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 $f:X \to \mathbb{R}$, 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.