Selection of calibrated subaction when temperature goes to zero in the discounted problem

Consider $T(x)= d \, x$ (mod 1) acting on $S^1$, a Lipschitz potential $A:S^1 \to \mathbb{R}$, $0<\lambda<1$ and the unique function $b_\lambda:S^1 \to \mathbb{R}$ satisfying $ b_\lambda(x) = \max_{T(y)=x} \{ \lambda \, b_\lambda(y) + A(y)\}.$ We will show that, when $\lambda \to 1$, the function $b_\lambda- \frac{m(A)}{1-\lambda}$ converges uniformly to the calibrated subaction $V(x) = \max_{\mu \in \mathcal{ M}} \int S(y,x) \, d \mu(y)$, where $S$ is the Ma\~ne potential, $\mathcal{ M}$ is the set of invariant probabilities with support on the Aubry set and $m(A)= \sup_{\mu \in \mathcal{M}} \int A\,d\mu$. For $\beta>0$ and $\lambda \in (0,1)$, there exists a unique fixed point $u_{\lambda,\beta} :S^1\to \mathbb{R}$ for the equation $e^{u_{\lambda,\beta}(x)} = \sum_{T(y)=x}e^{\beta A(y) +\lambda u_{\lambda,\beta}(y)}$. It is known that as $\lambda \to 1$ the family $e^{[u_{\lambda,\beta}- \sup u_{\lambda,\beta}]}$ converges uniformly to the main eigenfuntion $\phi_\beta $ for the Ruelle operator associated to $\beta A$. We consider $\lambda=\lambda(\beta)$, $\beta(1-\lambda(\beta))\to+\infty$ and $\lambda(\beta) \to 1$, as $\beta \to\infty$. Under these hypothesis we will show that $\frac{1}{\beta}(u_{\lambda,\beta}-\frac{P(\beta A)}{1-\lambda})$ converges uniformly to the above $V$, as $\beta\to \infty$. The parameter $\beta$ represents the inverse of temperature in Statistical Mechanics and $\beta \to \infty$ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when $\beta \to \infty$.


Introduction
Consider T (x) = d x (mod 1) acting on S 1 and a Lipschitz potential A : S 1 → R. We denote by Any invariant µ which attains this supremum is called an A-maximizing probability.
A subaction for A is a continuous function D : S 1 → R such that for all x ∈ S 1 , It is called a calibrated subaction if for all y ∈ S 1 , We refer the reader to [2,13,18,20] for general results on Ergodic Optimization and Thermodynamic Formalism.
Maximizing probabilities and calibrated subactions are dual objects in Ergodic Optimization. On the one hand m(A) satisfies (1) for any calibrated subaction D. Furthermore, it is known that a calibrated subaction can help to identify the support of the maximizing probabilities for A (see [6] or [2]). A natural problem is: how to find subactions? Note that we need to have at hand the exact value m(A) in order to verify if a specific candidate D is indeed a calibrated subaction. The discounted method, which is described below, can be quite useful in order to get a good approximation (via iteration of a contraction) of a calibrated subaction without the knowledge of the value m(A).
For each fixed λ ∈ (0, 1), consider the function b = b λ : S 1 → R satisfying the equation This function is unique and we call b λ the λ-calibrated subaction for A (see for instance Theorem 1 in [4] or [16]). The solution b λ can be obtained in the following way: consider τ j , j = 1, ..., d the inverse branches of T . For λ < 1, consider where a ∈ {1, 2, ..., d} N (see expression (6) in [16] or [22]). Then, the solution of (2) is given by b λ (x) = sup a∈{1,2,...,d} N S λ (x, a) (see section 3 in [16].). For fixed x, as the function S λ,A (x, .) is continuous in the second coordinate, there exists some a realizing the supremum, which is called a b λ (x) realizer. Denoting a = a 0 a 1 ... we have that for any k: The above family b λ , 0 < λ < 1, is equicontinuous. It is known that any convergent subsequence b λn − sup b λn , λ n → 1, determines in the limit a calibrated subaction for A (see Theorem 1 in [4], [1] or [22]). This procedure, known as the discounted method, is borrowed from Control Theory. The function b λ is obtained as a fixed point of a contraction (see [16]), which is, of course, computationally convenient (note that m(A) does not appear on expression (2)).
In this work the constant sup b λ will be replaced by m(A) 1−λ . Our first aim is to describe a certain calibrated subaction as the limit of b λ − m(A) 1−λ , as λ → 1. In this way the limit will not depend of the convergent subsequence. Later we will consider eigenfunctions of the Ruelle operator and selection of subaction via the limit when temperature goes to zero (see [6] and [2]).
A point x in S 1 is called A-non-wandering, if for any ǫ > 0, there exists z ∈ S 1 and n ≥ 1, such that, d(z, x) < ǫ, T n (z) = x, and | n−1 j=0 (A − m(A)) (T j (z))| < ǫ. The Aubry set for A is the set of A-non-wandering points in S 1 and is denoted by Ω(A). Any invariant probability with support inside the Aubry set is A-maximizing (see [6] or section 3 in [8]). We denote by M the set of A-maximizing probabilities.
Consider the function S given by Definition 22 in [6] (see also [12] and [13]): This function is called the Mañe potential.
The function S(x, y) is not continuous as a function of (x, y) (see Proposition 3.5 in [7]) We define, for each x ∈ S 1 , the subaction Our first goal is to prove the following theorem.
The above theorem assures that the subaction V is very special among the possible ones. We will prove that V is calibrated (see corollary 8).
This first part of the paper follows the ideas in [10] and [11] and obtains analogous results in the framework of ergodic optimization.
From now on we describe the second part of our paper which considers a limit when temperature goes to zero -this result has a quite different nature when compared with [10] and [11].
The parameter β represents the inverse of temperature in Statistical Mechanics From [6] we get that 1 β log φ β is equicontinuous and any limit of a convergent subsequence of 1 βn log φ βn , β n → ∞, is a calibrated subaction. The limit when β → ∞ is known as the limit when temperature goes to zero (see [2]). For the case of locally constant potentials a precise description is [5]. The most comprehensive result of this kind of convergence for Lipschitz potentials is Theorem 16 in [14].
In the standard terminology we say that there exists selection of subaction at zero temperature if the limit of the function 1 β log φ β exists, when β → ∞ (see [2] for general results).
For a fixed value β, the function u λ,β is obtained as a fixed point of a contraction. In this way, one can get a computable method (via iteration of the contraction) for getting good approximations of the main eigenfunction φ β (taking λ close to 1).
We denote by α(β) the main eigenvalue which is associated to φ β . The pressure P (βA) = log(α(β)) is equal to where h(µ) is the Kolmogorov entropy of µ (see [20]). Our second goal is to show:

Theorem 2. Suppose A is Lipchitz and there exists a relation between
The above result requires a control of the velocity such that (1 − λ(β)) goes to zero, vis-à-vis, the growth of β to infinity. We will show on the end of the paper a counterexample proving that 1 β log φ β does not always converge to V . This shows that is really necessary some kind of joint control of the parameters β and λ (as considered in the above theorem).
The last theorem shows the existence of selection of subaction when temperature goes to zero (but in a sense which is not the standard form).
In the next section we will prove Theorem 1 and in the last section we will prove Theorem 2.
The results that we get here are also true if one considers T (x) = σ(x) acting on Bernoulli space {1, ..., d} N .
Proof. From Remark 1 we get that, for each y ∈ Ω(A), the function S(y, ·) is Lipschitz continuous and the Lipschitz constant does not depend of y. Therefore V is Lipschitz. For any x, y ∈ S 1 we have Proof of 1.: It is known (Prop. 23 page 1395 [6]) that S(y, x) + S(x, y) ≤ 0. We say that x and y in Ω(A) are in the same static class if S(y, x) + S(x, y) = 0.
For an ergodic maximizing measure µ we claim that for µ-almost every pair x and y these two points are in the same static class.
First note that as A is Lipschitz the function S can be also written as Now we will show the claim: given an ergodic probability µ, an integrable function F and a Borel set B, such that µ(B) > 0, denote byB the set of points p in B, such that, for all ǫ, there exists an integer N > 0, with the properties: It is known that µ(B) = µ(B) (see [19]). This results is known as Atkinson's Theorem.
Let x, y ∈ supp(µ) where x is a Birkhoff point for A and fix ρ > 0. As µ(B(y, ρ)) > 0, for some n we have T n (x) ∈ B(y, ρ). Let ρ ′ < ρ be such that for any x ′ ∈ B(x, ρ ′ ) we have d(T j (x), T j (x ′ )) < ρ, for all j ∈ {0, ..., n}. Consider the set B = B(x, ρ ′ ). There exists p ∈ B(x, ρ ′ ), such that, for all ǫ, there exists an integer N > 0, satisfying As this is true for a. e. p ∈ B(x, ρ ′ ) we can suppose that N > n, replacing p by T N (p) in the case it is necessary. It follows that S(x, y) + S(y, x) = 0 which proves the claim.
We have S(x, y) + S(y, x) dµ(x) dµ(y) = 0, for any µ ∈ M which is ergodic, then S(x, y) dµ(x)dµ(y) = 0, using Fubini's Theorem (S(x, y) is integrable by last paragraph). It follows that for any ergodic probability µ ∈ M. The same inequality for a general µ ∈ M follows from the ergodic decomposition theorem.
We will show that any limit probability of µ y λ , as λ → 1, belongs to M.
Lemma 4. For any continuous function w : S 1 → R, and probability µ y λ as above, we get Proof. Indeed, Lemma 5. Given y ∈ S 1 , any accumulation probability µ ∞ , in the weak* topology, of a convergent subsequence µ y λ i , λ i → 1, belongs to M.
Proof. It follows from above lemma that µ ∞ is invariant. Moreover, by (3) and definition of µ y λ we have that When, i → ∞ the left integral converges to zero. Therefore, It is known (see for instance end of Theorem 11 in [3] or [18]) that (1 − λ) inf b λ → m(A), uniformly with λ → 1. Therefore, proving the claim.
On the other hand, if w is a subaction we have that therefore, using (6), for any λ and y we have Therefore, the functions U λn (x) are uniformly bounded above. As the functions b λ are equicontinuous in λ < 1 (see [16]), the family of functions U λ is equicontinuous. As U λ are uniformly bounded above and U λ (x λ ) ≥ 0 we conclude that this family is also uniformly bounded. Proof. Let U be the limit of the subsequence U λn = b λn − m(A) 1−λn , when n → ∞. From (2) we get Then, as λ n → 1 we conclude that U is a subaction. Furthermore, for any point x ∈ S 1 , there is some point y 0 ∈ T −1 (x) attaining the supremum of sup T (y)=x λ n U λn (y) + A(y) − m(A), for infinitely many values of n. In this way we get This proves that U is calibrated.
Proof of Theorem 1: We denote U any limit of U λn := b λn − m(A) 1−λn , when n → ∞. We know that U is a calibrated subaction and we want to show that U = V .
From lemma 6, for any maximizing probability µ ∈ M, we have that U dµ ≥ 0. If follows from lemma 3 that U ≥ V . Now we will show that U ≤ V . From lemma 3 the subaction V satisfies V dµ ≥ 0, for any µ ∈ M, and from lemma 6 we get, for any y and λ, the inequality If λ n i is a subsequence of λ n , such that, µ y λn i → µ ∞ , then, from lemma 5 we have that µ ∞ ∈ M. Therefore, we finally get that

Corollary 8. V is a calibrated subaction.
Proof. It is a consequence of lemma 7 and Theorem 1.

Selection for the zero temperature case
Now we will prove Theorem 2.
Proof. By definition Then, it follows that Let µ 0 be the equilibrium probability for β A + λ u λ,β − u λ,β • T . Then, It follows that On the other hand, if µ 1 is the equilibrium probability of β A, then Therefore, When λ → 1, any accumulation function of e u * λ,β will be an eigenfunction of the Ruelle Operator L β A associated to the maximal eigenvalue e P (β A) .
Proof. As u λ,β is equicontinuous the same is true for u * λ,β . Using the equicontinuity (with constant H) we have that for any x: Furthermore, If u β is an accumulation function of the family u * λ,β (when, λ → 1), then, we have: and moreover P (βA) This Remark will be used on the proof of Lemmas 11 and 12.
Lemma 11. Consider a fixed λ. Then, when β → ∞, the unique possible accumulation point of the family 1 β u λ,β is the function b λ defined in (2). Moreover, we get that the unique accumulation point of 1 Proof. As u * λ,β has Lipschitz constant β Lip(A) and is bounded by −β Lip(A) and β Lip(A), the family 1 β u * λ,β is equicontinuous and uniformly bounded by Lip(A). From the limit P (β A) β(1−λ) → m(A) 1−λ (as β → ∞) we conclude that (for fixed λ) the family 1 β u λ,β is equicontinuous (with a constant Lip(A)) and uniformly bounded. As there is a unique accumulation point b of 1 β u λ,β which satisfies In the previous section we study the limit of b λ − m(A) 1−λ . Now, we are interested in the limit of 1 β (u λ,β − P (β A) 1−λ ), when β → ∞ and λ → 1.
In order to prove the second claim we fix a point x ∈ S 1 . Let we get, for any b 0 ∈ {1, ..., d}, Dividing the right side by β n , taking β n → ∞ and λ n → 1, we get This shows that U is a subaction.
In order to show that U is calibrated, we fix for each λ and β a point a = a λ,β maximizing βA(τ a x) When β n → ∞ and λ n → 1, some a = a λn,βn will be chosen infinitely many times. When β n → ∞ and λ n → 1, this a will satisfy In the last section we proved that the function is the unique limit of the family b λ − m(A) 1−λ . In the present setting, in order to get a similar result, we will assume a certain condition: β → ∞ faster than λ → 1, in the sense that β(1 − λ) → ∞.
First we need a Lemma.
We finish this section introducing an example (on the symbolic space) where it is studied the limit of 1 β log(φ β ) in a particular case. This limit is not V and this shows that some joint control of β and λ is really necessary.