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

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 $t\ge 0$ 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 $L_{tf}^{\ast }$, 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 $tf$. By selection of a subaction $V$, when the temperature goes to zero (or, $t\to \infty$), we mean the existence of the limit $$V:=\underset{t\to \infty }{\lim }\frac{1}{t}\log (h_t).$$ 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${}^{\ast }$ sense) $$\mu :=\underset{t\to \infty }{\lim }{\mu }_t.$$ 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.