The Involution Kernel and the Dual Potential for Functions in the Walters’ Family

Abstract

First, we set a suitable 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 $A: \{0,1\}^{\mathbb {N}}=\Omega ^+\rightarrow {\mathbb {R}}$ we express A in the variable x, like A(x). In a similar way, given $B: \{0,1\}^{\mathbb {N}}=\Omega ^{-}\rightarrow {\mathbb {R}}$ we express B in the variable y, like B(y). Finally, given $W: \Omega ^{-} \times \Omega ^{+}\rightarrow {\mathbb {R}}$, 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 $A: \{0,1\}^{\mathbb {N}}\rightarrow {\mathbb {R}}$. Given a continuous potential $A: \Omega ^+\rightarrow {\mathbb {R}}$, we say that the continuous potential $A^*: \Omega ^{-}\rightarrow {\mathbb {R}}$ is the dual potential of A, if there exists a continuous $W: \Omega ^{-} \times \Omega ^{+}\rightarrow {\mathbb {R}}$, such that, for all $(y|x) \in \{0,1\}^{{\mathbb {Z}}-\{0\}}$

$$\begin{aligned} A^* (y) = \left[ A \circ {\hat{\sigma }}^{-1} + W \circ {\hat{\sigma }}^{-1} - W \right] (y|x). \end{aligned}$$

We say that W is an involution kernel for A. It is known that the function W allows to define a 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. Denote by $\theta : \Omega ^{-} \times \Omega ^{+} \rightarrow \Omega ^{-} \times \Omega ^{+}$ the function $\theta (...,y_2,y_1|x_1,x_2,...)= (...,x_2,x_1|y_1,y_2,...).$ We say that A is symmetric if $A^* (\theta (x|y))= A(y|x)= A(x).$ We present conditions for A to be symmetric and to be of twist type. It is known that if A is symmetric then $\mu _A$ has zero entropy production.