The Aubry set for the XY model and typicality of periodic optimization for $2$ -locally constant potentials

Yuika Kajihara Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto, 606-8502, Japan kajihara.yuika.6f@kyoto-u.ac.jp , Shoya Motonaga Department of Mathematical Sciences, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan motonaga@fc.ritsumei.ac.jp and Mao Shinoda Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo, 112-8610, Japan shinoda.mao@ocha.ac.jp

Abstract.

We consider the Aubry set for the XY model, symbolic dynamics $([0,1]^{\mathbb{N}_{0}},\sigma)$ with the uncountable symbol $[0,1]$ , and study its action-optimizing properties. Moreover, for a potential function that depends on the first two coordinates we obtain an explicit expression of the set of optimal periodic measures and a detailed description of the Aubry set. We also show the typicality of periodic optimization for 2-locally constant potentials with the twist condition. Our approach combines the weak KAM method for symbolic dynamics and variational techniques for twist maps.

2020 Mathematics Subject Classification:

Primary 37B10, 37E40, 37A99, 37J51

1. Introduction

This paper serves as a bridge between ergodic optimization for symbolic dynamics and variational problems for twist maps. More precisely, we consider symbolic dynamics with the uncountable symbols [0,1], the so-called XY model, and investigate the associated action-optimizing sets for Lipschitz continuous potentials, in particular 2-locally constant functions, from the view points of the weak KAM method and variational approaches. It is well known that, in Aubry-Mather theory for Euler-Lagrange flows [Mather91, Mane1997], optimizing invariant probability measures are closely related with optimizing curves through the principle of least action. Although our system has no variational structure, we will see the advantages of the concept of “optimizing orbits” as in [BLL13] and of variational techniques based on [Ban88] in ergodic optimization. Before presenting our main results, we briefly give the background of our study and some key notions.

For a continuous map $T$ on a compact metric space $\mathcal{X}$ and a continuous function (called as a potential) $\varphi:\mathcal{X}\rightarrow\mathbb{R}$ , ergodic optimization investigates the optimal (minimum) ergodic average

\displaystyle\alpha_{\varphi}=\inf_{\mu\in\mathcal{M}_{T}(\mathcal{X})}\int\varphi d\mu

where $\mathcal{M}_{T}(\mathcal{X})$ is the set of $T$ -invariant Borel probability measures on $\mathcal{X}$ endowed with the weak*-topology. An invariant measure which attains the minimum is called an optimizing (minimizing) measure for $\varphi$ and denote by $\mathcal{M}_{{\rm min}}(\varphi)$ the set of optimizing measures for $\varphi$ . Since $\mathcal{M}_{T}(\mathcal{X})$ is compact and $\int\varphi d(\cdot):\mathcal{M}_{T}(\mathcal{X})\to\mathbb{R}$ is continuous, $\mathcal{M}_{{\rm min}}(\varphi)\neq\emptyset$ . Note that we consider the minimum ergodic average instead of the maximum one in order to describe a more natural connection with the minimizing method in variational problems (see Section 4). We also remark that Jenkinson’s formula [Jenkinson19] for the optimal ergodic average:

(1)

\displaystyle\alpha_{\varphi}=\inf_{x\in\mathcal{X}}\liminf_{n\to\infty}\frac{1}{n}S_{n}\varphi(x)=\liminf_{n\to\infty}\inf_{x\in\mathcal{X}}\frac{1}{n}S_{n}\varphi(x)

where

(2)

\displaystyle S_{n}\varphi=\sum_{i=0}^{n-1}\varphi\circ T^{i}.

The uniqueness of optimizing measures and the “shape (complexity)” of their support are fundamental questions of ergodic optimization. This leads to the definition of the Mather set

\displaystyle\mathscr{M}_{\varphi}=\bigcup_{\mu\in\mathcal{M}_{{\rm min}}(\varphi)}{\rm supp}(\mu),

where ${\rm supp}(\mu)$ is the intersection of all compact sets with full measure with respect to $\mu$ . There are several results on the uniqueness of the optimizing measures and on the low complexity of the Mather set for “typical” functions (see [Jenkinson19] and the references therein for more details). However, it is worth pointing out that there are few specific examples whose optimizing measures are well understood.

One of the fundamental approaches to extract the detailed description of the Mather set is to consider (calibrated) subactions for potentials. We do not touch the details of this notion here (see Section for the precise definition and properties). We only remark that a certain value of the level set of an associated function contains the Mather set, and thus the existence and uniqueness of (calibrated) subactions are also interesting problems.

Another (but closely related) approach to obtain more precise information about the Mather set is to investigate “optimizing orbits” for potentials. Inspired by the weak KAM theory [Fathi14] due to Fathi, [BLL13] introduced several important notions related to “optimizing orbits” for symbolic dynamics with a finite set of symbols. Borrowing their ideas presented in [BLL13], we first study an “action-optimizing set” of Lipschitz continuous potentials for symbolic dynamics whose symbol is the interval $[0,1]$ . Let $\mathbb{N}_{0}$ be the set of non-negative integers and let $X=[0,1]^{\mathbb{N}_{0}}$ . Set the metric on $X$ by

\displaystyle d(\underline{x},\underline{y})=\sum_{i=0}^{\infty}\frac{|x_{i}-y_{i}|}{2^{i}}

for $\underline{x},\underline{y}\in X$ where $|\cdot|$ is the Euclidean distance in the interval $[0,1]$ . Let $\sigma:X\rightarrow X$ be the left shift, i.e., $(\sigma(\underline{x}))_{i}=x_{i+1}$ for all $i\in\mathbb{N}_{0}$ , and we call the topological dynamical system $(X,\sigma)$ the XY model. See [CR, LM14] for its details. From now on, we consider ergodic optimization for the case $\mathcal{X}=X$ and $T=\sigma$ . For a Lipschitz continuous potential $\varphi:X\to\mathbb{R}$ , we define two functions the Mañé potential $S_{\varphi}(\cdot,\cdot)$ and Peierl’s barrier $H_{\varphi}(\cdot,\cdot)$ on $X\times X$ as

	$\displaystyle S_{\varphi}(\underline{x},\underline{y})$	$\displaystyle=\lim_{\varepsilon\to 0}\inf\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z})\mid n\in\mathbb{N},\underline{z}\in B(\underline{x},\underline{y},n;\varepsilon)\},$
	$\displaystyle H_{\varphi}(\underline{x},\underline{y};\varepsilon)$	$\displaystyle=\lim_{\varepsilon\to 0}\liminf_{n\to\infty}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z})\mid\underline{z}\in B(\underline{x},\underline{y},n;\varepsilon)\},$

where

\displaystyle B(\underline{x},\underline{y},n;\varepsilon)

\displaystyle=\{\underline{z}\in X\mid d(\underline{x},\underline{z})<\varepsilon,d(\sigma^{n}(\underline{z}),\underline{y})<\varepsilon\}.

See Section 2 for the details of $S_{\varphi}$ and $H_{\varphi}$ . Then we define the Aubry set $\Omega_{\varphi}$ as the zero-level set of $\tilde{S}_{\varphi}$ , where $\tilde{S}_{\varphi}(\underline{x})=S_{\varphi}(\underline{x},\underline{x})$ . We remark that $\Omega_{\varphi}$ includes the Mather set $\mathscr{M}_{\varphi}$ of $\varphi$ .

Now we give our first main results concerning Peierl’s barrier. A positive-semi orbit $\{\sigma^{n}(x)\}_{n\in\mathbb{N}_{0}}$ is said to be $\varphi$ -static if for any non-negative integers $i<j$ it holds that

\sum_{n=i}^{j-1}\Big{(}\varphi\circ\sigma^{n}(\underline{x})-\alpha_{\varphi}\Big{)}=-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x})).

Define

\displaystyle A_{\varphi}

\displaystyle:=\{\sigma^{k}(\underline{x})\in X\mid\{\sigma^{n}(\underline{x})\}_{n\in\mathbb{N}_{0}}\ \text{is}\ \varphi\text{-static},k\in\mathbb{N}_{0}\}.

Note that this definition is motivated by Mañé’s work [Mane1997] for Lagrangian systems. See Section 3 for their details. We then obtain the following characterizations of the Aubry set and Peierl’s barrier as in [BLL13, Mane1997].

Main Theorem 1.

Let $\varphi:X\rightarrow\mathbb{R}$ be a Lipschitz function and define $\tilde{H}_{\varphi}(\underline{x})={H}_{\varphi}(\underline{x},\underline{x})$ . Then, we have

\Omega_{\varphi}=\tilde{H}_{\varphi}^{-1}(\{0\})=A_{\varphi}.

Main Theorem 2.

For any $\underline{x}\in\Omega_{\varphi}$ , ${H}_{\varphi}(\underline{x},\cdot)\colon X\to\mathbb{R}$ is a Lipschitz calibrated subaction. Moreover, the relation $\underline{x}\sim\underline{y}$ given by

H_{\varphi}(\underline{x},\underline{y})+H_{\varphi}(\underline{y},\underline{x})=0

is an equivalence relation if both $\underline{x}$ and $\underline{y}$ belong to $\Omega_{\varphi}$ .

Next, we restrict our attention to 2-locally constant potentials and make use of variational techniques. Although “action-optimizing sets” play important roles in the study of Lagrangian systems as well as symbolic dynamics with finite symbols, it is difficult to obtain explicit information about these sets in most cases. On the other hand, for the case of area-preserving twist maps on an annulus, Aubry and Mather originally developed their theory and it provides much detailed descriptions of “minimal” orbits (originally appeared in [Morse24] under the name “Class A”), i.e., minimizers under arbitrary two-point boundary conditions. There are several papers by them related to twist maps; for example, see [Aubry83, AD83, Mather82, Mather89, Mather91]. The best general reference here is Bangert’s survey article [Ban88]. Following [Ban88], we assume the twist condition for our 2-locally constant potentials and give the following result.

Main Theorem 3.

Suppose that $\varphi(\underline{x})=h(x_{0},x_{1})$ with $D_{2}D_{1}h<0$ , where $h:[0,1]^{2}\to\mathbb{R}$ is a $C^{2}$ -function on $[0,1]^{2}$ and $D_{i}$ means derivative for the $i$ -th component for $i=1,2$ . Let $h^{\ast}=\min_{x\in[0,1]}h(x,x)$ and $\mathrm{m}=\{a\in[0,1]\mid h(a,a)=h^{\ast}\}$ . Then we have (1)-(3):

(1)

$\alpha_{\varphi}=h^{\ast}$ .
(2)

$\mathcal{M}_{{\rm min}}(\varphi)\cap\mathcal{M}^{\mathrm{p}}=\{\delta_{a^{\infty}}\mid a\in\mathrm{m}\}$ , where $\delta_{\underline{x}}$ is the Dirac measure supported at $\underline{x}$ and $\mathcal{M}^{\mathrm{p}}$ stands for the set of invariant probability measures supported on a single periodic orbit.
(3)

$\Omega_{\varphi}\subset\mathrm{m}^{\mathbb{N}_{0}}$ , i.e., for any $\underline{x}=\{x_{i}\}_{i\in\mathbb{N}_{0}}\in\Omega_{\varphi}$ we have

$h(x_{i},x_{i})=h^{\ast}\ \text{for all}\ i\in\mathbb{N}_{0}.$

If, in addition, $h(x,x)$ has a unique minimum point $a_{\ast}$ in $[0,1]$ , then the Mather set of $\varphi$ coincides with the Aubry set of $\varphi$ and it consists of the single fixed point $a_{\ast}^{\infty}$ .

We call the assumption $D_{2}D_{1}h<0$ the twist condition. Note that Main theorem 3 holds under weaker assumptions for Lipschitz continuous $h$ on $[0,1]^{2}$ (see Section 4 for the details). We emphasize that, for 2-locally constant potentials with the twist condition, Main theorem 3 provides the explicit formulas of the optimal ergodic average and of the set of optimizing periodic measures, and in some cases it also completely determines the Mather set and the Aubry set.

Finally we turn to the typically periodic optimization (TPO) problem in the class of $2$ -locally constant potentials for the XY model. In the field of ergodic optimization, for “chaotic” systems, it is conjectured that the optimizing measure for a “typical” potential with a suitable regularity is a periodic measure, i.e., supported on a single periodic orbit. Many authors investigate the “typicality” of the periodic minimizing measures in many contexts (see [Jenkinson19] for more details). Recently Gao et.al established a TPO property of real analytic expanding circle maps for smooth potentials [GSZ]. Our result is also formulated for smooth potentials as described below. Consider the set of $C^{r}$ -functions ( $r\geq 2$ ) with the twist condition,

\mathscr{H}^{r}=\{h\in C^{r}([0,1]^{2};\mathbb{R})\mid D_{2}D_{1}h<0\},

equipped with the $C^{r}$ -norm. Using Theorem 3, we obtain the following TPO property.

Main Theorem 4 (TPO property for the XY model in the class of 2-locally constant functions).

Let $r\geq 2$ be an integer. For the XY model, we have the followings:

(i)

There is a $C^{r}$ open dense subset $\mathscr{O}$ in $\mathscr{H}^{r}$ such that for each $h\in\mathscr{O}$ the Mather set and the Aubry set of $h$ consist of a single fixed point.
(ii)

For arbitrary $h\in\mathscr{H}^{r}$ there is a $C^{r}$ open dense subset $\mathscr{V}_{h}$ in $C^{r}([0,1];\mathbb{R})$ such that for each $V\in\mathscr{V}_{h}$ the Mather set and the Aubry set of $h+V$ consist of a single fixed point.

The structure of this paper is as follows. In Section 2, we extend the definitions of the Mañé potential and Peierl’s barrier for symbolic dynamics with finite alphabets to our setting. We also confirm that these functions satisfy similar properties presented in [BLL13]. In Section 3, we derives a characterization of the Aubry set in terms of Mañé’s action-optimizing approach [Mane1997]. In Section 4, we consider 2-locally constant potentials under several assumptions and determine the set of optimizing periodic measures as well as the elements in the Aubry set. Finally, we verify that the TPO property holds within our framework in Section 5.

2. Mañé potential and Peierl’s barrier

In this section, following [BLL13], we consider the Mañé potential, Peierl’s barrier, and the Aubry set. We begin with the definition of the Mañé potential.

Definition 2.1 (Mañé potential).

For $\varphi:X\rightarrow\mathbb{R}$ and $\varepsilon>0$ define $S_{\varphi}:X\times X\rightarrow\mathbb{R}\cup\{\infty\}$ by

\displaystyle S_{\varphi}(\underline{x},\underline{y};\varepsilon)=\inf\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z})\mid n\in\mathbb{N},\underline{z}\in B(\underline{x},\underline{y},n;\varepsilon)\}

where

\displaystyle B(\underline{x},\underline{y},n;\varepsilon)=\{\underline{z}\in X\mid d(\underline{x},\underline{z})<\varepsilon,d(\sigma^{n}(\underline{z}),\underline{y})<\varepsilon\}.

We define the Mañé potential $S_{\varphi}$ by

\displaystyle S_{\varphi}(\underline{x},\underline{y})=\lim_{\varepsilon\to 0}S_{\varphi}(\underline{x},\underline{y};\varepsilon),

Remark 2.2.

The Mañé potential originates from the context of the Aubry-Mather theory for Euler-Lagrange flows. Mañé [Mane1997] considered a function $\phi\colon M\times M\to\mathbb{R}$ defined by

(3)

\displaystyle\phi(x,y)=\inf_{T>0}\inf_{\gamma\in C(x,y;T)}\int_{0}^{T}(L(\gamma,\dot{\gamma})-\tilde{\alpha}_{\varphi})dt,

where $M$ is a closed Riemannian manifold, $L:TM\to\mathbb{R}$ is a Tonelli Lagrangian (see [Mane1997] for the precise definition), $C(x,y;T)$ is the set of absolutely continuous curves $\gamma\colon\mathbb{R}\to M$ with $\gamma(0)=x,\gamma(T)=y$ , and $\tilde{\alpha}_{\varphi}$ is given by

\tilde{\alpha}_{\varphi}=\inf_{\mu\in\mathcal{M}_{\Phi^{t}(L)}}\int L(\gamma,\dot{\gamma})d\mu,

with the Euler-Lagrange flow $\Phi^{t}(L)$ . The right-side of $\eqref{eq:mane-el}$ corresponds to a minimizing method that provides trajectories with the energy $\tilde{\alpha}_{\varphi}$ in the two-point boundary value problem. [BLL13] has rewritten this concept in the context of ergodic optimization for symbolic dynamics with finite symbols, and Definition 2.1 is an analogy of their definition.

Definition 2.3 (Aubry set).

The set $\Omega_{\varphi}=\{\underline{x}\in X\mid S_{\varphi}(\underline{x},\underline{x})=0\}$ is called the Aubry set of $\varphi$ .

Note that we will see that $S_{\varphi}$ does not take $-\infty$ in Lemma 2.6 if $\varphi$ is Lipschitz. The following notion, called (calibrated) subation, is important as a technical tool for ergodic optimization.

Definition 2.4 (Subaction, calibrated subaction).

A continuous function $u:X\rightarrow\mathbb{R}$ is called a subaction of $\varphi$ if

\displaystyle u(\underline{x})+\varphi(\underline{x})\geq u(\sigma\underline{x})+\alpha_{\varphi}

for every $\underline{x}\in X$ . Moreover, a subaction $u$ is called calibrated if

\displaystyle\min_{\sigma(\underline{y})=\underline{x}}(\varphi(\underline{y})+u(\underline{y}))=u(\underline{x})+\alpha_{\varphi}

for every $\underline{x}\in X$ .

Remark 2.5.

There exists a calibrated subaction for a Walters function $\varphi$ on a weakly-expanding topological dynamical system by [Bou01]. Here, ‘Walters function’ is a broader class of functions that includes Holder continuous functions. Moreover, we can get Lipschits calibrated subaction for Lipschitz $\varphi$ . See [BCLMS11] for the details.

The next lemma gives the lower bound of $S_{\varphi}$ using subactions.

Lemma 2.6.

Assume $\varphi:X\rightarrow\mathbb{R}$ be Lipschitz. For a Lipschitz subaction $u$ of $\varphi$ we have

(4)

\displaystyle S_{\varphi}(\underline{x},\underline{y})\geq u(\underline{y})-u(\underline{x})

for every $\underline{x},\underline{y}\in X$ .

Proof.

Since $\varphi$ is Lipschitz, its calibrated subaction $u$ is also Lipschitz. Then we have

\varphi(\underline{x})-\alpha_{\varphi}\geq u\circ\sigma(\underline{x})-u(\underline{x})

for all $\underline{x}\in X$ . Fix $\varepsilon>0$ . Take $n\geq 1$ and $\underline{z}\in B(\underline{x},\underline{y},n;\varepsilon)$ . Then we have

	$\displaystyle S_{n}(\varphi-\alpha_{\varphi})(\underline{z})$	$\displaystyle\geq S_{n}(u\circ\sigma-u)(\underline{z})$
		$\displaystyle=u(\sigma^{n}(\underline{z}))-u(\underline{z})$
		$\displaystyle\geq u(\underline{y})-u(\underline{x})-L_{u}\varepsilon$

and

S_{\varphi}(\underline{x},\underline{y};\varepsilon)\geq u(\underline{y})-u(\underline{x})-L_{u}\varepsilon,

where $L_{u}$ is the Lipschitz constant of $u$ . Letting $\varepsilon\to 0$ , we have

S_{\varphi}(\underline{x},\underline{y})\geq u(\underline{y})-u(\underline{x}).

∎

Proposition 2.7.

$S_{\varphi}(\underline{x},\underline{y})$ is lower semicontinuous on $X\times X$ .

Proof.

Fix $\underline{y},\underline{z},\underline{w}\in X$ and $\varepsilon>0$ . Since

d(\sigma^{n}(\underline{w}),\underline{y})\leq d(\sigma^{n}(\underline{w}),\underline{z})+d(\underline{y},\underline{z})\leq\varepsilon+d(\underline{y},\underline{z}),

it holds that $B(\underline{x},\underline{y},n;\varepsilon+d(\underline{y},\underline{z}))\supset B(\underline{x},\underline{z},n;\varepsilon)$ and thus we obtain

S_{\varphi}(\underline{x},\underline{y};\varepsilon+d(\underline{y},\underline{z}))\leq S_{\varphi}(\underline{x},\underline{z};\varepsilon).

Moreover, we have

\liminf_{\underline{z}\to\underline{y}}\left(\lim_{\varepsilon\to 0}S_{\varphi}(\underline{x},\underline{z};\varepsilon)\right)=\liminf_{\underline{z}\to\underline{y}}S_{\varphi}(\underline{x},\underline{z})

and

\liminf_{\underline{z}\to\underline{y}}\left(\lim_{\varepsilon\to 0}S_{\varphi}(\underline{x},\underline{y};\varepsilon+d(\underline{y},\underline{z}))\right)=\lim_{\varepsilon^{\prime}\to 0}S_{\varphi}(\underline{x},\underline{y};\varepsilon^{\prime}))=S_{\varphi}(\underline{x},\underline{y}).

Thus it holds that

S_{\varphi}(\underline{x},\underline{y})\leq\liminf_{\underline{z}\to\underline{y}}S_{\varphi}(\underline{x},\underline{z}).

Therefore, the map $\underline{y}\mapsto S_{\varphi}(\underline{x},\underline{y})$ is lower semicontinuous for each $\underline{x}\in X$ . Similarly, we can verify that the map $\underline{y}\mapsto S_{\varphi}(\underline{y},\underline{x})$ is also lower semicontinuous for each $\underline{x}\in X$ . ∎

The following proposition asserts that the Mather set is included in the Aubry set.

Proposition 2.8.

Let $\varphi:X\to\mathbb{R}$ be a Lipschitz function. Then the Mather set $\mathscr{M}_{\varphi}$ is a subset of the Aubry set $\Omega_{\varphi}$ .

Proof.

By the ergodic decomposition, it is sufficient to show that $\mathrm{supp}(\mu)\subset\Omega_{\varphi}$ for each ergodic $\mu\in\mathcal{M}_{{\rm min}}(\varphi)$ . Take arbitrary ergodic $\mu\in\mathcal{M}_{{\rm min}}(\varphi)$ and a subaction $u$ for $\varphi$ . Note that $\int\varphi d\mu=\alpha_{\varphi}$ . Letting $\varphi^{u}=\varphi-\alpha_{\varphi}+u-u\circ\sigma$ , we obtain

\displaystyle\int\varphi^{u}d\mu=\int(\varphi-\alpha_{\varphi})d\mu=0

since $\int u\circ\sigma\ d\mu=\int u\ d\mu$ by the $\sigma$ -invariance of $\mu$ . From the definition of $u$ , it holds that $\varphi^{u}\geq 0$ , which implies that $\varphi^{u}(\underline{x})=0$ for $\mu$ -a.e. $\underline{x}\in X$ . Since $\mu$ is ergodic and $\varphi^{u}$ is continuous, we obtain $\varphi^{u}(\underline{x})=0$ on $\mathrm{supp}(\mu)$ . Hence, we see that

S_{n}(\varphi-\alpha_{\varphi})(\underline{x})=u\circ\sigma^{n}(\underline{x})-u(\underline{x})

for each $\underline{x}\in\mathrm{supp}(\mu)$ and $n\in\mathbb{N}$ . Note that $\sigma$ -invariance of $\mathrm{supp}(\mu)$ implies $\sigma^{n}(\underline{x})\in\mathrm{supp}(\mu)$ for $n\in\mathbb{N}$ if $\underline{x}\in\mathrm{supp}(\mu)$ . By Poincaré’s recurrence theorem, for $\mu$ -a.e. $\underline{x}$ , there exists a monotone increasing sequence $\{n_{k}\}_{k\in\mathbb{N}}$ with $n_{k}\to+\infty$ as $k\to+\infty$ such that $d(\underline{x},\sigma^{n_{k}}(\underline{x}))<\varepsilon$ . This implies that, for $\mu$ -a.e. $\underline{x}$ , we have

S_{\varphi}(\underline{x},\underline{x};\varepsilon)\leq S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{x})=u\circ\sigma^{n_{k}}(\underline{x})-u(\underline{x})\leq L_{\varphi}d(\underline{x},\sigma^{n_{k}}\underline{x})<L_{\varphi}\varepsilon,

i.e., $S_{\varphi}(\underline{x},\underline{x})\leq 0$ . Therefore, by the lower semicontinuity of $S_{\varphi}$ (Proposition 2.7) and the density of $\mu$ -a.e. points in $\mathrm{supp}(\mu)$ , it holds that $S_{\varphi}(\underline{x},\underline{x})\leq 0$ on $\mathrm{supp}(\mu)$ . Using Lemma 2.6, we have $S_{\varphi}(\underline{x},\underline{x})=0$ on $\mathrm{supp}(\mu)$ , which completes the proof. ∎

Next, let us define Peierl’s barrier.

Definition 2.9 (Peierl’s barrier).

For a Lipschitz function $\varphi:X\rightarrow\mathbb{R}$ and $\varepsilon>0$ , define $H_{\varphi}:X\times X\rightarrow\mathbb{R}\cup\{\infty\}$ by

\displaystyle H_{\varphi}(\underline{x},\underline{y};\varepsilon)=\liminf_{n\to\infty}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z})\mid\underline{z}\in B(\underline{x},\underline{y},n;\varepsilon)\}

and

\displaystyle H_{\varphi}(\underline{x},\underline{y})=\lim_{\varepsilon\to 0}H_{\varphi}(\underline{x},\underline{y};\varepsilon).

Remark 2.10.

Note that the Peierl’s barrier defined as above may take $\infty$ . Indeed, for $\varphi(\underline{x})=x_{0}$ , we see that $H_{\varphi}(1^{\infty},1^{\infty})=\infty$ in the following way. Fix $k\geq 1$ . Take $n\geq k+1$ and $\underline{z}\in X$ such that $d(1^{\infty},\underline{z})<2^{-(k+1)}$ and $d(\sigma^{n}(\underline{z}),1^{\infty})<2^{-(k+1)}$ . Since $\alpha_{\varphi}=0$ , we have

	$\displaystyle S_{n}\varphi(\underline{z})-n\alpha_{\varphi}$	$\displaystyle=\sum_{i=0}^{k-1}(z_{i}-1)+\sum_{i=0}^{k-1}(1-\alpha_{\varphi})+\sum_{i=k}^{n-1}(z_{i}-\alpha_{\varphi})$
		$\displaystyle\geq-1+k$

and $H_{\varphi}(1^{\infty},1^{\infty};2^{-k})\geq-1+k$ for all $k\geq 1$ . Letting $k\to\infty$ , we have $H_{\varphi}(1^{\infty},1^{\infty})=\infty$ . Similarly, we can show that any point $(\underline{x},\underline{y})\in X\times X$ of the form $(\underline{x},\underline{y})=(a_{0}\ldots a_{l}1^{\infty},b_{0}\ldots b_{m}1^{\infty})$ provides $H_{\varphi}(\underline{x},\underline{y})=\infty$ . Note that any cylinder set

[a_{0},\ldots,a_{l}]\times[b_{0},\ldots,b_{m}]=\{(\underline{x},\underline{y})\mid x_{i}=a_{i}\ (i=0,\ldots,l),y_{j}=b_{j}\ (j=0,\ldots,m)\}

contains $\{(a_{1}\ldots a_{l}1^{\infty},b_{1}\ldots b_{m}1^{\infty})\}$ and this implies that the function

(\underline{x},\underline{y})\in X\times X\mapsto H_{\varphi}(\underline{x},\underline{y})\in\mathbb{R}\cup\{+\infty\}

takes $+\infty$ on a dense set in $X\times X$ for the case $\varphi(\underline{x})=x_{0}$ . We remark that a similar phenomenon occurs even for a subshift with a finite alphabet, a point which seems to have been not explicitly discussed in the literature, e.g., [BLL13]. This omission, however, does not affect other arguments in [BLL13]. Below, we provide a sufficient condition for the finiteness of the Peierls barrier.

Now let us prove a part of Main Theorem 1. Note that the first equality of Main Theorem 1 says that the Aubry set defined in Definition 2.3 coincides with the zero-level set derived from Peierl’s barrier.

Theorem 2.11 (cf. Main Theorem 1).

Let $\varphi:X\rightarrow\mathbb{R}$ be a Lipschitz function. For $\underline{x}\in\Omega_{\varphi}$ and $\underline{y}\in X$ we have $H_{\varphi}(\underline{x},\underline{y})\leq L_{\varphi}d(\underline{x},\underline{y})$ . In particular, $\underline{x}\in\Omega_{\varphi}$ holds if and only if $H_{\varphi}(\underline{x},\underline{x})=0$ .

Proof.

The second claim immediately follows from the first claim and

H_{\varphi}(\underline{x}^{\prime},\underline{x}^{\prime})\geq S_{\varphi}(\underline{x}^{\prime},\underline{x}^{\prime})\geq 0

for all $\underline{x}^{\prime}\in X$ by Lemma 2.6. Now we prove the first claim. Since $\underline{x}\in\Omega_{\varphi}$ , we have $S_{\varphi}(\underline{x},\underline{x})=0$ . It suffices to consider the following two cases:

(1)

For any $\theta>0$ , there exists $\varepsilon\in(0,\theta)$ such that

$\displaystyle H_{\varphi}(\underline{x},\underline{x};\varepsilon)=S_{\varphi}(\underline{x},\underline{x};\varepsilon)$

(2)

There exists $\theta>0$ such that for any $\varepsilon\in(0,\theta)$ , there exists a finite number $N=N(\varepsilon)$ such that

\displaystyle S_{\varphi}(\underline{x},\underline{x};\varepsilon)=\inf\{S_{N}(\varphi-\alpha_{\varphi})(\underline{z}):\underline{z}\in B(\underline{x},\underline{x},N;\varepsilon)\}.

The proof of Case $(1)$ : Fix $\theta>0$ . By (1) and $S_{\varphi}(\underline{x},\underline{x})=0$ , we may assume there exists $\varepsilon\in(0,\theta/2)$ such that $H_{\varphi}(\underline{x},\underline{x};\varepsilon)=S_{\varphi}(\underline{x},\underline{x};\varepsilon)<\theta/2$ . Hence there exist an increasing sequence $\{n_{i}\}$ with $2^{-n_{1}}<\varepsilon$ and $\underline{z}^{(i)}\in B(\underline{x},\underline{x},n_{i};\varepsilon)$ such that $S_{n_{i}}(\varphi-\alpha_{\varphi})(\underline{z}^{(i)})<\theta$ . Define $\underline{w}^{(n_{i})}\in X$ by

\displaystyle\underline{w}^{(n_{i})}=z_{0}^{(n_{i})}z_{1}^{(n_{i})}\cdots z_{n_{i}-1}^{(n_{i})}\underline{y}.

Then for each $i\geq 1$ we have

	$\displaystyle S_{n_{i}}(\varphi-\alpha_{\varphi})(\underline{w}^{(n_{i})})$	$\displaystyle=S_{n_{i}}(\varphi-\alpha_{\varphi})(\underline{z})+\sum_{i=0}^{n_{i}-1}(\varphi\circ\sigma^{i}(\underline{w}^{(n_{i})})-\varphi\circ\sigma^{i}(\underline{z}))$
		$\displaystyle\leq\theta+L_{\varphi}\sum_{i=0}^{n_{i}-1}d(\sigma^{i}(\underline{w}^{(n_{i})}),\sigma^{i}(\underline{z}))$
		$\displaystyle\leq\theta+L_{\varphi}\sum_{i=1}^{n_{i}}\frac{d(\underline{y},\sigma^{n_{i}}\underline{z})}{2^{i}}$
		$\displaystyle\leq\theta+L_{\varphi}(d(\underline{x},\underline{y})+\varepsilon)$
		$\displaystyle<(1+L_{\varphi}/2)\theta+L_{\varphi}d(\underline{x},\underline{y}).$

It is easy to see $d(\underline{x},\underline{w}^{(n_{i})})\leq 2\varepsilon$ , $d(\sigma^{n_{i}}\underline{w}^{(n_{i})},\underline{y})=0$ and $\underline{w}^{(n_{i})}\in B(\underline{x},\underline{y},n_{i};2\varepsilon)\subset B(\underline{x},\underline{y},n_{i};\theta)$ . Hence we have

\displaystyle H_{\varphi}(\underline{x},\underline{y};\theta)\leq(1+L_{\varphi}/2)\theta+L_{\varphi}d(\underline{x},\underline{y}).

Then letting $\theta\to 0$ we have

\displaystyle H_{\varphi}(\underline{x},\underline{y})\leq L_{\varphi}d(\underline{x},\underline{y}).

The proof of Case $(2)$ : Fix $\varepsilon\in(0,\theta)$ . Set a monotone decreasing positive sequence $\{\varepsilon_{i}\}$ satisfying

\sum_{i\in\mathbb{N}}\varepsilon_{i}<\varepsilon.

For $\{\varepsilon_{i}\}$ , we define a positive integer sequence $\{N_{i}\}$ by $N_{i}=N(\varepsilon_{i})$ .

If $\{N_{i}\}$ is bounded, then there exist an integer $M$ and an infinite subsequence $\{N_{i_{j}}\}$ such that $M=N_{i_{j}}$ for any $j\in\mathbb{N}$ . Then we can take a sequence $\{\underline{z}^{(j)}\}$ such that $\underline{z}^{(j)}\in B(\underline{x},\underline{x},M;\varepsilon_{i_{j}})$ and

\lim_{j\to\infty}S_{M}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})=0

since $\underline{x}\in\Omega_{\varphi}$ and $\varepsilon_{i_{j}}\to 0$ as $j\to\infty$ . This yields $\sigma^{M}(\underline{x})=\underline{x}$ because if $d(\sigma^{M}(\underline{x}),\underline{x})>\delta$ for some $\delta>0$ , then for $\varepsilon_{i_{j}}<\delta 2^{-(M+2)}$ , we obtain

\delta<d(\sigma^{M}(\underline{x}),\sigma^{M}\underline{z}^{(j)})+d(\sigma^{M}\underline{z}^{(j)},\underline{x})<2^{M+1}d(\sigma^{M}\underline{z}^{(j)},\underline{x})<\delta/2,

which is contradiction. Then we get

	$\displaystyle\|S_{M}(\varphi-\alpha_{\varphi})(\underline{x})-S_{M}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})\|$	$\displaystyle\leq L_{\varphi}\sum_{k=0}^{M-1}d(\sigma^{k}(\underline{x}),\sigma^{k}(\underline{z}^{(j)}))$
		$\displaystyle\leq L_{\varphi}\sum_{k=0}^{M-1}2^{k}d(\underline{x},\underline{z}^{(j)})<L_{\varphi}2^{M}\varepsilon_{i_{j}}.$

Combining this inequality and $\varepsilon_{i_{j}}\to 0$ as $j\to\infty$ , we have

S_{M}(\varphi-\alpha_{\varphi})(\underline{x})=0.

For a $M$ -periodic sequence $\underline{x}$ , we define $\underline{w}^{(k)}$ by

w^{(k)}=(x_{0}\cdots x_{M-1})^{k}\underline{y}.

Notice that for $n_{k}=kM$ ,

S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{x})=0

and we get

	$\displaystyle S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})$	$\displaystyle=S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})-S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{x})$
		$\displaystyle\leq L_{\varphi}\sum_{j=0}^{n_{k}-1}d(\sigma^{j}(\underline{w}^{(k)}),\sigma^{j}(\underline{x}))$
		$\displaystyle\leq L_{\varphi}d(\underline{y},\underline{x}).$

Since $\underline{w}^{(k)}\in B(\underline{x},\underline{y},n_{k};2^{-n_{k}})$ for every $k\geq 1$ , we have $H_{\varphi}(\underline{x},\underline{y})\leq L_{\varphi}d(\underline{x},\underline{y})$ .

Next, we assume that $\{N_{i}\}$ is not bounded. Then we can take an increasing subsequence $\{N_{i_{j}}\}$ with $\max\{2^{-N_{i_{j}}},\varepsilon_{i_{j}}\}<\varepsilon_{j}$ and a sequence $\underline{z}^{(j)}$ satisfying

(5)

\displaystyle S_{N_{i_{j}}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})<\frac{\varepsilon}{2^{j}}

and $\underline{z}^{(j)}\in B(\underline{x},\underline{x},N_{i_{j}};\varepsilon_{i_{j}})$ for any $j\in\mathbb{N}$ . Set $\{M_{j}\}_{j\geq 0}$ by $M_{0}=0$ and $M_{j}=N_{i_{j}}$ . Set $\underline{w}^{(n)}$ by

\underline{w}^{(n)}=(z_{0}^{(1)}\cdots z^{(1)}_{M_{1}-1})(z_{0}^{(2)}\cdots z^{(2)}_{M_{2}-1})\cdots(z_{0}^{(n)}\cdots z^{(n)}_{M_{n}-1})\underline{y}.

Notice that for $j\leq n$

d(\sigma^{M_{j-1}+\cdots+M_{0}}(\underline{w}^{(n)}),\underline{z}^{(j)})\leq 2^{-N_{i_{j}}}<\varepsilon_{j}

since the first $M_{j}(=N_{i_{j}})$ coordinates of $\sigma^{M_{j-1}+\cdots M_{0}}\underline{w}^{(n)}$ coincide with that of $\underline{z}^{(j)}$ . Let $m_{k}=M_{1}+\cdots+M_{k}$ . For any $k$ ,

	$\displaystyle S_{m_{k}}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})=\sum_{j=1}^{k-1}S_{M_{j}}(\varphi-\alpha_{\varphi})(\sigma^{{M_{j-1}+\cdots+M_{0}}}(\underline{w}^{(k)}))$
	$\displaystyle=\sum_{j=1}^{k}S_{M_{j}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})$
	$\displaystyle\quad+\sum_{j=1}^{k}S_{M_{j}}(\varphi-\alpha_{\varphi})(\sigma^{M_{j-1}+\cdots+M_{0}}(\underline{w}^{(k)}))-S_{M_{j}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)}).$

Here by (5)

\sum_{j=1}^{k}S_{M_{j}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})\leq\sum_{j=1}^{k}\frac{\varepsilon}{2^{i}}<\varepsilon

for any $k$ . If $j<k$ ,

	$\displaystyle S_{M_{j}}(\varphi-\alpha_{\varphi})(\sigma^{{M_{j-1}+\cdots+M_{0}}}(\underline{w}^{(k)}))-S_{M_{j}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})$
	$\displaystyle=\sum_{i=0}^{M_{j}}\left(\varphi(\sigma^{i+{M_{j-1}+\cdots+M_{0}}}(\underline{w}^{(k)}))-\varphi(\sigma^{i}(\underline{z}^{(j)}))\right)$
	$\displaystyle\leq L_{\varphi}\sum_{i=0}^{M_{j}}d(\sigma^{i+M_{j-1}+\cdots+M_{0}}(\underline{w}^{(k)}),\sigma^{i}(\underline{z}^{(j)}))$
	$\displaystyle\leq L_{\varphi}\sum_{i=1}^{M_{j}}\frac{1}{2^{i}}d(\sigma^{M_{j}+M_{j-1}+\cdots+M_{0}}(\underline{w}^{(k)}),\sigma^{M_{j}}(\underline{z}^{(j)}))$
	$\displaystyle\leq L_{\varphi}d(\sigma^{M_{j}+M_{j-1}+\cdots+M_{0}}(\underline{w}^{(k)}),\sigma^{M_{j}}(\underline{z}^{(j)})$
	$\displaystyle\leq L_{\varphi}(d(\sigma^{M_{j}+M_{j-1}+\cdots+M_{0}}(\underline{w}^{(k)}),\underline{z}^{(j+1)})+d(\underline{z}^{(j+1)},\underline{x})+d(\underline{x},\sigma^{M_{j}}(\underline{z}^{(j)})))$
	$\displaystyle<L_{\varphi}(\varepsilon_{j+1}+\varepsilon_{i_{j+1}}+\varepsilon_{i_{j}})<2L_{\varphi}\varepsilon.$

If $j=k$ , Thus we get:

	$\displaystyle S_{M_{k}}(\varphi-\alpha_{\varphi})(\sigma^{{M_{k-1}+\cdots+M_{0}}}(\underline{w}^{(k)}))-S_{M_{k}}(\varphi-\alpha_{\varphi})(\underline{z}^{(k)})$
	$\displaystyle\leq L_{\varphi}\sum_{i=1}^{n}\frac{1}{2^{i}}d(\underline{y},\sigma^{M_{k}}(\underline{z}^{(k)}))$
	$\displaystyle\leq L_{\varphi}(d(\underline{x},\underline{y})+d(\underline{x},\sigma^{M_{k}}(\underline{z}^{(k)})))$
	$\displaystyle\leq L_{\varphi}d(\underline{x},\underline{y})+L_{\varphi}\varepsilon_{i_{k}}$
	$\displaystyle<L_{\varphi}d(\underline{x},\underline{y})+L_{\varphi}\varepsilon.$

Hence we get

\displaystyle S_{m_{k}}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})

\displaystyle<\varepsilon+3L_{\varphi}\varepsilon+L_{\varphi}d(\underline{x},\underline{y}).

Since $\underline{w}^{(k)}\in B(\underline{x},\underline{y},m_{k};\varepsilon_{i_{1}})\subset B(\underline{x},\underline{y},m_{k};\varepsilon)$ for every $k$ , we have

\displaystyle H_{\varphi}(\underline{x},\underline{y};\varepsilon)\leq(1+3L_{\varphi})\varepsilon+L_{\varphi}d(\underline{x},\underline{y}).

Letting $\varepsilon\to 0$ , we have

H_{\varphi}(\underline{x},\underline{y})\leq L_{\varphi}d(\underline{x},\underline{y}).

∎

Next, we prove the first half of Main Theorem 2.

Theorem 2.12 (Analogy of Theorem 4.1 in [BLL13], cf. Main Theorem 2).

For any $\underline{x}\in\Omega_{\varphi}$ , the map $X\ni\underline{y}\mapsto H_{\varphi}(\underline{x},\underline{y})$ is a Lipschitz calibrated subaction.

Proof.

Note that $H_{\varphi}(\underline{x},\cdot):X\to\mathbb{R}\cup\{+\infty\}$ does not take $+\infty$ by Theorem 2.11 and $\underline{x}\in\Omega_{\varphi}$ . We first check the Lipschitz property. Fix $\varepsilon>0$ . Take sequences $\{\underline{w}^{(n)}\},\{\underline{w}^{\prime(n)}\}\subset X$ and a monotone increasing sequence $\{N_{n}\}$ satisfying

	$\displaystyle\lim_{n\to\infty}S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{(n)})$	$\displaystyle=H_{\varphi}(\underline{x},\underline{y};\varepsilon),$
	$\displaystyle\underline{w}^{(n)}$	$\displaystyle\in B(\underline{x},\underline{y},N_{n};\varepsilon),$

and

	$\displaystyle\lim_{n\to\infty}S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{\prime(n)})$	$\displaystyle=H_{\varphi}(\underline{x},\underline{y}^{\prime};2\varepsilon),$
	$\displaystyle\underline{w}^{\prime(n)}$	$\displaystyle\in B(\underline{x},\underline{y}^{\prime},N_{n};2\varepsilon)$

respectively. Set

\underline{z}^{(n)}=w_{0}^{(n)}w_{1}^{(n)}\cdots w_{n-1}^{(n)}\underline{y}^{\prime}.

Since $\underline{z}^{(n)}\in B(\underline{x},\underline{y}^{\prime},N_{n};2\varepsilon)$ , (by replacing a subsequence if necessary) we take $N$ such that if $n\geq N$ ,

S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{z}^{({n})})\geq S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{\prime({n})})-\varepsilon.

Then we obtain

	$\displaystyle S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{(n)})$	$\displaystyle=S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{z}^{(n)})+\sum_{i=0}^{N_{n}-1}(\varphi(\sigma^{i}(\underline{w}^{(n)}))-\varphi(\sigma^{i}(\underline{z}^{(n)})))$
		$\displaystyle\geq S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{z}^{(n)})-L_{\varphi}d(\sigma^{N_{n}}(\underline{w}^{(n)}),y^{\prime})$
		$\displaystyle\geq S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{\prime(n)})-\varepsilon-L_{\varphi}(d(y,y^{\prime})+d(\sigma^{N_{n}}(\underline{w}^{(n)}),y))$
		$\displaystyle\geq S_{N_{n}}(\varphi-\alpha_{\varphi})(\underline{w}^{\prime(n)})-L_{\varphi}d(y,y^{\prime})-(L_{\varphi}+1)\varepsilon.$

Letting $n\to\infty$ and $\varepsilon\to 0$ , we have

\displaystyle H_{\varphi}(\underline{x},\underline{y^{\prime}})-H_{\varphi}(\underline{x},\underline{y})\leq L_{\varphi}(\underline{y},\underline{y^{\prime}}).

Since the opposite inequality can be obtained by swapping the roles of $\{\underline{w}^{(n)}\}$ and $\{\underline{w^{\prime}}^{(n)}\}$ , the map $X\ni\underline{y}\mapsto H_{\varphi}(\underline{x},\underline{y})$ is Lipschitz for any $\underline{x}\in\Omega_{\varphi}$ .

Next, we check the property of calibrated subaction. Fix $\varepsilon>0$ and we take a sequence $\underline{w}^{(k)}$ and a monotone increasing sequence $\{n_{k}\}$ such that

\underline{w}^{(k)}\in B(\underline{x},\underline{y},n_{k};\varepsilon)

satisfying

H_{\varphi}(\underline{x},\underline{y},\varepsilon)=\lim_{n_{k}\to\infty}S_{n_{k}}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)}).

Here, we can assume that $H_{\varphi}(\underline{x},\underline{y},\varepsilon)$ is finite since $\underline{x}\in\Omega_{\varphi}$ . Notice that

\underline{w}^{(k)}\in B(\underline{x},\sigma\underline{y},n_{k}+1;2\varepsilon).

Taking sufficiently large $N$ , for any $k\geq N$ ,

	$\displaystyle H_{\varphi}(\underline{x},\underline{y},\varepsilon)+\varepsilon$
	$\displaystyle>S_{n_{k}+1}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})-(\varphi(\sigma^{n_{k}}\underline{w}^{(k)})-\alpha_{\varphi})$
	$\displaystyle>S_{n_{k}+1}(\varphi-\alpha_{\varphi})(\underline{w}^{(k)})-(\varphi(\underline{y})-\alpha_{\varphi})-L_{\varphi}\varepsilon$
	$\displaystyle>H_{\varphi}(\underline{x},\sigma\underline{y},2\varepsilon)-\varepsilon-(\varphi(\underline{y})-\alpha_{\varphi})-L_{\varphi}\varepsilon$

Hence we get

H_{\varphi}(\underline{x},\sigma(\underline{y}))\leq H_{\varphi}(\underline{x},\underline{y})+\varphi(\underline{y})-\alpha_{\varphi}.

for any $\underline{y}\in X$ , and it implies

\displaystyle H_{\varphi}(\underline{x},{\underline{y}})\leq\min_{\hat{\underline{y}}\in\sigma^{-1}\{{\underline{y}}\}}\{H_{\varphi}(\underline{x},\hat{\underline{y}})+\varphi(\hat{\underline{y}})-\alpha_{\varphi}\}.

To show the opposite inequality, let $z\in[0,1]$ be a limit of a convergent subsequence $\{w^{(k_{j})}_{k_{j}-1}\}$ of $\{w^{(k)}_{k-1}\}\subset[0,1]$ . Letting $\underline{y^{\prime}}=z\underline{y}$ , for sufficiently large $j$ , we have

\underline{w}^{(k_{j})}\in B(\underline{x},\underline{y}^{\prime},k_{j}-1;2\varepsilon)

and

H_{\varphi}(\underline{x},\underline{y^{\prime}};2\varepsilon)\leq S_{k_{j}-1}(\varphi-\alpha_{\varphi})(\underline{w}^{(k_{j})})+\varepsilon.

Moreover, for large $j$ , we have

	$\displaystyle H_{\varphi}(\underline{x},\underline{y},\varepsilon)+\varepsilon$	$\displaystyle>S_{k_{j}-1}(\varphi-\alpha_{\varphi})(\underline{w}^{(k_{j})})+\varphi(\underline{y}^{\prime})-\alpha_{\varphi}-L_{\varphi}\varepsilon$
		$\displaystyle>H_{\varphi}(\underline{x},\underline{y}^{\prime},2\varepsilon)-\varepsilon+\varphi(\underline{y}^{\prime})-\alpha_{\varphi}-L_{\varphi}\varepsilon.$

Letting $\varepsilon\to 0$ we have

\displaystyle H_{\varphi}(\underline{x},\underline{y})

\displaystyle\geq H_{\varphi}(\underline{x},\underline{y^{\prime}})+\varphi(\underline{y^{\prime}})-\alpha_{\varphi}

\displaystyle\geq\min_{\hat{\underline{y}}\in\sigma^{-1}\{{\underline{y}}\}}\{H_{\varphi}(\underline{x},\hat{\underline{y}})+\varphi(\hat{\underline{y}})-\alpha_{\varphi}\},

which completes the proof. ∎

The next theorem is the second half of Main Theorem 2.

Theorem 2.13 (cf. Main Theorem 2).

For $\underline{x}$ and $\underline{y}\in\Omega_{\varphi}$ define $\underline{x}\sim\underline{y}$ by

\displaystyle H_{\varphi}(\underline{x},\underline{y})+H_{\varphi}(\underline{y},\underline{x})=0.

Then this is an euqivalence relation on $\Omega$ .

Before the proof of Theorem 2.13, we show the following lemma.

Lemma 2.14 (Analogy of Lemma 4.2 in [BLL13]).

For any $\underline{x},\underline{y},\underline{z}\in X$

(6)

\displaystyle H_{\varphi}(\underline{x},\underline{y})\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y}).

Proof.

We remark that both sides of (6) may become $+\infty$ .

Fix $\theta>0$ . Let $\varepsilon>0$ and $N\geq 1$ s.t. $2L_{\varphi}\varepsilon\leq\theta$ ,

	$\displaystyle H_{\varphi}(\underline{x},\underline{y})$	$\displaystyle\leq\inf_{n\geq N}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{w}):\underline{w}\in B(\underline{x},\underline{y},n;2\varepsilon)+\theta,$
	$\displaystyle H_{\varphi}(\underline{x},\underline{z})$	$\displaystyle\geq\inf_{n\geq N}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{w}):\underline{w}\in B(\underline{x},\underline{z},n;\varepsilon)\}-\theta$

and

\displaystyle H_{\varphi}(\underline{z},\underline{y})\geq\inf_{n\geq N}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{w}):\underline{w}\in B(\underline{z},\underline{y},n;\varepsilon)\}-\theta.

Then there exist $n_{1}\geq N$ and $\underline{w}^{(1)}\in B(\underline{x},\underline{z},n_{1};\varepsilon)$ s.t.

\displaystyle H_{\varphi}(\underline{x},\underline{z})\geq S_{n_{1}}(\varphi-\alpha_{\varphi})(\underline{w}^{(1)})-2\theta

and there exist $n_{2}\geq N$ and $\underline{w}^{(2)}\in B(\underline{z},\underline{y},n_{2};\varepsilon)$ s.t.

\displaystyle H_{\varphi}(\underline{z},\underline{y})\geq S_{n_{2}}(\varphi-\alpha_{\varphi})(\underline{w}^{(2)})-2\theta.

Let

\displaystyle\underline{w}=w^{(1)}_{0}\cdots w^{(1)}_{n_{1}-1}\underline{w}^{(2)},\quad\mbox{i.e.}\quad\underline{w}\in[w^{(1)}_{0}\cdots w^{(1)}_{n_{1}-1}]\cap\sigma^{-n_{1}}\{\underline{w}^{(2)}\}.

Then we have

	$\displaystyle S_{n_{1}+n_{2}}(\varphi-\alpha_{\varphi})(\underline{w})$
	$\displaystyle\leq S_{n_{1}}(\varphi-\alpha_{\varphi})(\underline{w}^{(1)})+S_{n_{2}}(\varphi-\alpha_{\varphi})(\underline{w}^{(2)})+L_{\varphi}\sum_{i=0}^{n_{1}-1}d(\sigma^{i}(\underline{w}),\sigma^{i}\underline{w}^{(1)})$
	$\displaystyle\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+L_{\varphi}\sum_{i=1}^{n_{1}}2^{-i}d(\sigma^{n_{1}}\underline{w},\sigma^{n_{1}}\underline{w}^{(1)})+4\theta$
	$\displaystyle\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+L_{\varphi}\left(d(\underline{w}^{(2)},\underline{z})+d(\underline{z},\sigma^{n_{1}}\underline{w}^{(1)})\right)+4\theta$
	$\displaystyle\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+2L_{\varphi}\varepsilon+4\theta$
	$\displaystyle\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+5\theta.$

Moreover $d(\sigma^{n_{1}+n_{2}}\underline{w},\underline{y})=d(\sigma^{n_{2}}\underline{w}^{(2)},\underline{y})<\varepsilon$ and

	$\displaystyle d(\underline{w},\underline{x})$	$\displaystyle\leq d(\underline{w},\underline{w}^{(1)})+d(\underline{w}^{(1)},\underline{x})$
		$\displaystyle\leq 2^{-n_{1}}d(\sigma^{n_{1}}\underline{w},\sigma^{n_{1}}\underline{w}^{(1)})+\varepsilon$
		$\displaystyle\leq 2^{-n_{1}}\left(d(\underline{w}^{(2)},\underline{z})+d(\underline{z},\sigma^{n_{1}}\underline{w}^{(1)})\right)+\varepsilon$
		$\displaystyle\leq\left(2^{-n_{1}+1}+1\right)\varepsilon\leq 2\varepsilon,$

which implies

	$\displaystyle H_{\varphi}(\underline{x},\underline{y})$	$\displaystyle\leq S_{n_{1}+n_{2}}(\varphi-\alpha_{\varphi})(\underline{w})+\theta$
		$\displaystyle\leq H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+6\theta.$

Since $\theta>0$ is arbitrary, we complete the proof. ∎

Proof of Theorem 2.13.

It suffices to show the transitive relation. Take $\underline{x},\underline{y}$ and $\underline{z}\in\Omega$ with $\underline{x}\sim\underline{y}$ and $\underline{y}\sim\underline{z}$ . By (6) we have

\displaystyle H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{x})

\displaystyle\leq H_{\varphi}(\underline{x},\underline{y})+H_{\varphi}(\underline{y},\underline{z})+H_{\varphi}(\underline{z},\underline{y})+H_{\varphi}(\underline{y},\underline{x})=0.

It also yields

\displaystyle H_{\varphi}(\underline{x},\underline{z})+H_{\varphi}(\underline{z},\underline{x})\geq H_{\varphi}(\underline{x},\underline{x})=0.

∎

Proof of Main Theorem 2.

It follows immediately from Theorem 2.12 and 2.13. ∎

At the end of this section, we present the invariance of the above equivalent classes.

Proposition 2.15.

For any $\underline{x}\in\Omega_{\varphi}$ we have

\displaystyle H_{\varphi}(\underline{x},\sigma(\underline{x}))+H_{\varphi}(\sigma(\underline{x}),\underline{x})=0.

In particular a equivalence class $[\underline{x}]$ of the relation satisfies $\sigma[\underline{x}]\subset[\underline{x}]$ .

Proof.

By Theorem 2.11 and Lemma 2.14 we have

\displaystyle H_{\varphi}(\underline{x},\sigma(\underline{x}))+H_{\varphi}(\sigma(\underline{x}),\underline{x})\geq H_{\varphi}(\underline{x},\underline{x})=0.

Thus we will show that it is non-negative. Since $H_{\varphi}(\underline{x},\cdot)$ is a subaction, we have

	$\displaystyle H_{\varphi}(\underline{x},\sigma(\underline{x}))$	$\displaystyle=\min_{\sigma(\underline{y})=\sigma(\underline{x})}\{H_{\varphi}(\underline{x},\underline{y})+\varphi(\underline{y})-\alpha_{\varphi}\}$
(7)			$\displaystyle\leq H_{\varphi}(\underline{x},\underline{x})+\varphi(\underline{x})-\alpha_{\varphi}=\varphi(\underline{x})-\alpha_{\varphi}.$

Let $\theta>0$ . Take $\displaystyle 0<\varepsilon<\theta/L_{\varphi}$ and $N\geq 1$ s.t.

\displaystyle H_{\varphi}(\sigma(\underline{x}),\underline{x})\leq\inf_{n\geq N}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z}):\underline{z}\in B(\sigma(\underline{x}),\underline{x},n;2\varepsilon)\}+\theta

and

\displaystyle H_{\varphi}(\underline{x},\underline{x})\geq\inf_{n\geq N+1}\{S_{n}(\varphi-\alpha_{\varphi})(\underline{z}):\underline{z}\in B(\underline{x},\underline{x},n;\varepsilon)\}-\theta.

Then there exist $n\geq N+1$ and $\underline{z}\in B(\underline{x},\underline{x},n;\varepsilon)$ s.t.

\displaystyle H_{\varphi}(\underline{x},\underline{x})\geq S_{n}(\varphi-\alpha_{\varphi})(\underline{z})-2\theta.

Then we have $\displaystyle d(\sigma(\underline{z}),\sigma(\underline{x}))=2\left(d(\underline{z},\underline{x})-\frac{|z_{0}-x_{0}|}{2}\right)\leq 2\varepsilon$ and

	$\displaystyle S_{n-1}(\varphi-\alpha_{\varphi})(\sigma(\underline{z}))$	$\displaystyle=S_{n}(\varphi-\alpha_{\varphi})(\underline{z})-\varphi(\underline{z})+\alpha_{\varphi}$
		$\displaystyle\leq H_{\varphi}(\underline{x},\underline{x})-\varphi(\underline{z})+\alpha_{\varphi}+2\theta$
		$\displaystyle=-\varphi(\underline{x})+\alpha_{\varphi}-\varphi(\underline{z})+\varphi(\underline{x})+2\theta$
		$\displaystyle\leq-\varphi(\underline{x})+\alpha_{\varphi}+L_{\varphi}d(\underline{x},\underline{z})+2\theta$
		$\displaystyle\leq-\varphi(\underline{x})+\alpha_{\varphi}+L_{\varphi}\varepsilon+2\theta$
		$\displaystyle\leq-\varphi(\underline{x})+\alpha_{\varphi}+3\theta.$

Combining (7), we have

\displaystyle H_{\varphi}(\sigma(\underline{x}),\underline{x})+H_{\varphi}(\underline{x},\sigma(\underline{x}))\leq+4\theta,

which complete the proof. ∎

3. Another characterization of the Aubry set

In this section, we describe another characterization of the Aubry set. We assume that the potential $\varphi:X\to\mathbb{R}$ is Lipschitz. A positive-semi orbit $\{\sigma^{n}(x)\}_{n\in\mathbb{N}_{0}}$ is said to be

•

$\varphi$ -semi-static: if for any non-negative integers $i<j$

\sum_{n=i}^{j-1}\Big{(}\varphi\circ\sigma^{n}(\underline{x})-\alpha_{\varphi}\Big{)}=S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x})).

•

$\varphi$ -static: if for any non-negative integers $i<j$

\sum_{n=i}^{j-1}\Big{(}\varphi\circ\sigma^{n}(\underline{x})-\alpha_{\varphi}\Big{)}=-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x})).

We call the sets

	$\displaystyle N_{\varphi}$	$\displaystyle:=\{\sigma^{k}(\underline{x})\in X\mid\{\sigma^{n}(\underline{x})\}_{n\in\mathbb{N}_{0}}\ \text{is}\ \varphi\text{-semi-static},k\in\mathbb{N}_{0}\},$
	$\displaystyle A_{\varphi}$	$\displaystyle:=\{\sigma^{k}(\underline{x})\in X\mid\{\sigma^{n}(\underline{x})\}_{n\in\mathbb{N}_{0}}\ \text{is}\ \varphi\text{-static},k\in\mathbb{N}_{0}\}$

as the $\varphi$ -semi-static set and the $\varphi$ -static set respectively. It is trivial that $N_{\varphi}$ and $A_{\varphi}$ are $\sigma$ -invariant since they are sets of positive semi-orbits. We will see that $A_{\varphi}\subset N_{\varphi}$ (Proposition 3.2), and that $A_{\varphi}$ (and hence $N_{\varphi}$ ) is not empty (Theorem 3.5). Moreover, by Proposition 2.7, we deduce that the $\varphi$ -static set $A_{\varphi}$ is closed and hence compact. We begin with the following.

Lemma 3.1.

For any $\underline{x},\underline{y},\underline{z}\in X$ , we have

\displaystyle S_{\varphi}(\underline{x},\underline{y})\leq S_{\varphi}(\underline{x},\underline{z})+S_{\varphi}(\underline{z},\underline{y}).

In particular,

\displaystyle S_{\varphi}(\underline{x},\underline{y})+S_{\varphi}(\underline{y},\underline{x})\geq 0

holds for all $\underline{x},\underline{y}\in X$ .

Proof.

We can prove the claim as in the proof of Lemma 2.14. Note that we do not assume that $N$ is sufficiently large in the discussion. ∎

Proposition 3.2.

For each $\varphi\in C(X)$ , it holds that $A_{\varphi}\subset N_{\varphi}$ .

Proof.

Fix $\underline{x}\in X$ . Since $\sigma^{i}(\underline{x})\in B(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}),j-i;\varepsilon)$ holds for any $\varepsilon>0$ , it is trivial that

\sum_{n=i}^{j-1}\Big{(}\varphi\circ\sigma^{n}(\underline{x})-\alpha_{\varphi}\Big{)}\geq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x})).

By Lemma 3.1, we have $S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))\geq-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))$ . Hence, for any $\underline{x}\in X$ , we have

\sum_{n=i}^{j-1}\Big{(}\varphi\circ\sigma^{n}(\underline{x})-\alpha_{\varphi}\Big{)}\geq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))\geq-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x})).

Therefore $\underline{x}\in A_{\varphi}$ implies $\underline{x}\in N_{\varphi}$ . ∎

Remark 3.3.

In the Aubry-Mather theory for Euler-Lagrange flows, the corresponding object of $A_{\varphi}$ (resp. $N_{\varphi}$ ) is called as the Aubry set (resp. the Mañé set), which looks different from our terminology “Aubry set” (Definition 2.3). In Theorem 3.5 below, we see that these notions are equivalent in our setting.

Before we state our main result in this section, we give the following lemma.

Lemma 3.4.

If $\underline{x}\in X$ satisfies $S_{\varphi}(\underline{x},\underline{x})=0$ , then

\displaystyle S_{\varphi}(\sigma(\underline{x}),\underline{x})\leq-\varphi(\underline{x})+\alpha_{\varphi}.

Proof.

When $\underline{x}$ is a fixed point of $\sigma$ , the Dirac measure at $\underline{x}$ must be the optimal measure for $\varphi$ and thus we have $\varphi(\underline{x})=\alpha_{\varphi}$ , which implies

S_{\varphi}(\sigma(\underline{x}),\underline{x})=S_{\varphi}(\underline{x},\underline{x})=0=-\varphi(\underline{x})+\alpha_{\varphi}.

Now we consider the case that $\underline{x}$ satisfies $\sigma(\underline{x})\neq\underline{x}$ . Fix $\varepsilon>0$ . Take $n^{(j)}\in\mathbb{N}$ and $\underline{z}^{(j)}\in B(\underline{x},\underline{x},n^{(j)};\varepsilon)$ s.t.

\lim_{j\to+\infty}S_{n^{(j)}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})=S_{\varphi}(\underline{x},\underline{x};\varepsilon)\leq S_{\varphi}(\underline{x},\underline{x})=0.

Since $\underline{z}^{(j)}\in B(\underline{x},\underline{x},n^{(j)};\varepsilon)$ , we have

d(\underline{x},\underline{z}^{(j)})<\varepsilon,\quad d(\sigma^{n_{j}}(\underline{z}^{(j)}),x)<\varepsilon.

By the property of the shift map, the inequality

d(\sigma(\underline{x}),\sigma(\underline{z}^{(j)}))<2\varepsilon

holds and thus we obtain

\sigma(\underline{z}^{(j)})\in B(\sigma(\underline{x}),\underline{x},n^{(j)}-1;2\varepsilon).

Note that $n^{(j)}$ is greater than 1 since $\sigma(\underline{x})\neq\underline{x}$ . Moreover, the Lipschitz continuity of $\varphi$ implies that

|\varphi(\underline{x})-\varphi(\underline{z}^{(j)})|\leq L_{\varphi}d(\underline{x},\underline{z}^{(j)}).

We compute

	$\displaystyle S_{\varphi}(\sigma(\underline{x}),\underline{x};2\varepsilon)$	$\displaystyle\leq\lim_{j\to+\infty}S_{n^{(j)}-1}(\varphi-\alpha_{\varphi})(\sigma(\underline{z}^{(j)}))$
		$\displaystyle=\lim_{j\to+\infty}\Big{(}S_{n^{(j)}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})-\varphi(\underline{z}^{(j)})+\alpha_{\varphi}\Big{)}$
		$\displaystyle\leq\lim_{j\to+\infty}\Big{(}S_{n^{(j)}}(\varphi-\alpha_{\varphi})(\underline{z}^{(j)})-\varphi(x)+L_{\varphi}d(\underline{x},\underline{z}^{(j)})+\alpha_{\varphi}\Big{)}$
		$\displaystyle\leq-\varphi(\underline{x})+L_{\varphi}\varepsilon+\alpha_{\varphi},$

which yields

S_{\varphi}(\sigma(\underline{x}),\underline{x})\leq-\varphi(\underline{x})+\alpha_{\varphi}.

∎

Theorem 3.5 (cf. Main Theorem 1).

For each Lipschitz function $\varphi:X\to\mathbb{R}$ , we have

\displaystyle A_{\varphi}=\Omega_{\varphi}.

Proof.

We first prove that $\underline{x}\in A_{\varphi}$ implies $S_{\varphi}(\underline{x},\underline{x})=0$ . For each $\underline{x}\in A_{\varphi}$ , we have

\displaystyle S_{n}(\varphi-\alpha_{\varphi})(\underline{x})+S_{\varphi}(\sigma^{n}(\underline{x}),\underline{x})=0

for $n\in\mathbb{N}$ by the definition of $\varphi$ -static set. Moreover, by the definition of the Mañé potential, it holds that

S_{n}(\varphi-\alpha_{\varphi})(\underline{x})\geq S_{\varphi}(\underline{x},\sigma^{n}(\underline{x})).

Therefore, using the triangle inequality for the Mañé potential (Lemma 3.1), we obtain

	$\displaystyle 0$	$\displaystyle=S_{n}(\varphi-\alpha_{\varphi})(\underline{x})+S_{\varphi}(\sigma^{n}(\underline{x}),\underline{x})$
		$\displaystyle\geq S_{\varphi}(\underline{x},\sigma^{n}(\underline{x}))+S_{\varphi}(\sigma^{n}(\underline{x}),\underline{x})$
		$\displaystyle\geq S_{\varphi}(\underline{x},\underline{x})\geq 0.$

Note that in the last inequality we use the fact that $S_{\varphi}(\underline{y},\underline{y})\geq 0$ for any $\underline{y}\in X$ . Thus we have $S_{\varphi}(\underline{x},\underline{x})=0$ .

Next we see that $S_{\varphi}(\underline{x},\underline{x})=0$ implies $x\in A_{\varphi}$ . Assume that $S_{\varphi}(\underline{x},\underline{x})=0$ . Since each $x\in\Omega_{\varphi}$ satisfies

H_{\varphi}(\underline{x},\sigma(\underline{x}))+H_{\varphi}(\sigma(\underline{x}),\underline{x})=0,

we obtain

\displaystyle 0\leq H_{\varphi}(\sigma(\underline{x}),\sigma(\underline{x}))\leq H_{\varphi}(\underline{x},\sigma(\underline{x}))+H_{\varphi}(\sigma(\underline{x}),\underline{x})=0

By the equivalence between $H_{\varphi}(\sigma(\underline{x}),\sigma(\underline{x}))=0$ and $S_{\varphi}(\sigma(\underline{x}),\sigma(\underline{x}))=0$ , we conclude that $S_{\varphi}(\sigma(\underline{x}),\sigma(\underline{x}))=0$ . Repeating the same discussion, we obtain $S_{\varphi}(\sigma^{k}(\underline{x}),\sigma^{k}(\underline{x}))=0$ for $k\in\mathbb{N}_{0}$ .

By Lemma 3.4, we have

S_{\varphi}(\sigma(\underline{x}),\underline{x}))\leq-\varphi(\underline{x})+\alpha_{\varphi}.

Trivially it holds that

S_{\varphi}(\underline{x},\sigma(\underline{x}))\leq\varphi(\underline{x})-\alpha_{\varphi},

since $\underline{x}\in B(\underline{x},\sigma(\underline{x}),n=1;\varepsilon)$ holds for any $\varepsilon>0$ and thus we obtain

S_{\varphi}(\underline{x},\sigma(\underline{x}))+S_{\varphi}(\sigma(\underline{x}),\underline{x})\leq 0.

Combining Lemma 3.1, we see that

S_{\varphi}(\underline{x},\sigma(\underline{x}))+S_{\varphi}(\sigma(\underline{x}),\underline{x})=0

and deduce that

S_{\varphi}(\underline{x},\sigma(\underline{x}))=\varphi(\underline{x})-\alpha_{\varphi},\quad S_{\varphi}(\sigma(\underline{x}),\underline{x})=-\varphi(\underline{x})+\alpha_{\varphi}.

Similarly, from the identities $S_{\varphi}(\sigma^{k}(\underline{x}),\sigma^{k}(\underline{x}))=0$ for $k\in\mathbb{N}_{0}$ , we have

	$\displaystyle S_{\varphi}(\sigma^{k}(\underline{x}),\sigma^{k+1}(\underline{x}))$	$\displaystyle=\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi},$
	$\displaystyle S_{\varphi}(\sigma^{k+1}(\underline{x}),\sigma^{k}(\underline{x}))$	$\displaystyle=-\varphi(\sigma^{k}(\underline{x}))+\alpha_{\varphi}$

for $k\in\mathbb{N}_{0}$ . Take arbitrary non-negative integers $i,j$ with $i<j$ . From the triangle inequality for the Mañé potential (Lemma 3.1),

S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))\leq\sum_{k=i}^{j-1}S_{\varphi}(\sigma^{k+1}(\underline{x}),\sigma^{k}(\underline{x}))=-\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}

holds and thus we obtain

\displaystyle\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}\leq-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))\leq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x})).

Since

S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))\leq\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}

trivially holds, we deduce that

\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}=-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))=S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x})),

which yields that $\underline{x}\in A_{\varphi}$ . ∎

Proof of Main Theorem 1.

It follows immediately from Theorem 2.11 and 3.5. ∎

Lastly, we show the relationship between Lipschitz subactions and $A_{\varphi}$ .

Proposition 3.6.

Let $u$ be a Lipschitz subaction of a Lipschitz function $\varphi:X\to\mathbb{R}$ . If $\underline{x}\in A_{\varphi}$ , then

(8)

\displaystyle u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))=\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}

for all $k\in\mathbb{N}_{0}$ . Conversely, if (8) holds for each $k\in\mathbb{N}_{0}$ , then $\underline{x}\in N_{\varphi}$ .

Proof.

Assume that $\underline{x}\in A_{\varphi}$ . Since $\underline{x}\in A_{\varphi}\subset N_{\varphi}$ , we have

\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}=-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))=S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))

for all non-negative integers $i<j$ . By Lemma 2.6, we obtain

u(\sigma^{j}(\underline{x}))-u(\sigma^{i}(\underline{x}))\leq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))

and

-S_{\varphi}(\sigma^{j}(\underline{x}),\sigma^{i}(\underline{x}))\leq u(\sigma^{j}(\underline{x}))-u(\sigma^{i}(\underline{x})).

Therefore, it holds that

\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}=u(\sigma^{j}(\underline{x}))-u(\sigma^{i}(\underline{x})).

We rewrite

u(\sigma^{j}(\underline{x}))-u(\sigma^{i}(\underline{x}))=\sum_{k=i}^{j-1}\big{(}u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))\big{)}

and compute

(9)

\displaystyle\sum_{k=i}^{j-1}\Big{\{}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}-\big{(}u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))\big{)}\Big{\}}=0.

Since $u$ is a calibrated subaction of $\varphi$ , we have

\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}-\big{(}u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))\big{)}\geq 0,\quad k\in\mathbb{N}_{0}

and thus we deduce that

\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}-\big{(}u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))\big{)}=0,\quad k\in\mathbb{N}_{0}

from (9).

Conversely, assume that (8) holds for each $k\in\mathbb{N}_{0}$ . Then we have

	$\displaystyle\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}$	$\displaystyle=\sum_{k=i}^{j-1}\big{(}u(\sigma^{k+1}(\underline{x}))-u(\sigma^{k}(\underline{x}))\big{)}$
		$\displaystyle=u(\sigma^{j}(\underline{x}))-u(\sigma^{i}(\underline{x}))\leq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))$

by Lemma 2.6. Since $\sum_{k=i}^{j-1}\big{(}\varphi(\sigma^{k}(\underline{x}))-\alpha_{\varphi}\big{)}\geq S_{\varphi}(\sigma^{i}(\underline{x}),\sigma^{j}(\underline{x}))$ is trivial, we see that $\underline{x}\in N_{\varphi}$ . ∎

4. Variational method applied to the Aubry set

In this section, we consider the case that the potential function $\varphi:X\to\mathbb{R}$ is 2-locally constant, i.e.,

\varphi(\underline{x})=h(x_{0},x_{1}),\quad\underline{x}=x_{0}x_{1}x_{2}\ldots,

for some Lipschitz continuous function $h:[0,1]^{2}\to\mathbb{R}$ under suitable assumptions (see below for the precise assumptions for $h$ ). In this case, we can obtain much more explicit information about the elements in the Mather set and the Aubry set for $\varphi$ by using variational techniques developed in [Ban88, Yu22].

Firstly, the following proposition is easily shown:

Proposition 4.1.

Let $\varphi:X\to\mathbb{R}$ be a function depending on only the first two coordinates, i.e., for all $\underline{x}=x_{0}x_{1}x_{2}\ldots$ , $\varphi(\underline{x})=\varphi(x_{0},x_{1})$ . Then $\varphi$ is Lipschitz continuous as a function on $X$ with respect to the metric $d$ on $X$ if and only if $\varphi$ is Lipschitz continuous as a function on $[0,1]^{2}$ with respect to the Euclidian metric $d_{\mathbb{R}^{2}}$ on $[0,1]^{2}$ .

Proof.

Suppose that there exists $L_{\varphi}>0$ such that $|\varphi(\underline{x})-\varphi(\underline{y})|\leq L_{\varphi}d(\underline{x},\underline{y})$ for all $\underline{x},\underline{y}\in X$ . Taking arbitrary $x_{0},x_{1},y_{0},y_{1}\in[0,1]$ , we have

	$\displaystyle\|\varphi(x_{0},x_{1})-\varphi(y_{0},y_{1})\|$	$\displaystyle=\|\varphi(x_{0}x_{1}0^{\infty})-\varphi(y_{0}y_{1}0^{\infty})\|$
		$\displaystyle\leq L_{\varphi}(\|x_{0}-y_{0}\|+\|x_{1}-y_{1}\|/2)$
	$\displaystyle L_{\varphi}\sqrt{2}d_{\mathbb{R}^{2}}((x_{0},x_{1}),(y_{0},y_{1})).$

Conversely, suppose that there exists $L_{\varphi}>0$ such that $|\varphi(x_{0},x_{1})-\varphi(y_{0},y_{1})|\leq L_{\varphi}d_{\mathbb{R}^{2}}((x_{0},x_{1}),(y_{0},y_{1}))$ for all $(x_{0},x_{1}),(y_{0},y_{1}))\in[0,1]^{2}$ . Then

	$\displaystyle\|\varphi(\underline{x})-\varphi(\underline{y})\|$	$\displaystyle=\|\varphi(x_{0},x_{1})-\varphi(y_{0},y_{1})\|$
		$\displaystyle\leq L_{\varphi}d_{\mathbb{R}^{2}}((x_{0},x_{1}),(y_{0},y_{1}))$
		$\displaystyle\leq 2L_{\varphi}(\|x_{0}-y_{0}\|+\|x_{1}-y_{1}\|/2)\leq 2L_{\varphi}d(\underline{x},\underline{y}),$

which complete the proof. ∎

Remark 4.2.

For a subshift of finite type with a finite set of symbols, locally constant functions are always Lipschitz continuous with respect to a natural metric on its symbolic space. However, for the symbolic dynamics with uncountable symbols $[0,1]$ , locally constant functions are not always Lipschitz continuous with respect to the metric $d$ on $X$ .

Hereafter, we always assume $(H_{3})$ and $(H_{4})$ for the Lipschitz continuous function $\varphi(\underline{x})=h(x_{0},x_{1})$ , where the assumptions $(H_{3})$ and $(H_{4})$ are defined by:

$(H_{3})$

If $\xi_{1}<\xi_{2}$ and $\eta_{1}<\eta_{2}$ , then

$h(\xi_{1},\eta_{1})+h(\xi_{2},\eta_{2})<h(\xi_{1},\eta_{2})+h(\xi_{2},\eta_{1}).$
$(H_{4})$

If both $(x_{-1},x_{0},x_{1})$ and $(x^{{}^{\prime}}_{-1},x_{0},x^{{}^{\prime}}_{1})$ with $(x_{-1},x_{0},x_{1})\neq(x^{{}^{\prime}}_{-1},x_{0},x^{{}^{\prime}}_{1})$ are minimal, then

$(x_{-1}-x^{{}^{\prime}}_{-1})(x_{1}-x^{{}^{\prime}}_{1})<0.$

Here, we give the definition of the word minimal :

Definition 4.3 (Minimal).

Fix $k,l\in\mathbb{N}_{0}$ with $k<l$ arbitrarily. A finite word $\{x_{i}\}_{i=k}^{l}$ is said to be minimal if, for any $\{y_{i}\}_{i=k}^{l}$ with $y_{k}=x_{k}$ and $y_{l}=x_{l}$ , we have:

\sum_{i=k}^{l-1}h(x_{i},x_{i+1})\leq\sum_{i=k}^{l-1}h(y_{i},y_{i+1}).

Moreover, an infinite word $\{x_{i}\}_{i\in\mathbb{N}_{0}}$ is said to be minimal if $\{x_{i}\}_{i=k}^{l}$ is minimal for any $k,l$ with $k<l$ .

Remark 4.4.

We state some remarks about the settings mentioned above.

(1)

In $(H_{3})$ , the equality $h(\xi_{1},\eta_{1})+h(\xi_{2},\eta_{2})=h(\xi_{1},\eta_{2})+h(\xi_{2},\eta_{1})$ holds if $\xi_{1}=\xi_{2}$ or $\eta_{1}=\eta_{2}$ .

(2)

The assumptions $(H_{3})$ and $(H_{4})$ hold if $h$ satisfies the twist condition $D_{1}D_{2}h<0$ . In fact, for $\xi_{1}<\xi_{2}$ and $\eta_{1}<\eta_{2}$ , we have:

	$\displaystyle 0$	$\displaystyle>\int_{\eta_{1}}^{\eta_{2}}\int_{\xi_{1}}^{\xi_{2}}D_{1}D_{2}H(x,y)dxdy$
		$\displaystyle=\int_{\eta_{1}}^{\eta_{2}}D_{2}H(\xi_{2},y)-D_{2}H(\xi_{1},y)dy$
		$\displaystyle=H(\xi_{1},\eta_{1})+H(\xi_{2}.\eta_{2})-H(\xi_{1},\eta_{2})-H(\xi_{2},\eta_{1}).$

This inequality implies $(H_{3})$ . Next, we show $(H_{4})$ . Suppose that both $(x_{-1},x_{0},x_{1})$ and $(x^{\ast}_{-1},x_{0},x^{\ast}_{1})$ are minimal and

(x_{-1},x_{0},x_{1})\neq(x^{\ast}_{-1},x_{0},x^{\ast}_{1}).

Clearly,

	$\displaystyle D_{1}H(x_{0},x_{1})+D_{2}H(x_{-1},x_{0})$	$\displaystyle=0,and$
	$\displaystyle D_{1}H(x_{0},x_{1}^{\ast})+D_{2}H(x^{\ast}_{-1},x_{0})$	$\displaystyle=0.$

Since $D_{1}D_{2}H<0$ , $D_{1}H(x,y)$ is monotonically decreasing with respect to $y$ , and $D_{2}H(x,y)$ is monotonically decreasing with respect to $x$ . Hence, if two minimal segment satisfies $x_{-1}-x_{-1}^{\ast}<0$ and $x_{1}-x_{1}^{\ast}<0$ , then

\displaystyle D_{1}H(x_{0},x_{1})+D_{2}H(x_{-1},x_{0})>D_{1}H(x_{0},x_{1}^{\ast})+D_{2}H(x^{\ast}_{-1},x_{0}),

which is contradiction.

(3)
The above notations and labels are derived from [Ban88]. In his paper, $h$ is considered as a fuction on $\mathbb{R}^{2}$ satisfying $(H_{1})-(H_{4})$ , where $(H_{1})$ and $(H_{2})$ are given by:
- $(H_{1})$
  
  $h(\xi,\eta)=h(\xi+1,\eta+1)$ for all $(\xi,\eta)\in\mathbb{R}^{2}$ , and
- $(H_{2})$
  
  $\displaystyle{\lim_{|\eta|\to\infty}h(\xi,\xi+\eta)=\infty}$ uniformly in $\xi$ .
Note that the condition $(H_{2})$ holds if $h$ satisfies $(H_{1})$ and the twist condition. For the proof, integrate $D_{2}D_{1}h$ over the triangular region bounded by three points $(\xi,\xi)$ , $(\xi,\xi+\eta)$ , and $(\xi+\eta,\xi+\eta)$ . In addition, the fact that the differentiability of $h$ is no longer needed is useful when considering geodesics on $\mathbb{T}^{2}$ , for example (see Section $6$ in [Ban88] for the detail).

Let $h^{\ast}=\min_{x\in[0,1]}h(x,x)$ . Set $X(n){(\subset X=[0,1]^{\mathbb{N}_{0}})}$ and $\mathrm{m}\ {(\subset[0,1])}$ by

X(n)=\{\underline{x}\in X\mid x_{i}=x_{n+i}\ \text{for all}\ i\in\mathbb{N}_{0}\},

and

\mathrm{m}=\{a\in[0,1]\mid h(a,a)=h^{\ast}\}.

Since $h$ is non-constant by $(H_{3})$ and $h$ is continuous, the set $\mathrm{m}$ is nonempty, compact and $\mathrm{m}\subsetneq[0,1]$ . Firstly, we introduce the result of [Ban88] and [Yu22].

Lemma 4.5.

Fix $n\in\mathbb{N}$ . Then $\displaystyle{\sum_{i=0}^{n-1}(h(x_{i},x_{i+1})-h^{\ast})\geq 0}$ for any $\underline{x}\in X(n)$ . Moreover, the equality is true if and only if there exists $a\in\mathrm{m}$ satisfying $\underline{x}_{i}=a$ for $i=0,\cdots,n-1$ .

This claim is essentially the same as Lemma 2.5 of [Yu22]. The proof is almost the same as the proof of the case $p=0$ in Theorem 3.3 of [Ban88].

Proof of Lemma 4.5.

Fix $n\in\mathbb{N}$ arbitrarily. It is easily seen that if $\underline{x}^{\ast}$ is a minimizer in $X(n)$ , i.e., satisfies:

S_{n}\varphi(\underline{x}^{\ast})=\min_{\underline{x}\in X(n)}S_{n}\varphi(\underline{x}),

so is $\sigma^{k}(\underline{x}^{\ast})$ for any $k\in\mathbb{N}$ . Set $m$ and $M$ with $0\leq m,M\leq n-1$ for each $x\in X(n)$ by:

(10)

\displaystyle x_{m}=\min_{0\leq i\leq n-1}x_{i},\ x_{M}=\max_{0\leq i\leq n-1}x_{i}

For the proof of the claim, it suffices to show the following claim:

Claim 4.6.

$x_{m}=x_{M}$ if $\underline{x}$ is a minimizer in $X(n)$ .

Suppose that the claim is failed, i.e., $\underline{x}$ is a minimizer in $X(n)$ and $x_{m}<x_{M}$ . We consider only the case $m=0$ . The other cases can be shown in a similar way.

Firstly, we introduce the definition of cross. We say the segments $\{\xi_{i}\}_{0}^{n-1}$ and $\{\eta_{i}\}_{0}^{n-1}$ cross if $(\xi_{l}-\eta_{l})(\xi_{l+1}-\eta_{l+1})\leq 0$ for some $l\in\{0,\ldots,n-1\}$ . Let $\underline{y}=\sigma^{M}(\underline{x})$ , i.e., $y_{i}=x_{i+M}$ for $i\in\mathbb{N}_{0}$ . Note that $\{x_{i}\}_{0}^{n-1}$ and $\{y_{i}\}_{0}^{n-1}$ cross at least once because if not we have $x_{i}\leq y_{i}$ for all $i\in\{1,\ldots,n-1\}$ and thus the inequality $x_{0}<y_{0}$ yields $\sum_{j=0}^{n-1}x_{j}<\sum_{j=0}^{n-1}y_{j}$ , but the periodicity of $\underline{x}$ and $\underline{y}$ yields $\sum_{j=0}^{n-1}y_{i}=\sum_{j=0}^{n-1}x_{i+M}=\sum_{j=0}^{n-1}x_{i}$ .

Suppose that $\{x_{i}\}_{0}^{n-1}$ and $\{y_{i}\}_{0}^{n-1}$ cross only once. The other cases can be shown by repeating the following discussion. Let $l^{*}$ be the largest number among $\{0,\ldots,n-1\}$ such that $x_{k}\leq y_{k}$ for all $k\in\{1,\ldots,l^{*}\}$ . Then we have $x_{l^{*}}\leq y_{l^{*}}$ and $x_{l^{*}+1}>y_{l^{*}+1}$ . Define $\underline{w},\underline{z}\in X(n)$ by $w_{i}=\min\{x_{i},y_{i}\},z_{i}=\max\{x_{i},y_{i}\}$ for all $i\in\mathbb{N}_{0}$ . When $x_{l^{*}}\neq y_{l^{*}}$ , by $(H_{3})$ , $x_{l^{*}}<y_{l^{*}}$ and $y_{l^{*}+1}<x_{l^{*}+1}$ give

	$\displaystyle h(w_{l^{}},w_{l^{}+1})+h(z_{l^{}},z_{l^{}+1})$	$\displaystyle=h(x_{l^{}},y_{l^{}+1})+h(y_{l^{}},x_{l^{}+1})$
		$\displaystyle<h(x_{l^{}},x_{l^{}+1})+h(y_{l^{}},y_{l^{}+1}).$

As a result, it holds that:

S_{n}\varphi(\underline{w})+S_{n}\varphi(\underline{z})<S_{n}\varphi(\underline{x})+S_{n}\varphi(\sigma^{M}(\underline{x}))=2\min_{\underline{x}^{\prime}\in X(n)}S_{n}\varphi(\underline{x}^{\prime}).

Therefore, at least one of the following inequalities holds:

S_{n}\varphi(\underline{w})<\min_{\underline{x}^{\prime}\in X(n)}S_{n}\varphi(\underline{x}^{\prime})\quad\text{or}\quad S_{n}\varphi(\underline{z})<\min_{\underline{x}^{\prime}\in X(n)}S_{n}\varphi(\underline{x}^{\prime}),

which is a contradiction (note that $\underline{w},\underline{z}\in X(n)$ ). When $x_{l^{*}}=y_{l^{*}}$ , we have $(x_{l^{*}-1}-y_{l^{*}-1})(x_{l^{*}+1}-y_{l^{*}+1})\leq 0$ , but the equality cannot occur by $(H_{4})$ and the minimality of $\underline{x}$ and $\underline{y}$ . Thus $x_{l^{*}-1}<y_{l^{*}-1}$ and $x_{l^{*}+1}>y_{l^{*}+1}$ hold and we have:

	$\displaystyle h(w_{l^{}-1},w_{l^{}})+h(z_{l^{}-1},z_{l^{}})$	$\displaystyle=h(x_{l^{}-1},x_{l^{}})+h(y_{l^{}},y_{l^{}}),\ \text{and}$
	$\displaystyle h(w_{l^{}},w_{l^{}+1})+h(z_{l^{}},z_{l^{}+1})$	$\displaystyle=h(y_{l^{}},y_{l^{}+1})+h(x_{l^{}},x_{l^{}+1}),$

as seen in Remark 4.4 (1). Therefore, we obtain

S_{n}\varphi(\underline{w})+S_{n}\varphi(\underline{z})=S_{n}\varphi(\underline{x})+S_{n}\varphi(\sigma^{M}(\underline{x}))=2\min_{\underline{x}^{\prime}\in X(n)}S_{n}\varphi(\underline{x}^{\prime}).

This implies that both $\underline{w}$ and $\underline{z}$ are also minimal, which is a contradiction for $(H_{4})$ . ∎

Using Lemma 4.5, we can easily show a part of our main theorem.

Lemma 4.7 (cf. Main Theorem 3. (1)).

$\alpha_{\varphi}=h^{\ast}$

Proof.

The continuity of $h$ and Lemma 4.5 imply that for any $\underline{x}\in X$ ,

	$\displaystyle\frac{1}{n}S_{n}\varphi(\underline{x})$	$\displaystyle=\frac{1}{n}(h(x_{0},x_{1})+\cdots+h(x_{n-1},x_{0})+h(x_{n-1},x_{n})-h(x_{n-1},x_{0}))$
		$\displaystyle\geq\frac{1}{n}(h(x_{0},x_{1})+\cdots+h(x_{n-1},x_{0}))-\frac{1}{n}(h_{\max}-h_{\min})$
		$\displaystyle\geq h^{\ast}-\frac{1}{n}(h_{\max}-h_{\min}).\$

By $\eqref{eq:Jenkinson}$ in Section 1, we have

\alpha_{\varphi}=\inf_{\underline{x}\in X}\liminf_{n\to\infty}\frac{1}{n}S_{n}\varphi(\underline{x})\geq h^{\ast}.

Moreover, taking $\underline{x}^{\ast}=a^{\infty}$ for some $a\in\mathrm{m}$ , we obtain

\displaystyle\frac{1}{n}S_{n}\varphi(\underline{x}^{\ast})=h^{\ast},

which implies that $\alpha_{\varphi}=h^{\ast}.$ ∎

Applying Lemma 4.5 and 4.7, we immediately obtain an explicit formula for optimizing periodic measures.

Theorem 4.8 (cf. Main Theorem 3. (2)).

Let $\varphi:X\to\mathbb{R}$ be a 2-locally constant function $\varphi(\underline{x})=h(x_{0},x_{1})$ where $h:[0,1]^{2}\to\mathbb{R}$ is a Lipschitz continuous function on $[0,1]^{2}$ with $(H_{3})$ and $(H_{4})$ . Then

\mathcal{M}_{{\rm min}}(\varphi)\cap\mathcal{M}^{\mathrm{p}}=\{\delta_{a^{\infty}}\mid a\in\mathrm{m}\},

where $\mathcal{M}^{\mathrm{p}}$ stands for the set of invariant probability measures supported on a single periodic orbit.

Next, we give an explicit characterization for elements in the Aubry set of $\varphi$ . Consider the distance between a point $x\in[0,1]$ to the closed set $\mathrm{m}$ :

d_{\mathbb{R}}(x,\mathrm{m})=\inf_{a\in\mathrm{m}}|x-a|.

The following lemma plays a key role in our statement.

Lemma 4.9 (A slightly extended version of Lemma 2.7 of [Yu22]).

Set

\phi(\delta)=\inf_{n\in\mathbb{N}}\phi(\delta;n)

where

\phi(\delta;n)=\inf\{S_{n}(\varphi-\alpha_{\varphi})(\underline{x})\mid\underline{x}\in X(n),\ \max_{0\leq i\leq n-1}d_{\mathbb{R}}({x}_{i},\mathrm{m})\geq\delta)\}.

Then $\phi(\delta)>0$ if $\delta>0$ .

Remark 4.10.

Let

X(n;\delta)=\{\underline{x}\in X(n)|\max_{0\leq i\leq n-1}d_{\mathbb{R}}({x}_{i},\mathrm{m})\geq\delta\}.

Then $\phi(\delta)$ should be defined as $+\infty$ if $X(n;\delta)=\emptyset$ . By the definition, if $\delta^{\prime}<\delta$ then $X(n;\delta^{\prime})\supset X(n;\delta)$ and $\phi(\delta^{\prime})<\phi(\delta)$ .

Proof of Lemma 4.9.

Lemma 2.7 in [Yu22] corresponds to the case of $\#\mathrm{m}=2$ , i.e., $\mathrm{m}=\{u_{0},u_{1}\}$ for some $u_{0},u_{1}\in[0,1]$ and

d_{\mathbb{R}}({x}_{i},\mathrm{m})=\min_{j=0,1}|x_{i}-u_{j}|,

but our proof is almost the same as his proof. It is sufficient to prove that for $\delta>0$ ,

(1)

$\phi(\delta;1)>0$ , and
(2)

$\phi(\delta;n)\geq\phi(\delta;1)$ for all $n\in\mathbb{N}$ .

The first claim is clear since $\underline{x}$ is given by $x_{0}^{\infty}$ for some $x_{0}\not\in\mathrm{m}$ . We show the second using induction. The case of $n=1$ is trivial. Assume that it holds if $n\leq m-1$ . Take any $\underline{x}\in X(m;\delta)$ . When $x_{j}\neq x_{0}$ for any $j\in\{1,\ldots,m-1\}$ , there exists an integer $k\in\{1,\ldots,m-1\}$ such that

(x_{k}-x_{k-1})(x_{k}-x_{k+1})\geq 0

since $x_{0}=x_{m}$ . From this inequality, $(H_{3})$ , and Remark 4.4 (1), we have

h(x_{k},x_{k})+h(x_{k-1},x_{k+1})\leq h(x_{k-1},x_{k})+h(x_{k},x_{k+1}).

Set

\underline{u}=x_{k}^{\infty},\qquad\underline{v}=(x_{0}\cdots x_{k-1}x_{k+1}\cdots x_{m-1})^{\infty}.

Then

S_{m}(\varphi-\alpha_{\varphi})(\underline{x})\geq S_{1}(\varphi-\alpha_{\varphi})(\underline{u})+S_{m-1}(\varphi-\alpha_{\varphi})(\underline{v}).

Clearly, $\underline{u}\in X(1)$ and $\underline{v}\in X(m-1)$ hold and thus we obtain the following two inequalities by Lemma 4.5:

S_{1}(\varphi-\alpha_{\varphi})(\underline{u})\geq 0,\qquad S_{m-1}(\varphi-\alpha_{\varphi})(\underline{v})\geq 0.

Moreover, at least one of

d_{\mathbb{R}}({x}_{k},\mathrm{m})\geq\delta

and

\max_{i\in[0,m]\backslash\{k\}}d_{\mathbb{R}}({x}_{i},\mathrm{m})\geq\delta

is true, which yields at least one of the following inequalities:

S_{1}(\varphi-\alpha_{\varphi})(\underline{u})\geq\phi(\delta;1),\qquad S_{m-1}(\varphi-\alpha_{\varphi})(\underline{v})\geq\phi(\delta;m-1).

By the assumption of induction $\phi(\delta;n)\geq\phi(\delta;1)>0$ for $n\in\{1,\ldots,m-1\}$ , we have

\displaystyle S_{m}(\varphi-\alpha_{\varphi})(\underline{x})

\displaystyle\geq S_{1}(\varphi-\alpha_{\varphi})(\underline{u})+S_{m-1}(\varphi-\alpha_{\varphi})(\underline{v})\geq\phi(\delta;1)>0.

When there exists $j$ such that $x_{j}=x_{0}$ , then we get

\displaystyle S_{m}(\varphi-\alpha_{\varphi})(\underline{x})\geq\phi(\delta;j)+\phi(\delta;m-j)\geq 2\phi(\delta;1)>0.

The proof is complete. ∎

Using these lemmata, we can show:

Theorem 4.11 (cf. Main Theorem 3. (3)).

$\Omega_{\varphi}\subset\mathrm{m}^{\mathbb{N}_{0}}$ .

Proof.

It suffices to show that $\underline{x}\not\in\Omega_{\varphi}$ if $\underline{x}\in X\backslash\mathrm{m}^{\mathbb{N}_{0}}$ . There exists an integer $N$ such that

\max_{0\leq i\leq N}d_{\mathbb{R}}({x}_{i},\mathrm{m})\geq{\delta}.

since $\underline{x}\in X\backslash\mathrm{m}^{\mathbb{N}_{0}}$ . It follows from the compactness of $X$ and continuity of $h$ that $\varphi$ is uniformly continuous. Thus we can take $\varepsilon_{0}>0$ satisfying if $|\eta_{1}-\eta_{2}|<\varepsilon_{0}$ , then:

|h(\xi,\eta_{1})-h(\xi,\eta_{2})|<\frac{1}{2}\phi(\delta/2),

where the function $\phi$ is given in Lemma 4.9. Fix $\varepsilon\in(0,\min\{{2^{-N}},{2^{-(N+2)}\delta},{\varepsilon_{0}}/{2}\})$ . Let $k$ be a large number satisfying $2^{-k}<\varepsilon$ and $B(\underline{x},\underline{x},k;\varepsilon)\neq\emptyset$ . For $\underline{z}\in B(\underline{x},\underline{x},k;\varepsilon)$ , set $\underline{w}=\underline{w}(k)$ by

\underline{w}(k)=(z_{0}\cdots z_{k-1})^{\infty}.

Immediately,

\displaystyle|z_{0}-z_{k}|\leq d(\underline{z},\sigma^{k}(\underline{z}))\leq d(\underline{z},\underline{x})+d(\underline{x},\sigma^{k}(\underline{z}))<2\epsilon<\epsilon_{0}

and

\displaystyle|S_{k}(\varphi-\alpha_{\varphi})(\underline{z})-S_{k}(\varphi-\alpha_{\varphi})(\underline{w})|

\displaystyle=|h(z_{k-1},z_{k})-h(z_{k-1},z_{0})|<\frac{1}{2}\phi(\delta/2).

Thus we get

(11)

\displaystyle S_{k}(\varphi-\alpha_{\varphi})(\underline{z})>S_{k}(\varphi-\alpha_{\varphi})(\underline{w}(k))-\frac{1}{2}\phi(\delta/2).

It is seen that $\underline{w}\in B(\underline{x},\underline{x},k;2\varepsilon)\cap X(k)$ since $k$ satisfies $\frac{1}{2^{k}}<\varepsilon$ . Moreover, by $\varepsilon<\min\{\frac{1}{2^{N}},\frac{\delta}{2^{N+2}}\}$ and $k$ with $2^{-k}<\varepsilon$ (so $k\geq N$ ), we see that

\max_{0\leq i\leq N}d_{\mathbb{R}}({w}_{i},\mathrm{m})\geq\frac{\delta}{2}.

Hence the estimate (11) shows:

S_{k}(\varphi-\alpha_{\varphi})(\underline{z})>\frac{1}{2}\phi(\delta/2)>0

and we obtain

H(\underline{x},\underline{x})>0.

This is the desired inequality. ∎

Proof of Main Theorem 3.

We already have (1)-(3) in Main Theorem 3. We now prove the rest part. Suppose that $h(x,x)$ has a unique minimum point $a_{*}$ in $[0,1]$ . Then $\mathrm{m}=\{a_{*}\}$ , and thus $\Omega_{\varphi}=\{a_{*}^{\infty}\}$ . Since the Mather set $\mathscr{M}_{\varphi}$ is not empty and included in $\Omega_{\varphi}$ , we obtain the desired result. ∎

5. TPO property

In this section, we discuss typical properties of optimal measures. We still consider the case that the potential function is 2-locally constant.

As stated in the last part of Section 4, by Theorem 4.11, we see that if $\underline{x}\in\Omega_{\varphi}$ then $\underline{x}\in\mathrm{m}^{\mathbb{N}_{0}}$ . Therefore, if $\mathrm{m}=\{a\in[0,1]\mid h(a,a)=\min_{x\in[0,1]}h(x,x)\}$ for $\varphi=h(x,y)$ is a singleton $\{a\}$ , we have $\Omega_{h}=\{a^{\infty}\}$ and it must coincide with the Mather set of $h$ . As described below, In this section, we will see that such $h$ is typical.

Let $r\geq 2$ be a positive integer. Consider the set of $C^{r}$ -variational structure with the twist condition,

\mathscr{H}^{r}=\{h\in C^{r}([0,1]^{2};\mathbb{R})\mid D_{2}D_{1}h<0\},

equipped with the $C^{r}$ -norm, i.e.,

||h||_{C^{r}}=\sum_{|\beta|\leq r}\sup_{(x,y)\in[0,1]^{2}}|\partial^{\beta}h(x,y)|.

For $h\in\mathscr{H}^{r}$ , let $\tilde{h}(x)=h(x,x)$ for $x\in[0,1]$ .

Proposition 5.1.

The subset $\mathscr{O}$ in $\mathscr{H}^{r}$ given by

\mathscr{O}=\{h\in\mathscr{H}^{r}|\ \tilde{h}\ \text{has a unique minimum point}\ x^{*}(h)\ s.t.\ \tilde{h}^{\prime\prime}(x^{*}(h))>0\}

is $C^{r}$ open and dense in $\mathscr{H}^{r}$ .

Proof.

(Dense): Take $h\in\mathscr{H}^{r}$ . Let $a\in[0,1]$ be a minimum point of $\tilde{h}$ . Take arbitrary $\varepsilon>0$ . Set

h_{\varepsilon}(x,y):={h}(x,y)+\varepsilon V(x)

where $V\in C^{r}([0,1];\mathbb{R})$ with a unique minimum point at $x=a$ s.t. $V(a)\geq 0,V^{\prime\prime}(a)>0$ (e.g., $V(x)=\cos(2\pi(x-a)+\pi)$ or $(x-a)^{2}$ ). Then $||{h}_{\varepsilon}-{h}||_{C^{2}}=\varepsilon||V||_{C^{2}}$ and $D_{2}D_{1}h_{\varepsilon}=D_{2}D_{1}h<0$ . Moreover, for $x\neq a$

{h}_{\varepsilon}(x,x)=h(x,x)+\varepsilon V(x)\geq h(a,a)+\varepsilon V(a)>h(a,a)=h_{\varepsilon}(a,a).

Therefore, $h_{\varepsilon}\in\mathscr{O}$ holds.

(Open): Let $h\in\mathscr{O}$ . Denote the corresponding minimum point $x_{h}$ of $\tilde{h}$ . Take $\varepsilon\in(0,\tilde{h}^{\prime\prime}(x_{h})/2)$ Let $h_{\varepsilon}\in\mathscr{H}$ s.t. $||h_{\varepsilon}-h||_{C^{r}}<\varepsilon$ . Then we have

	$\displaystyle\|(\tilde{h}_{\varepsilon}-\tilde{h})^{\prime\prime}(x_{h})\|$	$\displaystyle=\|D_{1}D_{1}({h}_{\varepsilon}-{h})(x_{h},x_{h})+D_{1}D_{2}({h}_{\varepsilon}-{h})(x_{h},x_{h})$
		$\displaystyle\qquad+D_{2}D_{1}({h}_{\varepsilon}-{h})(x_{h},x_{h})+D_{2}D_{2}({h}_{\varepsilon}-{h})(x_{h},x_{h})\|$
		$\displaystyle\leq\|\|h_{\varepsilon}-h\|\|_{C^{r}}<\varepsilon,$

i.e.,

\tilde{h}_{\varepsilon}^{\prime\prime}(x_{h})\geq{\tilde{h}^{\prime\prime}(x_{h})}-\varepsilon>\tilde{h}^{\prime\prime}(x_{h})/2>0.

This also implies that $h_{\varepsilon}$ has a unique minimum point at $x=x_{h}$ , which yields $h_{\varepsilon}\in\mathscr{O}$ . ∎

Similarly we easily obtain the following perturbation result in the sense of Mañé. Note that for any $h\in\mathscr{H}^{r}$ and $f\in C^{r}([0,1];\mathbb{R})$ the function $h(x,y)+f(x)$ satisfies the twist condition, i.e., $D_{2}D_{1}(h+f)=D_{2}D_{1}h<0$ .

Proposition 5.2.

For arbitrary $h\in\mathscr{H}^{r}$ , the set

	$\displaystyle\mathscr{V}_{h}$	$\displaystyle=\{V\in C^{r}([0,1];\mathbb{R})\|\ \widetilde{h}+V\ \text{has a unique minimum point}$
		$\displaystyle\qquad\qquad\qquad\qquad\qquad x^{}(\tilde{h}+V)\ s.t.\ (\widetilde{h}+V)^{\prime\prime}(x^{}(\tilde{h}+V))>0\}$

is $C^{r}$ open and dense in $\mathscr{V}_{h}$ in $C^{r}([0,1];\mathbb{R})$ .

Proof of Main Theorem 4.

Combining Propositions 5.1,5.2 and Main Theorem 3, we obtain Main Theorem 4. ∎

Acknowledgement. The first author was partially supported by JSPS KAKENHI Grant Number 23H01081 and 23K19009. The third author was partially supported by JSPS KAKENHI Grant Number 21K13816.

Data Availability. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

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	$\displaystyle\|\varphi(x_{0},x_{1})-\varphi(y_{0},y_{1})\|$	$\displaystyle=\|\varphi(x_{0}x_{1}0^{\infty})-\varphi(y_{0}y_{1}0^{\infty})\|$
		$\displaystyle\leq L_{\varphi}(\|x_{0}-y_{0}\|+\|x_{1}-y_{1}\|/2)$
	$\displaystyle L_{\varphi}\sqrt{2}d_{\mathbb{R}^{2}}((x_{0},x_{1}),(y_{0},y_{1})).$

	$\displaystyle\|\varphi(\underline{x})-\varphi(\underline{y})\|$	$\displaystyle=\|\varphi(x_{0},x_{1})-\varphi(y_{0},y_{1})\|$
		$\displaystyle\leq L_{\varphi}d_{\mathbb{R}^{2}}((x_{0},x_{1}),(y_{0},y_{1}))$
		$\displaystyle\leq 2L_{\varphi}(\|x_{0}-y_{0}\|+\|x_{1}-y_{1}\|/2)\leq 2L_{\varphi}d(\underline{x},\underline{y}),$

	$\displaystyle h(w_{l^{}-1},w_{l^{}})+h(z_{l^{}-1},z_{l^{}})$	$\displaystyle=h(x_{l^{}-1},x_{l^{}})+h(y_{l^{}},y_{l^{}}),\ \text{and}$
	$\displaystyle h(w_{l^{}},w_{l^{}+1})+h(z_{l^{}},z_{l^{}+1})$	$\displaystyle=h(y_{l^{}},y_{l^{}+1})+h(x_{l^{}},x_{l^{}+1}),$

	$\displaystyle\|(\tilde{h}_{\varepsilon}-\tilde{h})^{\prime\prime}(x_{h})\|$	$\displaystyle=\|D_{1}D_{1}({h}_{\varepsilon}-{h})(x_{h},x_{h})+D_{1}D_{2}({h}_{\varepsilon}-{h})(x_{h},x_{h})$
		$\displaystyle\qquad+D_{2}D_{1}({h}_{\varepsilon}-{h})(x_{h},x_{h})+D_{2}D_{2}({h}_{\varepsilon}-{h})(x_{h},x_{h})\|$
		$\displaystyle\leq\|\|h_{\varepsilon}-h\|\|_{C^{r}}<\varepsilon,$

The Aubry set for the XY model and typicality of periodic optimization for 22-locally constant potentials

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

Main Theorem 1.

Main Theorem 2.

Main Theorem 3.

Main Theorem 4 (TPO property for the XY model in the class of 2-locally constant functions).

2. Mañé potential and Peierl’s barrier

Definition 2.1 (Mañé potential).

Remark 2.2.

Definition 2.3 (Aubry set).

Definition 2.4 (Subaction, calibrated subaction).

Remark 2.5.

Lemma 2.6.

Proof.

Proposition 2.7.

Proof.

Proposition 2.8.

Proof.

Definition 2.9 (Peierl’s barrier).

Remark 2.10.

Theorem 2.11 (cf. Main Theorem 1).

Proof.

Theorem 2.12 (Analogy of Theorem 4.1 in [BLL13], cf. Main Theorem 2).

Proof.

Theorem 2.13 (cf. Main Theorem 2).

Lemma 2.14 (Analogy of Lemma 4.2 in [BLL13]).

Proof.

Proof of Theorem 2.13.

Proof of Main Theorem 2.

Proposition 2.15.

Proof.

3. Another characterization of the Aubry set

Lemma 3.1.

Proof.

Proposition 3.2.

Proof.

Remark 3.3.

Lemma 3.4.

Proof.

Theorem 3.5 (cf. Main Theorem 1).

Proof.

Proof of Main Theorem 1.

Proposition 3.6.

Proof.

4. Variational method applied to the Aubry set

Proposition 4.1.

Proof.

Remark 4.2.

Definition 4.3 (Minimal).

Remark 4.4.

Lemma 4.5.

Proof of Lemma 4.5.

Claim 4.6.

Lemma 4.7 (cf. Main Theorem 3. (1)).

Proof.

Theorem 4.8 (cf. Main Theorem 3. (2)).

Lemma 4.9 (A slightly extended version of Lemma 2.7 of [Yu22]).

Remark 4.10.

Proof of Lemma 4.9.

Theorem 4.11 (cf. Main Theorem 3. (3)).

Proof.

Proof of Main Theorem 3.

5. TPO property

Proposition 5.1.

Proof.

Proposition 5.2.

Proof of Main Theorem 4.

References

The Aubry set for the XY model and typicality of periodic optimization for $2$ -locally constant potentials