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

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.