Lyapunov optimization for non-generic one-dimensional expanding Markov maps

TL;DR

This paper investigates Lyapunov optimizing measures for certain non-generic expanding Markov maps, revealing the existence of both uncountably many ergodic measures with positive entropy and unique periodic orbit measures, using novel perturbation techniques.

Contribution

It introduces a new $C^1$ perturbation lemma enabling interpolation between expanding Markov maps and shift maps, advancing understanding of Lyapunov optimization in non-generic settings.

Findings

Existence of uncountably many Lyapunov optimizing measures with positive entropy.

Existence of unique periodic orbit optimizing measures in certain non-generic cases.

Development of a new perturbation lemma for $C^1$ expanding Markov maps.

Abstract

For a non-generic, yet dense subset of $C^1$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some H\"older continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^1$ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

Equations

$$\inf \left\{\int \varphi d\mu :\mu \in \mathcal{M}(T)\right\},$$

$$0={\alpha }_0<{\beta }_0<{\alpha }_1<{\beta }_1<{\alpha }_2<\cdots <{\beta }_{p-1}=1.$$

$$\Vert f{\Vert }_{C^1}=\underset{x\in X}{\sup }|f(x)|+\underset{x\in X}{\sup }|Df(x)|.$$

$$\Lambda (f)=\bigcap _{n\ge 0}{f}^{-n}\left(X\right).$$

$${\chi }_{\inf }(f)=\inf \left\{\int \log |Df|d\mu :\mu \in \mathcal{M}(f)\right\}$$

$${\chi }_{\sup }(f)=\sup \left\{\int \log |Df|d\mu :\mu \in \mathcal{M}(f)\right\}.$$

$$\int \log |Df|d\mu ={\chi }_{\inf }.$$

$$\Vert f_0-f_{\infty }{\Vert }_{C^1}\le \epsilon \text{and}\log |D{f}_{\infty }|\circ {\pi }_{f_{\infty }}=\varphi .$$

$$\Vert f{\Vert }_{C^{1+\mathrm{Lip}}}=\Vert f{\Vert }_{C^1}+\underset{\stackrel{x,y\in X}{x\ne y}}{\sup }\frac{|Df(x)-Df(y)|}{|x-y|}.$$

$$\mathcal{P}(t)=\sup \left\{h_{\mu }(f)-t\int \log |Df|d\mu :\mu \in \mathcal{M}(f)\right\},$$

$$[a_0\cdots a_{n-1}]=\{\{b_k{\}}_{k\ge 0}\in {\Sigma }_p:a_k=b_k\text{ for every k∈{0,1,…,n−1}}\}.$$

$$E_{a_0\cdots a_{n-1}}(\varphi )=\exp \left(\min \{\varphi (\underline{a}):\underline{a}\in [a_0\cdots a_{n-1}]\}\right),$$

$$F_{a_0\cdots a_{n-1}}(\varphi )=\exp \left(\max \{\varphi (\underline{a}):\underline{a}\in [a_0\cdots a_{n-1}]\}\right)$$

$$V_{a_0\cdots a_{n-1}}(\varphi )=F_{a_0\cdots a_{n-1}}(\varphi )-E_{a_0\cdots a_{n-1}}(\varphi ).$$

$$M(\varphi )=\underset{\underline{a}\in {\Sigma }_p}{\sup }|\varphi (\underline{a})|.$$

$$|E_{a_0\cdots a_{n-1}}(\varphi )-E_{a_0\cdots a_{n-1}}({\varphi }^{\prime })|\le V_{a_0\cdots a_{n-1}}(\varphi )+M(e^{\varphi }-e^{{\varphi }^{\prime }});$$

$$|F_{a_0\cdots a_{n-1}}(\varphi )-F_{a_0\cdots a_{n-1}}({\varphi }^{\prime })|\le V_{a_0\cdots a_{n-1}}(\varphi )+M(e^{\varphi }-e^{{\varphi }^{\prime }}).$$

$$|E_{a_0\cdots a_{n-1}}(\varphi )-E_{a_0\cdots a_{n-1}}({\varphi }^{\prime })|$$

$$\le V_{a_0\cdots a_{n-1}}(\varphi )+M(e^{\varphi }-e^{{\varphi }^{\prime }}).$$

$$c_0=\frac{1}{2}\underset{\stackrel{y\in {\bigcap }_{k=0}^{l-1}{f}_0^{-k}(X)}{1\le l\le N_0-1}}{\min }|D{f}_0^l(y)|\text{ and }{\lambda }_0=2^{\frac{1}{N_0}}.$$

$$X_{a_0\cdots a_{n-1}}(f)=\{x\in X:f^k(x)\in [{\alpha }_{a_k},{\beta }_{a_k}]\text{ for every }k\in \{0,1,\dots ,n-1\}\}.$$

$$X_{a_0\cdots a_{n-1}{\square }_i}(f)=\{x\in X_{a_0\cdots a_{n-1}}(f):f^n(x)\in ({\beta }_{i-1},{\alpha }_i)\}$$

$$X_{a_0\cdots a_{n-1}\hat{i}}(f)=\{x\in X_{a_0\cdots a_{n-1}}(f):f^n(x)\in [{\alpha }_{i-1},{\beta }_i]\}.$$

$$\left|X_{a_0\cdots a_{n-1}\hat{i}}(f)\right|=\left|X_{a_0\cdots a_{n-1}(i-1)}(f)\right|+\left|X_{a_0\cdots a_{n-1}{\square }_i}(f)\right|+\left|X_{a_0\cdots a_{n-1}i}(f)\right|.$$

$${\Delta }_{a_0\cdots a_{n-1}}^{(i)}(f)=\frac{\left|X_{a_0\cdots a_{n-1}\hat{i}}(f)\right|}{\left|X_{a_0\cdots a_{n-1}{\square }_i}(f)\right|}-1.$$

$${\Delta }_{a_0\cdots a_{n-1}}^{(i)}(f_0)\le K(f_0).$$

$$\frac{|D{f}_0^n(x)|}{|D{f}_0^n(y)|}\le M_0.$$

$$\frac{\left|X_{a_0\cdots a_{n-1}{\square }_i}(f_0)\right|}{|X_{a_0\cdots a_{n-1}}(f_0)|}\ge M_0^{-1}\frac{\left|f_0^n(X_{a_0\cdots a_{n-1}{\square }_i}(f_0))\right|}{\left|f_0^n(X_{a_0\cdots a_{n-1}}(f_0))\right|}=M_0^{-1}({\alpha }_i-{\beta }_{i-1}),$$

$$\frac{|X_{a_0\cdots a_{n-1}(i-1)}(f_0)|+|X_{a_0\cdots a_{n-1}i}(f_0)|}{|X_{a_0\cdots a_{n-1}}(f_0)|}$$

$$\le 1-M_0^{-1}({\alpha }_i-{\beta }_{i-1}).$$

$$1⃝=X_{a_0\cdots a_{n-1}0}(h_n);2⃝=X_{a_0\cdots a_{n-1}{\square }_1}(h_n);3⃝=X_{a_0\cdots a_{n-1}1}(h_n);$$

Full text

Keywords: expanding Markov map; Lyapunov optimizing measure; non-generic property

M. S. was partially supported by the Grant-in-Aid for JSPS Research Fellows. H. T. was partially supported by the Grant-in-Aid for Young Scientists (A) of the JSPS 15H05435 and the Grant-in-Aid for Scientific Research (B) of the JSPS 16KT0021. Both authors were partially supported by the JSPS Core-to-Core Program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.

1. Introduction

Ergodic optimization aims to describe invariant probability measures of a dynamical system which optimize the integral of a given “performance” function. In its basic form, given a continuous self-map $T$ of a compact metric space $M$ and a real-valued continuous function $\phi$ on $M$ one seeks $\phi$-minimizing measures, i.e., measures in $\mathcal{M}(T)$ which attain the infimum

[TABLE]

where $\mathcal{M}(T)$ denotes the set of $T$-invariant Borel probability measures. A root of this theory is found in the works of Mather [20] and Mañé [18, 19] on the dynamics of the Euler-Lagrange flow: orbits with prescribed properties can be obtained by considering ergodic invariant probability measures which minimize the integral of the Lagrangian, and the orbits are obtained as typical points for these measures (called action minimizing measures, see Sorrentino [28]).

A general belief is that the minimizing measure is unique and supported on a periodic orbit, for “most” performance functions. The meaning of “most” is the genericity in the sense of Baire’s Category Theorem, the typicality in the sense of the Lebesgue measure in parameter space, and so on. For instance, see Mañé’s conjecture [19], and that of Yuan and Hunt [30] after numerical observations by Hunt and Ott [12, 13]. For the doubling map $x\mapsto 2x$ (mod $1$) and a parametrized performance function $\cos(2\pi(x-\theta))$ $(\theta\in\mathbb{R})$, Bousch [2] proved that for all $\theta$ the minimizing measure is unique, and is supported on a subset of a semi-circle. Further, he proved that for Lebesgue almost every $\theta$ the corresponding minimizing measure is supported on a periodic orbit. Contreras et al [11] studied $C^{1}$ expanding maps of the circle and performance functions in the Banach space $C^{0,\alpha}$ of $\alpha$-Hölder continuous functions. They considered its subspace consisting of those $\phi\in C^{0,\alpha}$ for which $|\phi(x)-\phi(y)|=o(|x-y|^{\alpha})$, and showed that for an open dense subset of this subspace the minimizing measure is unique and supported on a periodic orbit. For transitive and “expanding” dynamical systems, the uniqueness of minimizing measure was established for open dense subsets of suitable separable Banach spaces, see Bousch [3] and Contreras [9]. Quas and Siefken [25] proved the uniqueness of minimizing measure for the one-sided full shift and an open dense subset of a certain non-separable Banach space. In all these three results, the unique minimizing measure is supported on a periodic orbit. For generic functions in the space $C^{0}$ of continuous functions the uniqueness of minimizing measure still holds, but the measure is fully supported, see Bousch and Jenkinson [4], Jenkinson [15]. Brémont [7] proved that for generic $C^{0}$ functions, any minimizing measure has zero entropy.

In these developments, the non-uniqueness of minimizing measure was considered somewhat irrelevant, as it is considered to be a non-generic property. However, the non-uniqueness of action-minimizing measure is a universal phenomenon (the action minimizing measure is, in a sense, a generalization of the KAM tori [28]). In the context of the thermodynamic formalism, the study of the non-uniqueness of minimizing measure has a connection with the asymptotic behavior of Gibbs measures as the temperature drops to zero, see Baraviera, Leplaideur and Lopes [1]. Despite its importance, the non-uniqueness of minimizing measure has not yet received an adequate deal of attention.

In this paper we treat expanding Markov maps of the interval, and show that the uniqueness of Lyapunov optimizing measure fails in a severe way. Let $p\geq 2$ be an integer and $\{\alpha_{i}\}_{i=0}^{p-1}$, $\{\beta_{i}\}_{i=0}^{p-1}$ sequences in $[0,1]$ such that

[TABLE]

Put $X=\bigcup_{i=0}^{p-1}[\alpha_{i},\beta_{i}]$. Denote by $\mathscr{E}$ the set of $C^{1}$ maps on $X$ such that the following holds for every $f\in\mathscr{E}$:

• (E1)

$f$ maps each interval $[\alpha_{i},\beta_{i}]$, $i\in\{0,\ldots,p-1\}$ diffeomorphically onto $[0,1]$;

• (E2)

there exist constants $c>0$, $\lambda>1$ such that for every $n\geq 1$ and every $x\in\bigcap_{k=0}^{n-1}f^{-k}\left(X\right)$, $|Df^{n}(x)|\geq c\lambda^{n}$.

We endow $\mathscr{E}$ with the $C^{1}$ topology given by the norm

[TABLE]

The space $\mathscr{E}$ is an open subset of $C^{1}$ functions on $X$ (See Lemma 2.2 to check the openness of (E2)), and hence becomes a (non-complete) Baire space.

Define

[TABLE]

Restricting $f$ to $\Lambda(f)$ we obtain a dynamical system which is also denoted by $f$ with a slight abuse of notation. Then $\Lambda(f)$ is a Cantor set with a Markov partition given by the collection $\{[\alpha_{i},\beta_{i}]\}_{i=0}^{p-1}$ of intervals which topologically conjugates $f$ to the left shift on $p$ symbols.

Put

[TABLE]

and

[TABLE]

Since $\mathcal{M}(f)$ is compact and $\nu\in\mathcal{M}(f)\mapsto\int\log|Df|d\nu$ is continuous, the infimum and supremum are attained. A measure $\mu\in\mathcal{M}(f)$ is Lyapunov minimizing if

[TABLE]

Lyapunov maximizing measures is defined similarly, with $\chi_{\rm inf}$ replaced by $\chi_{\rm sup}$.

The notion of Lyapunov optimizing measures was introduced by Contreras et al [11]. They showed that for an open dense subset of the space $\bigcup_{\beta>\alpha}C^{1+\beta}$ of expanding maps of the circle in the $C^{1+\alpha}$ topology, the Lyapunov minimizing measure is unique and supported on a periodic orbit. For a generic $C^{1}$ expanding map of the circle, Jenkinson and Morris [16] proved that the Lyapunov minimizing measure is unique and has zero entropy. See Morita and Tokunaga [21], Tokunaga [29] for extensions of the results of [16] to higher dimension. With the method of [16] one can show that for generic maps in $\mathscr{E}$ the Lyapunov minimizing measure is unique, and it is fully supported, has zero entropy. In the realm of non-genericity the structure of Lyapunov minimizing measures is in contrast.

Theorem A**.**

There exists a dense subset $\mathscr{A}$ of $\mathscr{E}$ such that the following holds for every $f\in\mathscr{A}$:

$\chi_{\rm inf}(f)\neq\chi_{\rm sup}(f)$;

• -

*there exist uncountably many Lyapunov minimizing measures of $f$ which are ergodic, fully supported and have positive entropy; *

• -

$\log|Df|$* is not Hölder continuous.*

Theorem A has been inspired by the following result of the first-named author [27] on the ergodic optimization for the subshift of finite type. For an integer $p\geq 2$ let $\Sigma_{p}=\{0,1,\ldots,p-1\}^{\mathbb{Z}_{\geq 0}}$ denote the one-sided shift space on $p$ symbols, endowed with the product topology of the discrete topology on $\{0,1,\ldots,p-1\}$. Let $\sigma\colon\Sigma_{p}\circlearrowleft$ denote the left shift: if $\underline{a}$, $\underline{b}\in\Sigma_{p}$, $\underline{a}=\{a_{n}\}_{n\geq 0}$, $\underline{b}=\{b_{n}\}_{n\geq 0}$ and $\sigma(\underline{a})=\underline{b}$ then $b_{n}=a_{n+1}$ holds for every $n\geq 0$. For a $p\times p$ matrix $A=(A_{ij})$ whose each entry is [math] or $1$, define $\Sigma_{A}=\{\{a_{n}\}_{n\geq 0}\in\Sigma_{p}\colon A_{a_{n}a_{n+1}}=1\ \forall n\geq 0\}$. The subshift of finite type is the dynamical system $\sigma_{A}\colon\Sigma_{A}\circlearrowleft$ given by $\sigma_{A}=\sigma|_{\Sigma_{A}}$. We say $\sigma_{A}$ is topologically mixing if there exists an integer $M>0$ such that all the entries of $A^{M}$ are positive. Let $C(\Sigma_{A})$ denote the space of real-valued continuous functions on $\Sigma_{A}$ endowed with the supremum norm.

Theorem 1**.**

([27, Theorem A])* Let $\sigma_{A}\colon\Sigma_{A}\circlearrowleft$ be a topologically mixing subshift of finite type. There exists a dense subset $\mathscr{C}_{\sigma_{A}}$ of $C(\Sigma_{A})$ such that for every $\phi\in\mathscr{C}_{\sigma_{A}}$ there exist uncountably many $\phi$-minimizing measures which are ergodic, fully supported and have positive entropy. *

The argument in [27] is to view elements of $\mathcal{M}(\sigma_{A})$ as duals of $C(\Sigma_{A})$, and use the Bishop-Phelps Theorem [14, Theorem V.1.1] for a continuous path of ergodic Markovian measures. Our strategy for proving Theorem A is to use the Markov partition formed by the intervals $\{[\alpha_{i},\beta_{i}]\}_{i=0}^{p-1}$ to translate the Lyapunov optimization to the ergodic optimization for $\sigma$, and then appeal to Theorem Theorem 1. As a result, for every $f\in\mathscr{A}$ the uncountably many minimizing measures in the statement lie on a continuous path of ergodic measures which are equilibrium states of $f$ for some Hölder continuous potentials.

We have suppressed small generalizations of Theorem A for the brevity of presentation. An extension is possible to the case where $f$ is topologically conjugate the topologically mixing subshift of finite type. Moreover, we can drop the conditions $0=\alpha_{0}$ and $\beta_{p-1}=1$. Statements analogous to Theorem A hold for Lyapunov maximizing measures. Since both proofs are identical, we restrict ourselves to Lyapunov minimizing ones.

For $f\in\mathscr{E}$ define a coding map $\pi_{f}\colon\Sigma_{p}\to\Lambda(f)$ by $\pi_{f}(\underline{a})\in\bigcap_{n\geq 0}f^{-n}([\alpha_{a_{n}},\beta_{a_{n}}])$ where $\underline{a}=\{a_{n}\}_{n\geq 0}$. Then $\pi_{f}$ is a homeomorphism satisfying $\pi_{f}\circ\sigma=f\circ\pi_{f}$, and for each fixed $\phi\in C(\Sigma_{p})$ there is a one-to-one correspondence between $\phi$-minimizing measures in $\mathcal{M}(\sigma)$ and $\phi\circ\pi_{f}^{-1}$-minimizing ones in $\mathcal{M}(f)$. Theorem A is proved by combining Theorem 1 and the next lemma which allows one to realize a perturbation in $C(\Sigma_{p})$ as a perturbation in $\mathscr{E}$.

Lemma (the Realization Lemma)****.

Let $f_{0}\in\mathscr{E}$ be of class $C^{2}$. For every $\varepsilon>0$ there exists a neighborhood $\mathscr{U}$ of $\log|Df_{0}|\circ\pi_{f_{0}}$ in $C(\Sigma_{p})$ such that for every $\phi\in\mathscr{U}$ there exists $f_{\infty}\in\mathscr{E}$ such that

[TABLE]

We finish the proof of Theorem A assuming the Realization Lemma.

Proof of Theorem A.

For the shift map $\sigma_{p}\colon\Sigma_{p}\circlearrowleft$ consider the dense subset $\mathscr{C}_{\sigma_{p}}$ of $C(\Sigma_{p})$ in Theorem 1. Since $C^{2}$ maps are dense in $\mathscr{E}$, the Realization Lemma implies that the set $\widehat{\mathscr{A}}=\{f\in\mathscr{E}\colon\log|Df|\circ\pi_{f}\in\mathscr{C}_{\sigma_{p}}\}$ is dense in $\mathscr{E}$. From Theorem 1, if $f\in\widehat{\mathscr{A}}$ then there exist uncountable many Lyapunov minimizing measures which are ergodic, fully supported and have positive entropy.

Let $\widehat{\mathscr{E}}$ denote the set of elements of $\mathscr{E}$ for which there exist two periodic measures with different Lyapunov exponents. Clearly, if $f\in\widehat{\mathscr{E}}$ then $\chi_{\rm inf}(f)\neq\chi_{\rm sup}(f)$. Set $\mathscr{A}=\widehat{\mathscr{A}}\cap\widehat{\mathscr{E}}$. The set $\mathscr{A}$ satisfies the desired properties. Indeed, since $\widehat{\mathscr{E}}$ is an open dense subset of $\mathscr{E}$ and $\widehat{\mathscr{A}}$ is a dense subset of $\mathscr{E}$, $\mathscr{A}$ is a dense subset of $\mathscr{E}$. Let $f\in\mathscr{A}$ and suppose $\log|Df|$ is Hölder continuous. From the so-called Mañé-Conze-Guivarc’h lemma (See [23] and the references therein) there exists a continuous function $u$ on $\Lambda(f)$ such that $\log|Df|=u-u\circ f+\chi_{\rm inf}$. Hence $\chi_{\rm inf}(f)=\chi_{\rm sup}(f)$ holds and $f\notin\widehat{\mathscr{E}}$, a contradiction. ∎

For a proof of the Realization Lemma, we construct a Cauchy sequence $\{f_{n}\}_{n\geq 0}$ in $\mathscr{E}$ and obtain $f_{\infty}$ as a limit of this sequence. This construction has two main steps which are carried out in Sect.3. First, we construct by induction a sequence of continuous piecewise linear maps $h_{n}\colon X\to[0,1]$ $(n=0,1,\ldots)$. Then we perturb each $h_{n}$ and obtain the desired sequence $\{f_{n}\}_{n\geq 0}$. Although the constructions of $\{h_{n}\}_{n\geq 0}$ and $\{f_{n}\}_{n\geq 0}$ are rather intuitive, the difficulty is to ensure that the induction does not halt on the way. We do this by showing that the sizes of gaps at each step of induction have a definite proportion (See Lemma 3.1). This is the reason for the assumption of $C^{2}$ smoothness in the Realization Lemma. The realization lemma is clearly false for expanding circle maps: there is no space to absorb differences which stem from the perturbation. Indeed, our proof exploits the total disconnectedness of the Cantor set.

The Realization Lemma implies that any property of minimizing measures which holds for a dense subset of functions in $C(\Sigma_{p})$ transmits to a dense subset of $\mathscr{E}$. By the result of Brémont [7, Proposition 2.1], for a dense subset of $C(\Sigma_{p})$ the minimizing measure is unique, and it is supported on a periodic orbit. It follows that for a dense subset of $\mathscr{E}$ the Lyapunov minimizing measure is unique, and it is supported on a periodic orbit. Below we give a stronger statement which in particular indicates that a version of Theorem A does not hold in the $C^{1+{\rm Lip}}$ topology.

Let $\mathscr{E}_{\rm Lip}$ denote the space of maps in $\mathscr{E}$ with Lipschitz continuous derivative endowed with the topology given by the norm

[TABLE]

Theorem B**.**

There exists an open subset $\mathscr{O}$ of $\mathscr{E}_{\rm Lip}$ such that for every $f\in\mathscr{O}$ there exists a unique Lyapunov minimizing measure, and it is supported on a periodic orbit. In addition, $\mathscr{O}$ is a dense subset of $C(\Sigma_{p})$.

The first statement of Theorem B is a consequence of the result of Contreras [9]. A proof of the last statement of Theorem B is briefly outlined as follows. The total disconnectedness of the phase space implies that maps with locally constant derivative are dense in $\mathscr{E}$ (See Lemma 4.1). If $Df$ is a locally constant function on $\Lambda(f)$, then $\log|Df|\circ\pi_{f}$ becomes Lipschitz continuous with respect to the standard distance on $\Sigma_{p}$ of any scale. By the Realization Lemma and the result of Contreras [9], $f$ is approximated by another for which the Lyapunov minimizing measure is unique and supported on a periodic orbit. Choosing a distance of sufficiently small scale relative to the expansion rate of $f$ in (E2) we obtain the Lipschitz continuity of $Df$ (See Sect.4.3).

Theorem B has one important consequence on the zero-temperature limit in the thermodynamic formalism (See e.g. [1] and the references therein). For $f\in\mathscr{E}$ define a geometric pressure function $t\in\mathbb{R}\mapsto\mathscr{P}(t)$ by

[TABLE]

where $h_{\mu}(f)$ denotes the Kolmogorov-Sinaĭ entropy of $(f,\mu)$. An equilibrium state for the potential $-t\log|Df|$ is a measure in $\mathcal{M}(f)$ which attains this supremum. If $\log|Df|$ is Hölder continuous, then $t\in\mathbb{R}\mapsto\mathscr{P}(t)$ is real-analytic, and for every $t\in\mathbb{R}$ there exists a unique equilibrium state for the potential $-t\log|Df|$ [5, 26], which we denote by $\mu_{t}$. Lyapunov minimizing measures are obtained by freezing the system: any accumulation point of $\{\mu_{t}\}_{t\in\mathbb{R}}$ as $t\to+\infty$ is a Lyapunov minimizing measure. By the zero-temperature limit we mean the weak* limit of $\{\mu_{t}\}_{t\in\mathbb{R}}$ as $t\to+\infty$. The uniqueness of the Lyapunov minimizing measure implies the existence of the zero-temperature limit.

Corollary 3**.**

For every $f\in\mathscr{O}$ the zero-temperature limit $\displaystyle{\lim_{t\to+\infty}\mu_{t}}$ exists, and is supported on a periodic orbit.

If $\log|Df|$ is a locally constant function on $\Lambda(f)$, then the zero-temperature limit exists, see Brémont [6] and Leplaideur [17]. Such maps are dense in $\mathscr{E}$ (See Lemma 4.1). The non-existence of zero-temperature limit was treated by Chazottes and Hochman [8], Coronel and Rivera-Letelier [10] in the context of the subshift of finite type.

The rest of this paper consists of three sections. Sect.2 and Sect.3 are entirely dedicated to a proof of the Realization Lemma. In Sect.4 we prove Theorem B.

2. Preliminaries

In this section we develop fundamental estimates needed for the proofs of the main results.

2.1. Control of variations

For an integer $n\geq 1$, by a word of length $n$ we mean an $n$-string of integers in $\{0,1,\ldots,p-1\}$. For each integer $n\geq 1$ let $W_{n}$ denote the set of words of length $n$. For $a_{0}\cdots a_{n-1}\in W_{n}$ define

[TABLE]

For $\phi\in C(\Sigma_{p})$ and $a_{0}\cdots a_{n-1}\in W_{n}$ define

[TABLE]

[TABLE]

and

[TABLE]

Set

[TABLE]

Lemma 2.1**.**

For every $\phi,\phi^{\prime}\in C(\Sigma_{p})$, every $n\geq 1$ and every $a_{0}\cdots a_{n-1}\in W_{n}$ the following holds:

[TABLE]

[TABLE]

Proof.

Let $\underline{a},\underline{a^{\prime}}\in[a_{0}\cdots a_{n-1}]$ be such that $E_{a_{0}\cdots a_{n-1}}(\phi)=e^{\phi(\underline{a})}$ and $E_{a_{0}\cdots a_{n-1}}(\phi^{\prime})=e^{\phi^{\prime}(\underline{a^{\prime}})}$. Then

[TABLE]

A proof of the second inequality is analogous. ∎

2.2. Uniform expansion for nearby maps

The next lemma ensures that $\mathscr{E}$ is an open subset of $C^{1}$ functions on $X$.

Lemma 2.2**.**

Let $f_{0}\in\mathscr{E}$. There exist constants $\varepsilon_{0}>0$, $c_{0}>0$, $\lambda_{0}>1$ such that the following holds: if $f\in\mathscr{E}$ and $\|f_{0}-f\|_{C^{1}}\leq\varepsilon_{0}$, then for every $x\in X$ and every $n\geq 1$ such that $x,f(x),\ldots,f^{n-1}(x)\in X$, $|Df^{n}(x)|\geq c_{0}\lambda_{0}^{n}$.

Proof.

By (E2) there exist constants $c>0$, $\lambda>1$ such that if $x\in X$ and $n\geq 1$ are such that $x,f_{0}(x),\ldots,f_{0}^{n-1}(x)\in X$ then $|Df_{0}^{n}(x)|\geq c\lambda^{n}$. Choose an integer $N_{0}\geq 1$ such that $c\lambda^{N_{0}}\geq 3$. Choose $\varepsilon_{0}>0$ such that $|Df^{N_{0}}(x)|\geq 2$ holds for every $f\in\mathscr{E}$ satisfying $\|f_{0}-f\|_{C^{1}}\leq\varepsilon_{0}$ and $x\in X$ such that $x,f(x),\ldots f^{N_{0}-1}(x)\in X$. For such an $f$, let $x\in X$ and $n\geq 1$ be such that $x,f(x),\ldots,f^{n-1}(x)\in X$. Write $n=N_{0}k+l$ where $k,l$ are nonnegative integers with $0\leq l\leq N_{0}-1$. Put

[TABLE]

Splitting the orbit of $x$ into a concatenation of segments of length $N_{0}$ and then using the Chain Rule gives $|Df^{n}(x)|\geq c_{0}\lambda_{0}^{n}.$ ∎

2.3. Consequence of bounded distortion

Let $a_{0}\cdots a_{n-1}\in W_{n}$ and $f\colon X\to[0,1]$ a continuous map. Define

[TABLE]

For each $i\in\{1,2,\ldots,p-1\}$ define

[TABLE]

and

[TABLE]

Denote by $|I|$ the length of a bounded interval $I$. Note that

[TABLE]

Put

[TABLE]

Lemma 2.3**.**

Let $f_{0}\in\mathscr{E}$ be of class $C^{2}$. There exists a constant $K(f_{0})>0$ such that for every $i\in\{1,2,\ldots,p-1\}$, every $n\geq 1$ and every $a_{0}\cdots a_{n-1}\in W_{n}$,

[TABLE]

Proof.

Since $f_{0}\in\mathscr{E}$ is of class $C^{2}$ the bounded distortion holds: there exists $M_{0}>1$ such that for every $n\geq 1$, every $a_{0}\cdots a_{n-1}\in W_{n}$ and every $x,y\in X_{a_{0}\cdots a_{n-1}}(f_{0})$ we have

[TABLE]

Let $i\in\{1,2,\ldots,p-1\}$. For every $a_{0}\cdots a_{n-1}\in W_{n}$ we have

[TABLE]

and therefore

[TABLE]

Put $K(f_{0})=\max_{i\in\{1,2,\ldots,p-1\}}\frac{1-M_{0}^{-1}(\alpha_{i}-\beta_{i-1})}{M_{0}^{-1}(\alpha_{i}-\beta_{i-1})}.$ These two inequalities yield the desired one. ∎

3. On the proof of the Realization Lemma

In this section we complete the proof of the Realization Lemma. Throughout this section, let $f_{0}\in\mathscr{E}$ be of class $C^{2}$ and put $\phi_{0}=\log|Df_{0}|\circ\pi_{f_{0}}$.

3.1. Construction of a sequence of continuous piecewise linear maps

Let $\phi\in C(\Sigma_{p})$ be sufficiently close to $\phi_{0}$ and $N\geq 2$ an integer. We construct by induction a sequence $\{h_{n}(\phi,N)\}_{n\geq 0}$ of continuous piecewise linear maps on $X$ which maps each connected component of $X$ bijectively onto $[0,1]$. In what follows we will write $h_{n}$ for $h_{n}(\phi,N)$.

Start with $h_{n}=f_{0}$ for every $n\in\{0,1,\ldots,N-1\}$. Let $n\geq N$ and suppose $h_{n-1},h_{n-2}$ has been defined so that the following holds:

• (P$)_{n-1}$:

$h_{n-1}(X_{a_{0}\cdots a_{n-1}}(h_{n-1}))=X_{a_{1}\cdots a_{n-1}}(h_{n-2})$ for every $a_{0}\cdots a_{n-1}\in W_{n}$:

Put

[TABLE]

The open intervals $X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$ in the union are called a gap of of order $k$. The total number of gaps of order $k$ is $p^{k}(p-1)$. Note that

[TABLE]

where all unions are disjoint. Define $h_{n}$ so that the following holds (See FIGURE 1):

• (i)

$h_{n}\equiv h_{n-1}$ on $G_{n-1}$;

For each $a_{0}\cdots a_{n-1}\in W_{n}$, $h_{n}|_{X_{a_{0}\cdots a_{n-1}}(h_{n-1})}$ is defined as follows:

• (ii)

for every $i\in\{0,1,\ldots,p-1\}$,

[TABLE]

and

[TABLE]

where the sign is $+$ if $Df_{0}>0$ on $[\alpha_{a_{0}},\beta_{a_{0}}]$ and $-$ otherwise;

• (iii)

for every $i\in\{1,\ldots,p-1\}$, $Dh_{n}|_{X_{a_{0}\cdots a_{n-1}\Box_{i}}(h_{n})}$ is a constant function.

Since $h_{n}$ is required to be continuous, there is no ambiguity in this definition. Note that (P$)_{n}$ holds, which recovers the assumption of the induction.

There is a difference between the transition from $N-1$ to $N$ and that from $n-1$ to $n$, $n\geq N+1$. Since $h_{N-1}=f_{0}$, an “overhang” may happen, in which case the definition of $h_{N}|_{X_{a_{0}\cdots a_{N-1}}(f_{0})}$ does not make sense. As developed in the proof of Lemma 3.1 below. the overhang does hot happen for an appropriately chosen $\phi$ and $N$.

3.2. Analytic estimates on the sequence of continuous piecewise linear maps

In this subsection we develop three estimates on the sequence $\{h_{n}(\phi,N)\}_{n\geq 0}$. The next lemma states that $\{h_{n}\}_{n\geq 0}$ respects the proportions of gaps.

Lemma 3.1**.**

There exist a neighborhood $\hat{\mathscr{U}}$ of $\phi_{0}$ and $\hat{N}\geq 1$ such that the following holds for every $\phi\in\hat{\mathscr{U}}$ and every integer $N>\hat{N}$: the sequence $\{h_{n}(\phi,N)\}_{n\geq N}$ is well-defined, and for every $i\in\{1,\ldots,p-1\}$, every $n\geq 1$ and every $a_{0}\cdots a_{n-1}\in W_{n}$,

[TABLE]

Proof.

We argue in two steps.

Step 1: Well-definedness of $\{h_{n}(\phi,N)\}_{n\geq N}$. If $n\geq N+1$ then the transition from $n-1$ to $n$ makes sense. It suffices to show that the transition from $N-1$ to $N$ makes sense.

Fix $\delta\in(0,1)$ such that $\frac{(1+\delta)^{2}}{1+K(f_{0})}<1$. Choose a neighborhood $\hat{\mathscr{U}}$ of $\phi_{0}$ and $\hat{N}\geq 1$ such that for every integer $N\geq\hat{N}$, every $\phi\in\hat{\mathscr{U}}$, every $a_{0}\cdots a_{N-1}\in W_{N}$, every $i\in\{1,\ldots,p-1\}$ and every $j\in\{i-1,i\}$ the following holds:

[TABLE]

[TABLE]

The second condition follows from Lemma 2.1. Lemma 2.3 implies

[TABLE]

By the Mean Value Theorem, for each $j\in\{i-1,i\}$ there exists $\xi_{j}\in X_{a_{0}\cdots a_{N-1}j}(f_{0})$ such that $|Df_{0}(\xi)|\left|X_{a_{0}\cdots a_{N-1}j}(f_{0})\right|=\left|X_{a_{1}\cdots a_{N-1}j}(f_{0})\right|$. Hence

[TABLE]

(1) and (2) yield

[TABLE]

This condition prevents the overhang mentioned in the last paragraph of Sect.3.1. Hence the transition from $N-1$ to $N$ makes sense.

Step 2: Proof of the inequality. Let $N\geq\hat{N}$ be an integer and $\phi\in\hat{\mathscr{U}}$. If $n\in\{0,1,\ldots,N-1\}$ then $h_{n}=f_{0}$, and Lemma 2.3 gives $\Delta^{(i)}_{a_{0}\cdots a_{n-1}}(h_{n})\leq K(f_{0}).$ Suppose $n=N$. The definition of $h_{N}$ gives

[TABLE]

To estimate the denominator of this fraction, put

[TABLE]

Shrinking $\hat{\mathscr{U}}$ and enlarging $\hat{N}$ if necessary, we may assume

[TABLE]

We have $s^{-1}|X_{a_{1}\cdots a_{N-1}\widehat{i}}(f_{0})|\leq\int_{X_{a_{1}\cdots a_{N-1}\widehat{i}}(f_{0})}|Df_{0}(f_{0}^{-1}(y))|^{-1}dy$, and Lemma 2.1 gives $E_{a_{0}\cdots a_{N-1}j}(\phi)\geq s-\eta$ for every $j\in\{i-1,i\}$. Hence, the denominator of the fraction in (3) is bounded from below by

[TABLE]

where the first inequality follows from Lemma 2.3 and the last from (4). From (3) we have

[TABLE]

From Lemma 2.1 and the definition of $\eta$, for every $j\in\{i-1,i\}$ we have

[TABLE]

Substituting $E_{a_{0}\cdots a_{N-1}j}(\phi)\geq s-\eta$ into (5) yields

[TABLE]

It is left to treat the case $n\geq N+1$. The construction of $h_{n}$ from $h_{n-1}$ in Sect.3.1 implies

[TABLE]

Using this inductively yields $\Delta^{(i)}_{a_{0}\cdots a_{n-1}}(h_{n})\leq\Delta^{(i)}_{a_{n-N}\cdots a_{n-1}}(h_{N})\leq 2K(f_{0}).$ ∎

In what follows, let $\hat{\mathscr{U}}$ be the neighborhood of $\phi_{0}$ in $C(\Sigma_{p})$ and $\hat{N}\geq 1$ the number in the statement of Lemma 3.1. For an integer $n\geq 1$ and $a_{0}\cdots a_{n-1}\in W_{n}$ define $\tau_{a_{0}\cdots a_{n-1}\Box_{i}}$ to be the constant value of $|Dh_{n}|$ on $X_{a_{0}\cdots a_{n-1}\Box_{i}}(h_{n})$.

Lemma 3.2**.**

The following holds for every $\phi\in\hat{\mathscr{U}}$, every integer $N>\hat{N}$ and $\{h_{n}(\phi,N)\}_{n\geq 0}$:

• (a)

For every $i\in\{1,\ldots,p-1\}$, every $n\geq N+1$ and every $a_{0}\cdots a_{n-1}\in W_{n}$,

[TABLE]

Moreover, for every $i\in\{1,\ldots,p-1\}$ and every $a_{0}\cdots a_{N-1}\in W_{N}$,

[TABLE]

• (b)

For every integers $m$, $n$ with $m>n\geq N$, every $a_{0}\cdots a_{m-1}\in W_{m}$ and every $i$, $j\in\{1,2,\ldots,p-1\}$,

[TABLE]

Proof.

As for (a), let $i\in\{1,\ldots,p-1\}$ and $a_{0}\cdots a_{n-1}\in W_{n}$. We first consider the case $n\geq N+1$. From the definition of $h_{n}$,

[TABLE]

Hence

[TABLE]

This yields

[TABLE]

The last inequality follows from Lemma 3.1.

A proof for the case $n=N$ is analogous to the above argument modulo minor differences. We simply replace $E_{a_{0}\cdots a_{N-1}}(\phi)$ by $\sup_{x\in X_{a_{0}\cdots a_{N-1}}(f_{0})}|Df_{0}(x)|$ and argue in the same way. On the fraction in the summand, for every $j\in\{i-1,i\}$,

[TABLE]

where the last inequality follows from the second estimate in Lemma 2.1. This completes the proof of Lemma 3.2(a).

From the construction in Sect.3.1,

[TABLE]

Lemma 3.2(a) implies

[TABLE]

and

[TABLE]

If $\tau_{a_{0}\cdots a_{n-1}\Box_{i}}>\tau_{a_{0}\cdots a_{m-1}\Box_{j}}$ then

[TABLE]

In the case $\tau_{a_{0}\cdots a_{n-1}\Box_{i}}<\tau_{a_{0}\cdots a_{m-1}\Box_{j}}$ we get the same inequality. This completes the proof of Lemma 3.2(b). ∎

3.3. Perturbation to $C^{1}$ maps

We have constructed a sequence $\{h_{n}(\phi,N)\}_{n\geq 0}$ of continuous piecewise linear maps. For each $n\geq N$ we define a $C^{1}$ map $f_{n}\colon X\to[0,1]$ by perturbing $h_{n}$ on each gap of order $k=N,N+1,\ldots,n$.

Start with $f_{n}=h_{n}$ on $G_{N}$. Let $k\geq N$ and $a_{0}\cdots a_{k-1}\in W_{k}$. Define $f_{n}|_{X_{a_{0}\cdots a_{k-1}}(h_{n-1})}$ as follows. Recall that

[TABLE]

and for every $i\in\{1,2,\ldots,p-1\}$,

[TABLE]

For every $x\in X_{a_{0}\cdots a_{k-1}(i-1)}(h_{n-1})\bigcup X_{a_{0}\cdots a_{k-1}i}(h_{n-1})$, set $f_{n}(x)=h_{n}(x)$. In order to define $f_{n}$ on $X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$ we need the next lemma.

Lemma 3.3**.**

Let $E>0$, $E^{\prime}>0$, $\tau>0$ be such that $2\tau-(E+E^{\prime})/2>0$ and $I=[\alpha,\beta]$ a compact interval. There exists a $C^{1}$ diffeomorphism $\varphi\colon I\to\mathbb{R}$ such that $D\varphi(\alpha)=E$, $D\varphi(\beta)=E^{\prime}$, $|\varphi(I)|=\tau|I|$ and for every $x\in I$,

[TABLE]

Proof.

Define a continuous function $g\colon I\to\mathbb{R}$ by the following set of conditions: $g(\alpha)=E$, $g(\beta)=E^{\prime}$, $g((\alpha+\beta)/2)=2\tau-(E+E^{\prime})/2$, $g$ is linear on $[\alpha,(\alpha+\beta)/2]$ and $[(\alpha+\beta/2),\beta]$. Define $\varphi\colon I\to\mathbb{R}$ by $\varphi(x)=\int_{0}^{x}g(y)dy.$ It is easy to check that $f$ satisfies the desired properties apart from the last one. To show the last property, note that $2\tau-(E+E^{\prime})/2\leq D\varphi(x)\leq\max\{E,E^{\prime}\}$ for every $x\in I$. Let $x\in I$ and suppose $D\varphi(x)-\tau<0$. Then

[TABLE]

If $D\varphi(x)-\tau\geq 0$, then

[TABLE]

This completes the proof of the lemma. ∎

Let $\psi\colon\overline{X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})}\to\mathbb{R}$ be a $C^{1}$ diffeomorphism for which the conclusion of Lemma 3.3 holds with $E=E_{a_{0}\cdots a_{k-1}(i-1)}(\phi)$, $E^{\prime}=E_{a_{0}\cdots a_{k-1}i}(\phi)$, $\tau=\tau_{a_{0}\cdots a_{k-1}\Box_{i}}$, $I=\overline{X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})}$. Lemma 3.2 ensures the condition $2\tau-(E+E^{\prime})/2>0$ in Lemma 3.3. Define

[TABLE]

where $l$ denotes the left boundary point of $X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$. The sign is $+$ if $Df_{0}>0$ on $(r,s)$ and $-$ otherwise. This finishes the definition of $f_{n}$. Lemma 3.3 implies that $f_{n}\colon X\to[0,1]$ is $C^{1}$ and satisfies (E1).

Lemma 3.4**.**

The following holds for every $\phi\in\hat{\mathscr{U}}$, every integer $N>\hat{N}$ and the sequences $\{h_{n}(\phi,N)\}_{n\geq 0}$ and $\{f_{n}(\phi,N)\}_{n\geq 0}$: Let $n\geq N+1$. For every $k\in\{N,N+1,\ldots n\}$, every $a_{0}\cdots a_{k-1}\in W_{k}$, every $i\in\{1,2,\ldots,p-1\}$ and every $x\in X_{a_{0}\cdots a_{k-1}\widehat{i}}(h_{n-1})$ we have

[TABLE]

Moreover, for every $a_{0}\cdots a_{N-1}\in W_{N}$, every $i\in\{1,2,\ldots,p-1\}$ and every $x\in X_{a_{0}\cdots a_{N-1}\widehat{i}}(f_{0})$ we have

[TABLE]

Proof.

From Lemma 3.3 and Lemma 3.2(a) we have

[TABLE]

A proof of the second inequality in the lemma is analogous and hence we omit it. ∎

3.4. Cauchy property

Starting from a $C^{2}$ map $f_{0}\in\mathscr{E}$ we have constructed a sequence $\{f_{n}(\phi,N)\}_{n\geq 0}$ of $C^{1}$ maps on $X$. We show that $\{f_{n}\}_{n\geq 0}$ is a Cauchy sequence in $\mathscr{E}$ which is contained in a $C^{1}$ neighborhood of $f_{0}$.

Lemma 3.5**.**

For every $\varepsilon>0$ there exist a neighborhood $\mathscr{U}$ of $\phi_{0}=\log|Df_{0}|\circ\pi_{f_{0}}$ in $C(\Sigma_{p})$ and $N_{0}\geq 1$ such that the following holds for every $\phi\in\mathscr{U}$, every integer $N>N_{0}$ and the sequence $\{f_{n}(\phi,N)\}_{n\geq 0}$:

• (a)

for every $n\geq N$,

[TABLE]

• (b)

$\{f_{n}\}_{n\geq 0}$* is a Cauchy sequence in $\mathscr{E}$ with respect to the norm $\|\cdot\|_{C^{1}}$.*

Proof.

Let $\varepsilon>0$. Let $\phi\in\hat{\mathscr{U}}$ and $N>\hat{N}$ be an integer. Depending on $\varepsilon$ we will choose $\phi$ that is sufficiently close to $\phi_{0}$, and then choose a sufficiently large $N$.

We first estimate $\|f_{0}-f_{N}\|_{C^{1}}$. Let $a_{0}\cdots a_{N-1}\in W_{N}$ and $i\in\{1,2,\ldots,p-1\}$. From the construction, $|f_{0}(x)-f_{N}(x)|\leq|X_{a_{1}\cdots a_{N-1}\widehat{i}}(f_{0})|$ holds for every $x\in X_{a_{0}\cdots a_{N-1}\widehat{i}}(f_{0})$. (E2) for $f_{0}$ implies $|X_{a_{1}\cdots a_{N-1}\widehat{i}}(f_{0})|\leq c\lambda^{-N+1}$. Hence

[TABLE]

where the last inequality holds for sufficiently large $N$.

Let $a_{0}\cdots a_{N-1}\in W_{N}$ and $i\in\{1,2,\ldots,p-1\}$. For every $x\in X_{a_{0}\cdots a_{N-1}\widehat{i}}(f_{0})$ we have

[TABLE]

The second term is bounded by Lemma 3.4. We estimate the first term. If $|Df_{0}(x)|\leq\tau_{a_{0}\cdots a_{N-1}\Box_{i}}$, then from $\tau_{a_{0}\cdots a_{N-1}\Box_{i}}\leq E_{a_{0}\cdots a_{N-1}}(\phi)$ in Lemma 3.2(a) and from Lemma 2.1 we have

[TABLE]

If $|Df_{0}(x)|>\tau_{a_{0}\cdots a_{N-1}\Box_{i}}$, then from $|Df_{0}(x)|\leq F_{a_{0}\cdots a_{N-1}}(\phi_{0})$ and the second inequality in Lemma 3.2(a) we have

[TABLE]

It follows that

[TABLE]

provided $\phi$ is sufficiently close to $\phi_{0}$ and $N$ is sufficiently large. From (6) and (7) we obtain

[TABLE]

Let $\varepsilon_{0}$, $c_{0}$, $\lambda_{0}$ be the constants for which the conclusions of Lemma 2.2 holds with respect to $f_{0}$. Let $\varepsilon\in(0,\varepsilon_{0})$. Let $m$, $n$ be integers with $m>n\geq N$. We estimate $\|f_{n}-f_{m}\|_{C^{1}}$. If $x\in X$ is contained in a gap of order $\leq n$ we have $h_{m}(x)=h_{n}(x)$, and thus $f_{m}(x)=f_{n}(x)$. Suppose $x\in X$ is not contained in any gap of order $\leq n$. Then, there exist $a_{0}\cdots a_{n-1}\in W_{n}$ and $i\in\{1,2,\ldots,p-1\}$ such that $x\in X_{a_{0}\cdots a_{n-1}\widehat{i}}(h_{n-1})$. The construction of $\{h_{n}\}_{n\geq 0}$ in Sect.3.1 implies

[TABLE]

Since $x\in X_{a_{0}\cdots a_{n-1}\widehat{i}}(h_{n-1})$ and $a_{0}\cdots a_{n-1}\in W_{n}$ are arbitrary, we obtain

[TABLE]

Proceeding to the estimate of derivatives, again let $x\in X$ and let $a_{0}\cdots a_{n-1}\in W_{n}$, $i\in\{1,2,\ldots,p-1\}$ be such that $x\in X_{a_{0}\cdots a_{n-1}\widehat{i}}(h_{n-1})$. We treat two cases separately.

Case I. $x$ is not contained in a gap of order $\leq m$. We have $|Df_{n}(x)|=E_{a_{0}\cdots a_{n-1}(i-1)}(\phi)$ or $|Df_{n}(x)|=E_{a_{0}\cdots a_{n-1}i}(\phi)$, and $|Df_{m}(x)|=E_{a_{0}\cdots a_{m-1}}(\phi)$. Hence

[TABLE]

Case II. $x$ is contained in a gap of order $k\in[n,m]$. Let $a_{0}\cdots a_{k-1}\in W_{k}$ and $j\in\{1,2,\ldots,p-1\}$ be such that $x\in X_{a_{0}\cdots a_{k-1}\Box_{j}}(h_{m-1})$. We have

[TABLE]

The first and the third terms are bounded by Lemma 3.4. For the second term, Lemma 3.2(b) gives

[TABLE]

Hence we obtain

[TABLE]

Since $x\in X_{a_{0}\cdots a_{n-1}\widehat{i}}(h_{n-1})$ and $a_{0}\cdots a_{n-1}\in W_{n}$ are arbitrary, we obtain

[TABLE]

where the multiplicative constant only depends on $\phi$ and $f_{0}$. Overall, for every integers $m$, $n$ with $m>n\geq N$,

[TABLE]

Since $f_{N-1}=f_{0}$, for every $n\geq N+1$ we have

[TABLE]

where the last inequality holds provided $\phi$ is sufficiently close to $\phi_{0}$ and $N$ is sufficiently large depending on $\phi$. From this and (8) we obtain $\|f_{0}-f_{n}\|_{C^{1}}\leq\varepsilon$ for every $n\geq N$. Lemma 2.2 gives $|X_{a_{1}\cdots a_{n-1}\widehat{i}}(f_{n-1})|\leq c_{0}^{-1}\lambda_{0}^{-n}$, and so the first term of (9) converges to zero as $n\to\infty$. The convergence of the second term follows from the uniform continuity of $\phi$. (9) implies that $\{f_{n}\}_{n\geq 0}$ is a Cauchy sequence. ∎

3.5. End of the proof of the Realization Lemma

We complete the proof of the Realization Lemma.

Proof of the Realization Lemma.

Let $f_{\infty}$ denote the limit of the Cauchy sequence $\{f_{n}\}_{n\geq 0}$. Then $f_{\infty}$ satisfies (E1). By Lemma 3.5, $\|f_{0}-f_{\infty}\|_{C^{1}}\leq\varepsilon$ holds. Lemma 2.2 implies that $f_{\infty}$ satisfies (E2). If $x\in\Lambda(f_{\infty})$ and $\pi_{f_{\infty}}^{-1}(x)=\{a_{n}\}_{n\geq 0}$ then $x\in\bigcap_{n\geq 0}X_{a_{0}\cdots a_{n}}(f_{\infty})$. By construction, $|Df_{n}(x)|\equiv E_{a_{0}\cdots a_{n}}(\phi)$ holds for every $n\geq N$, and therefore

[TABLE]

which yields $\log|Df_{\infty}|\circ\pi_{f_{\infty}}=\phi$. ∎

4. On the proof of Theorem B

In this last section we prove Theorem B. In Sect.4.1 we recall the result of Contreras [9] on ergodic optimization for expanding maps. In Sect.4.2 we show that any map in $\mathscr{E}$ is approximated by another whose derivative is locally constant. In Sect.4.3 we refine the construction in Sect.3.1 and prove a Lipschitz version of the Realization Lemma. In Sect.4.4 we complete the proof of Theorem B.

4.1. The result of Contreras on optimization by periodic measures

Let $(Y,d)$ be a compact metric space and $\phi$ a real-valued Lipschitz continuous function on $Y$. The Lipschitz norm of $\phi$ is given by

[TABLE]

Let ${\rm Lip}(Y,d)$ denote the space of Lipschitz continuous functions on $Y$ endowed with the topology given by the norm $\|\cdot\|_{\rm Lip}$. A Lipschitz continuous map $T\colon Y\circlearrowleft$ is expanding if there exist an integer $m\geq 1$ and a constant $\lambda>1$ such that for every $y\in Y$ there exist $\ell_{y}\in\{1,2,\ldots,m\}$, a neighborhood $U_{y}$ of $y$ in $Y$ and continuous maps $S_{i}\colon U_{y}\to Y$, $i\in\{1,\ldots,\ell_{y}\}$ such that the following holds:

$S_{i}(U_{y})\cap S_{j}(U_{y})=\emptyset$ if $i\neq j$;

• -

$T^{-1}(U_{y})=\bigcup_{i=1}^{\ell_{y}}S_{i}(U_{y})$;

• -

$T(S_{i}(z))=z$ for every $i\in\{1,2,\ldots,\ell_{y}\}$ and every $z\in U_{y}$;

• -

$d(z,w)\geq\lambda d(S_{i}(z),S_{i}(w))$ for every $i\in\{1,2,\ldots,\ell_{y}\}$ and every $z,w\in U_{y}$.

Theorem 2**.**

([9, Theorem A])* Let $(Y,d)$ be a compact metric space and $T\colon Y\circlearrowleft$ an expanding map. There exists an open dense subset $\mathscr{O}_{(Y,d)}$ of ${\rm Lip}(Y,d)$ such that for every $\phi\in\mathscr{O}_{(Y,d)}$ there exists a unique $\phi$-minimizing measure, and it is supported on a periodic orbit.*

We will apply Theorem 2 to the left shift $\sigma\colon\Sigma_{p}\circlearrowleft$. The topology of $\Sigma_{p}$ is equivalent to the topology generated by the distance $d_{\theta}$ given by

[TABLE]

where $\theta\in(0,1)$, $\underline{a}=\{a_{n}\}_{n\geq 0}$, $\underline{b}=\{b_{n}\}_{n\geq 0}$ and $s(\underline{a},\underline{b})=\min\{n\geq 0\colon a_{n}\neq b_{n}\}$. Note that $\sigma$ is expanding in the above sense.

4.2. Approximation by maps with locally constant derivatives

For $f\in\mathscr{E}$ we introduce the following additional condition:

• (E3)

there exists an integer $n\geq 1$ such that the value of $Df$ is constant on each connected component of $\bigcap_{k=0}^{n-1}f^{-k}(X)$.

Lemma 4.1**.**

Let $f_{0}\in\mathscr{E}$. For any $\varepsilon>0$ there exists $f\in\mathscr{E}$ which satisfies $\|f_{0}-f\|_{C^{1}}\leq\varepsilon$ and (E3).

Proof.

Let $f_{0}\in\mathscr{E}$ and $\varepsilon>0$. From (E2) there exists an integer $n_{0}\geq 1$ such that for every $n\geq n_{0}$ and every $a_{0}\cdots a_{n-1}\in W_{n}$,

[TABLE]

Since $Df_{0}$ is uniformly continuous, there exists an integer $n_{1}\geq 1$ such that for every $n\geq n_{1}$ and every $a_{0}\cdots a_{n-1}\in W_{n}$,

[TABLE]

Put $n_{2}=\max\{n_{0},n_{1}\}$. Let $n\geq n_{2}$. Define $f\in\mathscr{E}$ so that the following holds:

(On the complement of the union of gaps of order $\leq n-1$) Let $a_{0}\cdots a_{n-1}\in W_{n}$. Then $f\left(X_{a_{0}\cdots a_{n-1}}(f_{0})\right)=X_{a_{1}\cdots a_{n-1}}(f_{0})$ and for every $x\in X_{a_{0}\cdots a_{n-1}}(f_{0})$,

[TABLE]

The sign is $+$ if $Df_{0}(x)>0$ and $-$ otherwise;

• -

(On gaps of low order) Let $k\in\{1,\ldots,n_{2}-1\}$, $i\in\{1,\ldots,p-1\}$ and let $G$ be a gap of order $k$. Then

[TABLE]

• -

(On gaps of high order) Let $k\in\{n_{2},\ldots,n-1\}$, $i\in\{1,\ldots,p-1\}$ and consider the gap $(l,r)=X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$ of order $k$. Let $\psi\colon[l,r]\to\mathbb{R}$ be a $C^{1}$ diffeomorphism for which the conclusion of Lemma 3.3 holds with $E=Df(l)$, $E^{\prime}=Df(r)$, $\tau=\frac{|f_{0}(l)-f_{0}(r)|}{l-r}$, $I=[l,r]$. Then $f(x)=f_{0}(l)+\psi(x)$ holds for every $x\in(l,r)$.

Let $a_{0}\cdots a_{n-1}\in W_{n}$. From the definition of $f$ and (10) we have

[TABLE]

From the definition of $f$ and (11) we have

[TABLE]

Let $k\in\{n_{2},\ldots,n-1\}$, $i\in\{1,\ldots,p-1\}$ and consider the gap $X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$ of order $k$. Let $x\in X_{a_{0}\cdots a_{k-1}\Box_{i}}(h_{n-1})$. From (10),

[TABLE]

From Lemma 3.3 and (11),

[TABLE]

From (12), (13), (14), (15), (16), $\|f_{0}-f\|_{C^{1}}\leq\varepsilon$ follows. By Lemma 2.2, (E2) holds for $f$. Hence $f\in\mathscr{E}$. (E3) follows from the definition of $f$. ∎

4.3. Lipschitz continuity of $Df_{\infty}$

The next lemma is a version of the Realization Lemma.

Lemma 4.2**.**

Let $f_{0}\in\mathscr{E}$ satisfy (E3) and let $\theta\in(0,1/\sup_{x\in X}|Df_{0}(x)|)$. For every $\varepsilon>0$ there exists a neighborhood $\mathscr{U}$ of $\log|Df_{0}|\circ\pi_{f_{0}}$ in $C(\Sigma_{p})$ such that for every $\phi\in\mathscr{U}\cap{\rm Lip}(\Sigma_{p},d_{\theta})$ there exists $f_{\infty}\in\mathscr{E}_{\rm Lip}$ such that

[TABLE]

Proof.

Let $\varepsilon>0$ be such that $\theta\leq 1/(\sup_{x\in X}|Df_{0}(x)|+\varepsilon)$. Although $f_{0}$ may not be of class $C^{2}$, from (E3) the estimate as in Lemma 2.3 remains to hold, and thus the construction in Sect.3.1 remains to work: there exists a neighborhood $\mathscr{U}$ of $\log|Df_{0}|\circ\pi_{f_{0}}$ in $C(\Sigma_{p})$ such that for every $\phi\in\mathscr{U}$ there exists a Cauchy sequence $\{f_{n}\}_{n\geq 0}$ in $\mathscr{E}$ such that $\|f_{0}-f_{n}\|_{C^{1}}\leq\varepsilon$ for every $n\geq 1$ and its $C^{1}$ limit $f_{\infty}$ belongs to $\mathscr{E}$ and satisfies and $\log|Df_{\infty}|\circ\pi_{f_{\infty}}=\phi.$ We have $\sup_{x\in X}|Df_{n}(x)|\leq\sup_{x\in X}|Df_{0}(x)|+\varepsilon$. Let $a_{0}\cdots a_{n-1}\in W_{n}$ and $i\in\{1,2,\ldots,p-1\}$. From $X_{a_{0}\cdots a_{n-1}\widehat{i}}(h_{n})=X_{a_{0}\cdots a_{n-1}\widehat{i}}(f_{n})$, $f^{n}\left(X_{a_{0}\cdots a_{n-1}\widehat{i}}(f_{n})\right)=[\alpha_{i-1},\beta_{i}]$ and the Mean Value Theorem we have

[TABLE]

Let $\phi\in{\rm Lip}(\Sigma_{p},d_{\theta})$. Let $L(\phi)$ denote the Lipschitz constant of $\phi$. Then

[TABLE]

and therefore

[TABLE]

Lemma 3.1 implies

[TABLE]

Hence

[TABLE]

From this estimate and the construction of $\{f_{n}\}_{n\geq 0}$ in Sect.3.3 it follows that the Lipschitz constant of $Df_{n}$ is uniformly bounded over all $n$. The Lipschitz continuity of $Df_{\infty}$ is a direct consequence of the uniform convergence of $Df_{n}$ to $Df_{\infty}$. ∎

4.4. End of proof of Theorem B

To finish, we need an auxiliary lemma.

Lemma 4.3**.**

Let $f_{0}\in\mathscr{E}$. There exist $\theta\in(0,1)$ and a neighborhood $\mathscr{W}$ of $f_{0}$ in $\mathscr{E}$ such that for every $f\in\mathscr{W}$, $\pi_{f}\colon\Sigma_{p}\to\Lambda(f)$ is Lipschitz continuous with respect to the distance $d_{\theta}$.

Proof.

By Lemma 2.2 there exist constants $c>0$, $\lambda>1$ and a neighborhood $\mathscr{W}$ of $f_{0}$ in $\mathscr{E}$ such that the following holds for every $f\in\mathscr{W}$: if $n\geq 1$ and $x\in\bigcap_{k=0}^{n-1}f^{-k}(X)$, then $|Df^{n}(x)|\geq c\lambda^{n}$. Let $\theta\in[\lambda^{-1},1)$. For every $\underline{a}$, $\underline{b}\in\Sigma_{p}$, $\underline{a}\neq\underline{b}$ we have $|\pi_{f}(\underline{a})-\pi_{f}(\underline{b})|\leq c\lambda^{-s(\underline{a},\underline{b})}\leq c\theta^{s(\underline{a},\underline{b})}=cd_{\theta}(\underline{a},\underline{b}).$ ∎

Proof of Theorem B.

Set

[TABLE]

From Theorem 2, for maps in $\mathscr{O}$ the Lyapunov minimizing measure is unique, and it is supported on a periodic orbit. From Lemma 4.3, $\mathscr{O}$ is an open subset of $\mathscr{E}_{\rm Lip}$.

It remains to show $\mathscr{O}$ is dense in $\mathscr{E}$. Let $f_{0}\in\mathscr{E}$ satisfy (E3). By virtue of Lemma 4.1 it suffices to show that $f_{0}$ is approximated by elements of $\mathscr{O}$. Let $\theta\in(0,1/\sup_{x\in X}|Df_{0}(x)|)$. Since $\phi_{0}=\log|Df_{0}|\circ\pi_{f_{0}}$ is a locally constant function, $\phi_{0}\in{\rm Lip}(\Sigma_{p},d_{\theta})$ holds. By Lemma 4.2, for every $\varepsilon>0$ there exists a neighborhood $\mathscr{U}$ of $\phi_{0}$ in $C(\Sigma_{p})$ such that for every $\phi\in\mathscr{U}\cap{\rm Lip}(\Sigma_{p},d_{\theta})$ there exists $f_{\infty}\in\mathscr{E}_{\rm Lip}$ such that $\|f_{0}-f_{\infty}\|_{C^{1}}\leq\varepsilon$ and $\log|Df_{\infty}|\circ\pi_{f_{\infty}}=\phi$. Since $\mathscr{O}_{(\Sigma_{p},d_{\theta})}$ is a dense subset of ${\rm Lip}(\Sigma_{p},d_{\theta})$ from Theorem 2, $\mathscr{U}\cap\mathscr{O}_{(\Sigma_{p},d_{\theta})}\neq\emptyset$ holds. If $\phi\in\mathscr{U}\cap\mathscr{O}_{(\Sigma_{p},d_{\theta})}$ then $f_{\infty}\in\mathscr{O}$. ∎