On the selection of subaction and measure for a subclass of potentials defined by P. Walters

TL;DR

This paper investigates the zero-temperature limits of Gibbs measures and eigenfunctions for a class of potentials on Bernoulli space, providing explicit formulas and examples within Walters' subclass.

Contribution

It introduces new explicit expressions for the zero-temperature limits of eigenfunctions and measures for Walters' potentials, expanding understanding of selection phenomena.

Findings

Existence of limits for eigenfunctions and measures as temperature approaches zero.

Explicit formulas for the limiting subactions and measures in Walters' subclass.

Identification of a large family of Walters' potentials with well-defined zero-temperature limits.

Abstract

Suppose $\sigma$ is the shift acting on Bernoulli space $X=\{0,1\}^\mathbb{N}$, and, consider a fixed function $f:X \to \mathbb{R}$, under the Waters's conditions (defined in a paper in ETDS 2007). For each real value $t\geq 0$ we consider the Ruelle Operator $L_{tf}$. We are interested in the main eigenfunction $h_t$ of $L_{tf}$, and, the main eigenmeasure $\nu_t$, for the dual operator $L_{tf}^*$, which we consider normalized in such way $h_t(0^\infty)=1$, and, $\int h_t \,d\,\nu_t=1, \forall t>0$. We denote $\mu_t= h_t \nu_t$ the Gibbs state for the potential $t\, f$. By selection of a subaction $V$, when the temperature goes to zero (or, $t\to \infty$), we mean the existence of the limit $$V:=\lim_{t\to\infty}\frac{1}{t}\log(h_{t}).$$ By selection of a measure $\mu$, when the temperature goes to zero (or, $t\to \infty$), we mean the existence of the limit (in the weak$^*$ sense) $$\mu:=\lim_{t\to\infty} \mu_t.$$ We present a large family of non-trivial examples of $f$ where the selection of measure exists. These $f$ belong to a sub-class of potentials introduced by P. Walters. In this case, explicit expressions for the selected $V$ can be obtained for a certain large family of potentials.