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

Suppose $\sigma $ is the shift acting on Bernoulli space $X=\{0,1\}^{\mathbb {N}}$, and consider a fixed function $f:X \to \mathbb {R}$ satisfying the Walters conditions (defined in [P. Walters. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys.27 (2007), 1323–1348]). For each real value $t\geq 0$ we consider the Ruelle operator $L_{\mathit {tf}}$. We are interested in the main eigenfunction $h_t$ of $L_{\mathit {tf}}$ and the main eigenmeasure $\nu _t$ for the dual operator $L_{\mathit {tf}}^*$, which we consider normalized in such a way that $h_t(0^\infty )=1$ and $\int h_t \,d\nu _t=1$ for all $t\gt 0$. We denote by $\mu _t= h_t \nu _t$ the Gibbs state for the potential $\mathit {tf}$. By the 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 the 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 a measure exists. These $f$ belong to a sub-class of potentials introduced by Walters. In this case, explicit expressions for the selected $V$can be obtained for a certain large family of parameters.