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

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