The involution kernel and the dual potential for functions in the Walters family

Abstract

Our notation: Points in $\{0,1{\}}^{\mathbb{Z}-\{0\}}=\{0,1{\}}^{\mathbb{N}}\times \{0,1{\}}^{\mathbb{N}}={\Omega }^{-}\times {\Omega }^{+}$, are denoted by $(y|x)=(...,y_2,y_1|x_1,x_2,...)$, where $(x_1,x_2,...)\in \{0,1{\}}^{\mathbb{N}}$, and $(y_1,y_2,...)\in \{0,1{\}}^{\mathbb{N}}$. The bijective map $\hat{\sigma }(...,y_2,y_1|x_1,x_2,...)=(...,y_2,y_1,x_1|x_2,...)$ is called the bilateral shift and acts on $\{0,1{\}}^{\mathbb{Z}-\{0\}}$. Given $A:\{0,1{\}}^{\mathbb{N}}={\Omega }^{+}\to \mathbb{R}$ we express $A$ in the variable $x$, like $A(x)$. In a similar way, given $B:\{0,1{\}}^{\mathbb{N}}={\Omega }^{-}\to \mathbb{R}$ we express $B$ in the variable $y$, like $B(y)$. Finally, given $W:{\Omega }^{-}\times {\Omega }^{+}\to \mathbb{R}$, we express $W$ in the variable $(y|x)$, like $W(y|x)$. By abuse of notation we write $A(y|x)=A(x)$ and $B(y|x)=B(y).$ The probability ${\mu }_A$ denotes the equilibrium probability for $A:\{0,1{\}}^{\mathbb{N}}\to \mathbb{R}$. Given a continuous potential $A:{\Omega }^{+}\to \mathbb{R}$, we say that the continuous potential $A^{\ast }:{\Omega }^{-}\to \mathbb{R}$ is the dual potential of $A$, if there exists a continuous $W:{\Omega }^{-}\times {\Omega }^{+}\to \mathbb{R}$, such that, for all $(y|x)\in \{0,1{\}}^{\mathbb{Z}-\{0\}}$ $$A^{\ast }(y)=\left[A\circ {\hat{\sigma }}^{-1}+W\circ {\hat{\sigma }}^{-1}-W\right](y|x).$$ We say that $W$ is an involution kernel for $A$. The function $W$ allows you to define an spectral projection in the linear space of the main eigenfunction of the Ruelle operator for $A$. Given $A$, we describe explicit expressions for $W$ and the dual potential $A^{\ast }$, for $A$ in a family of functions introduced by P. Walters. We present conditions for $A$ to be symmetric and to be of twist type.

Key words: Involution kernel, dual potential, symmetric potential, Walters' family, entropy production, twist condition