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

Abstract

Consider T (x) = d x (mod 1) acting on S 1 , a Lipschitz potential A : S 1 → R, 0 < λ < 1 and the unique function b λ :We will show that, when λ → 1, the function b λ − m(A) 1−λ converges uniformly to the calibrated subaction V (x) = max µ∈M S(y, x) dµ(y), where S is the Mañe potential, M is the set of invariant probabilities with support on the Aubry set and m(A) = sup µ∈M A dµ.For β > 0 and λ ∈ (0, 1), there exists a unique fixed point u λ,β : S 1 → R for the equation e u λ,β (x) = T (y)=x e βA(y)+λu λ,β (y) . It is known that as λ → 1 the family e [u λ,β −sup u λ,β ] converges uniformly to the main eigenfuntion φ β for the Ruelle operator associated to βA. We consider λ = λ(β), β(1 − λ(β)) → +∞ and λ(β) → 1, as β → ∞. Under these hypotheses we will show that 1 β (u λ,β − P (βA) 1−λ ) converges uniformly to the above V , as β → ∞. The parameter β represents the inverse of temperature in Statistical Mechanics and β → ∞ means that we are considering that the temperature goes to zero. Under these conditions we get selection of subaction when β → ∞.

Explicit examples in ergodic optimization

“…In [5] the expression was obtained via techniques related to the Peierls barrier (see Lema 2.2). References [2,10,13,16,22,23,28,33,38,41,42,46,48] present results which are related to the topics we consider in our paper.…”

On the Thin Boundary of the Fat Attractor

“…Even if the maximizing probability for A is not unique there exist anyway a unique special limit subaction when λ → 1 (see [23]). That is, there exist a selection on the discounted method for any Holder potential A (the potential do not have to be generic).…”