  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.
