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 $t\ge 0$ 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 $h_t(0^{\mathrm{\infty }})=1$, and, $\int h_{t}d{\nu }_t=1,\mathrm{\forall }t>0$. We denote $\mu_t= h_t \nu_t$theGibbsstateforthepotential$t, f$. By selection of a subaction $V$, when the temperature goes to zero (or, $t\to \mathrm{\infty }$), we mean the existence of the limit $$V:=\underset{t\to \mathrm{\infty }}{\lim }\frac{1}{t}\log (h_t)\mathrm{.}$$ By selection of a measure $\mu$, when the temperature goes to zero (or, $t\to \infty$),wemeantheexistenceofthelimit(intheweak$^*$sense)$$\mu :={\lim }_{t\to \mathrm{\infty }}{\mu }_t\mathrm{.}$$Wepresentalargefamilyofnon-trivialexamplesof$f$ 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.