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

TL;DR

This paper explores the involution kernel and dual potential for functions within Walters' family, providing explicit formulas and conditions for symmetry and twist type, advancing understanding of spectral properties in dynamical systems.

Contribution

It introduces explicit expressions for the involution kernel and dual potential for Walters' functions, and establishes criteria for symmetry and twist properties.

Findings

Explicit formulas for involution kernel and dual potential for Walters' functions

Conditions for symmetry and twist type in Walters' family

Enhanced understanding of spectral projections in Ruelle operators

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 $A^*: \Omega^{-}\to \mathbb{R}$ is the dual potential of $A$, if there exists a continuous $W: \Omega^{-} \times \Omega^{+}\to \mathbb{R}$, such that, for all $(y|x) \in \{0,1\}^{\mathbb{Z}-\{0\}}$ $$ A^* (y) = \left[ A \circ \hat{\sigma}^{-1} + W \circ \hat{\sigma}^{-1} - W \right] (y|x). $$ 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 $A^*$, 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.