangle$ denote the corresponding $1$-cylinder of $\mathbb{N}^{\mathbb{N}}$. For each $p\in\mathbb{N}$ we consider the shift-invariant subspace

[TABLE]

Define a projection $\pi_{p}\colon\mathbb{N}^{\mathbb{N}}\to\Sigma_{p}$ as follows: for each $x=x_{0}x_{1}x_{2}\cdots\in\mathbb{N}^{\mathbb{N}}$ define $\pi_{p}(x)\in\Sigma_{p}$ by replacing in the sequence $x_{0}x_{1}x_{2}\cdots$ all symbols greater than or equal to $p+1$ by the symbol $p$. For each $\mu\in\mathcal{M}(\sigma)$, write $\mu|_{p}$ for $\mu\circ\pi_{p}^{-1}$. Since $\pi_{p}$ commutes with the shift, $\mu|_{p}$ is a $\sigma|_{\Sigma_{p}}$-invariant measure. Put

[TABLE]

Notice that $\int\phi d\mu>-\infty$ if and only if $K_{p}(\mu)\to 0$ as $p\to\infty$.

Proposition 2.6**.**

If $\phi\colon X\to\mathbb{R}$ is constant on each $1$-cylinder of $X$, then for any $\delta>0$ there exists $p_{0}\geq 0$ such that for every $\mu\in\mathcal{M}_{\phi}(\sigma|_{X})$ and every $p\geq p_{0}$,

[TABLE]

Proof.

Before proceeding let us summarize basic facts on entropy. Let $\mathscr{A}=\{A_{k}\}_{k\in\mathbb{N}}$ be a countable partition of $X$ into Borel sets and let $\mu\in\mathcal{M}(\sigma|_{X})$. The entropy of $\mathscr{A}$ with respect to $\mu$ is the number

[TABLE]

with the convention $0\log 0=0$. If $H_{\mu}(\mathscr{A})<\infty$ then define

[TABLE]

where the symbols $\bigvee$ and $\vee$ denote the join of the partitions $\sigma^{-i}\mathscr{A}$ $(0\leq i\leq n-1)$. Since $n\mapsto H_{\mu}\left(\bigvee_{i=0}^{n-1}\sigma^{-i}\mathscr{A}\right)$ is sub-additive, this limit exists, is finite and $h_{\mu}(\mathscr{A})\leq H_{\mu}(\mathscr{A})$. We have $h(\mu)=\sup_{\mathscr{A}}h_{\mu}(\mathscr{A})$ where the supremum is taken over all countable partitions $\mathscr{A}$ of $X$ with $H_{\mu}(\mathscr{A})<\infty$. If $\mathscr{A}$ is a generator of the Borel sigma-field of $X$ and $H_{\mu}(\mathscr{A})<\infty$ then $h(\mu)=h_{\mu}(\mathscr{A})$.

Now, consider two partitions of $X$:

[TABLE]

It is easy to check that $\mathscr{A}_{p}\vee\mathscr{B}_{p}$ are generators of the Borel sigma-field of $X$.

Lemma 2.7**.**

For every $\mu\in\mathcal{M}(\sigma)$ with finite entropy, $H_{\mu}(\mathscr{B}_{p})<\infty$. Moreover, $h_{\mu}(\mathscr{A}_{p}\vee\mathscr{B}_{p})\leq h_{\mu}(\mathscr{A}_{p})+h_{\mu}(\mathscr{B}_{p}).$

Proof.

The first assertion follows from [29, Lemma 2.1]. For each integer $n\geq 1$ we introduce a finite partition

[TABLE]

Then $h_{\mu}(\mathscr{A}_{p}\vee\mathscr{B}_{p,n})\leq h_{\mu}(\mathscr{A}_{p})+h_{\mu}(\mathscr{B}_{p,n})$ holds. The $\{\mathscr{B}_{p,n}\}_{n=1}^{\infty}$ defines an increasing sequence of sub Borel sigma-fields of $X$ satisfying $\bigvee_{n=1}^{\infty}\mathscr{B}_{p,n}=\mathscr{B}_{p}.$ By [31, Theorem 4.22], $\lim_{n\to\infty}h_{\mu}(\mathscr{B}_{p,n})=h_{\mu}(\mathscr{B}_{p})$, and similarly $\lim_{n\to\infty}h_{\mu}(\mathscr{A}_{p}\vee\mathscr{B}_{p,n})=h_{\mu}(\mathscr{A}_{p}\vee\mathscr{B}_{p})$. ∎

Lemma 2.8**.**

For every $\mu\in\mathcal{M}(\sigma|_{X})$, $h_{\mu}(\mathscr{A}_{p})=h(\mu|_{p}).$

Proof.

Consider finite partitions $\mathscr{C}_{p}=\{\langle 0\rangle\cap\Sigma_{p},\ldots,\langle p\rangle\cap\Sigma_{p}\}$ of $\Sigma_{p}$, and $\pi_{p}^{-1}\mathscr{C}_{p}=\{\pi_{p}^{-1}\langle 0\rangle,\ldots,\pi_{p}^{-1}\langle p\rangle\}$ of $\mathbb{N}^{\mathbb{N}}$. Then $h_{\mu}(\mathscr{A}_{p})=h_{\mu}(\pi_{p}^{-1}\mathscr{C}_{p})$ holds. Since $\pi_{p}$ commutes with the shift, $h_{\mu}(\pi_{p}^{-1}\mathscr{C}_{p})=h_{\mu|_{p}}(\mathscr{C}_{p})$. Since $\mathscr{C}_{p}$ is a generator of the Borel sigma-field of $\Sigma_{p}$, $h_{\mu|_{p}}(\mathscr{C}_{p})=h(\mu|_{p})$ holds. ∎

Returning to the proof of Proposition 2.6, let $\mu\in\mathcal{M}_{\phi}(\sigma|_{X})$. Then $h(\mu)<\infty$ holds. In the case $c_{p}(\mu)=0$ there is nothing to prove since $h(\mu)-h(\mu|_{p})=0$. Hence we assume $c_{p}(\mu)>0$. Lemma 2.7 gives $h(\mu)=h_{\mu}(\mathscr{A}_{p}\vee\mathscr{B}_{p})\leq h_{\mu}(\mathscr{A}_{p})+h_{\mu}(\mathscr{B}_{p}).$ Using this and Lemma 2.8 we have

[TABLE]

To treat the last summand in (2.4), define a potential $\varphi\colon\mathbb{N}^{\mathbb{N}}\to\mathbb{R}$ which is constant on each $1$-cylinder of $\mathbb{N}^{\mathbb{N}}$ by $\varphi|_{\langle k\rangle}=\phi(k)$. The summability of $\phi$ implies that of $\varphi$. Denote by $\nu_{p}$ the Bernoulli measure on $\mathbb{N}^{\mathbb{N}}$ which assigns to each $1$-cylinder $\langle k\rangle$, $k\geq p+1$ the probability $\mu[k]/c_{p}(\mu)$. Notice that

[TABLE]

Since $\int\varphi d\nu_{p}\leq\sup_{k\geq p+1}\phi(k),$ the summability of $\phi$ implies $\int\varphi d\nu_{p}\to-\infty$ as $p\to\infty$. Lemma 2.2 applied to $(\mathbb{N}^{\mathbb{N}},\varphi)$ shows that for any $\delta>0$ there exists $p_{0}\geq 1$ independent of $\mu$ such that for every $p\geq p_{0}$, $-h(\nu_{p})/\int\varphi d\nu_{p}\leq\beta_{\infty}(\varphi)+\delta.$ Since $\phi$ is constant on each $1$-cylinder, $\beta_{\infty}(\varphi)=\beta_{\infty}$ holds, and we obtain

[TABLE]

Plugging (2.5) into (2.6) and then rearranging the result gives

[TABLE]

Plugging this inequality into (2.4) yields the desired one. ∎

In view of (2.1) fix $\delta>0$ and then $\gamma_{0}\in(0,1)$ such that

[TABLE]

Let $\gamma\in(\gamma_{0},1)$. Since $\{\mu_{j}\}_{j=1}^{\infty}$ converges in the weak*-topology to $\mu_{0}$, Portmanteau’s theorem implies $\lim_{p\to\infty}\varlimsup_{j\to\infty}c_{p}(\mu_{j})=0$. From this and (2.3), there exists $p_{0}\geq 0$ such that for each $p\geq p_{0}$ we have

[TABLE]

for sufficiently large $j$. Since $\int\phi d\mu_{j}>-\infty$ we have

[TABLE]

The equation (2.9) for $\mu_{0}$ in the place of $\mu_{j}$ implies $\int\phi d\mu_{0}|_{p}\to\int\phi d\mu_{0}$ as $p\to\infty$. We assume $p$ is large enough so that $-\int\phi d\mu_{0}|_{p}>0$. Since $\phi$ is bounded continuous on $\Sigma_{p}$, the weak*-convergence of $\mu_{j}|_{p}$ to $\mu_{0}|_{p}$ as $j\to\infty$ gives $\int\phi d\mu_{j}|_{p}\to\int\phi d\mu_{0}|_{p}$. In particular, for sufficiently large $j$ we have

[TABLE]

The first inequality is a consequence of Proposition 2.6, the second of (2.8) and (2.9). The last inequality is trivial if $K_{p}(\mu_{j})=0$. Otherwise we appeal to the following: for $a,b,c,d>0$ such that $c>d$ and $a/c\geq b/d$, $(a-b)/(c-d)\geq a/c$ holds. Apply this with $a=\gamma h(\mu_{j})$, $b=(\beta_{\infty}+\delta)K_{p}(\mu_{j})$, $c=-\int\phi d\mu_{j}$, $d=K_{p}(\mu_{j})$. The condition $a/c\geq b/d$ is fulfilled by virtue of (2.7) and $\gamma_{0}<\gamma<1$.

The weak*-convergence of $\mu_{j}|_{p}$ to $\mu|_{p}$ as $j\to\infty$ takes place in the space of shift-invariant measures on $\Sigma_{p}$ where the entropy is upper semi-continuous and $\phi$ is bounded continuous. Hence

[TABLE]

By Proposition 2.6 and (2.9) for $\mu_{0}$ in the place of $\mu_{j}$, we have

[TABLE]

Therefore, letting $p\to\infty$ and then $\gamma\to 1$ in (2.10) yields the desired inequality.

Case 2: $\phi$ is not constant on some $1$-cylinder of $X$. For each $q\geq 1$, let $(E^{q})^{\mathbb{N}}$ denote the full shift with symbols in $E^{q}$ and $\sigma_{q}$ the left shift on $(E^{q})^{\mathbb{N}}$. The map

[TABLE]

is a homeomorphism onto its image and satisfies $\theta\circ\sigma_{q}=\sigma^{q}\circ\theta$. Put

[TABLE]

The restriction of $\theta$ to $X_{q}$ is also denoted by $\theta$. The $1$-cylinder in $X_{q}$ corresponding to the symbol $\omega\in E^{q}$ is denoted by $(\omega)$. Define $\Phi\colon X_{q}\to\mathbb{R}$ by

[TABLE]

Note that $\Phi$ is constant on each $1$-cylinder of $X_{q}$.

Lemma 2.9**.**

For each $\mu\in\mathcal{M}(\sigma|_{X})$ the following holds:

• (a)

if $\int\phi d\mu>-\infty$ then $\left|q\int\phi d\mu-\int\Phi d(\mu\circ\theta)\right|\leq D_{q}(\phi);$

• (b)

if $\int\Phi d(\mu\circ\theta)>-\infty$ then $\int\phi d\mu>-\infty$.

Proof.

Since $q\int\phi\circ\theta d(\mu\circ\theta)=\int(S_{q}\phi)\circ\theta d(\mu\circ\theta)$ we have

[TABLE]

as required in Lemma 2.9(a). Lemma 2.9(b) follows from approximating $\phi$ with $\mu$-integrable functions, using Lemma 2.9(a) and then the Monotone Convergence Theorem. ∎

The uniform continuity of $\phi$ implies

[TABLE]

see [10, Proposition 6.2(b)]. From (2.11) and Lemma 2.9(a), for any $\epsilon\in(0,-\int\phi d\mu_{0})$ there exists $q\geq 1$ such that $|\int\Phi d(\mu\circ\theta)-q\int\phi d\mu|\leq q\epsilon/2$ holds for every $\mu\in\mathcal{M}_{\phi}(\sigma|_{X})$. Moreover, $h(\mu\circ\theta)=qh(\mu)$ holds. By Lemma 2.9, measures in the definition of $P(\Phi)$ and those in the definition of $P(\phi)$ are in one-to-one correspondence. It follows that $|P(\Phi)|\leq q\epsilon/2$ and

[TABLE]

From (2.11) we have $\log Z_{1}(\beta\Phi)=o\left(1/q\right)+\log Z_{q}(\beta\phi)$ for every $\beta>0$, which implies $\beta_{\infty}(\Phi)\leq\beta_{\infty}<1.$ Hence we obtain

[TABLE]

The second inequality follows from the result in Case 1 applied to $(X_{q},\Phi)$. Letting $\epsilon\to 0$ yields (2.2). This completes the proof of Theorem 2.4. ∎

Acknowledgments

This research was partially supported by the JSPS KAKENHI 19K21835, 20H01811. I thank Johannes Jaerisch for fruitful discussions.