The Dual Potential, the involution kernel and Transport in Ergodic Optimization

Consider the shift $\sigma$ acting on the Bernoulli space $\Sigma={1,2,...,n}^\mathbb{N}$. We denote $\hat{\Sigma}= {1,2,...,n}^\mathbb{Z}$. We analyze several properties of the maximizing probability $\mu_{\infty,A}$ of a Holder potential $A: \Sigma \to \mathbb{R}$. Associated to $A(x)$, via the involution kernel, $W: \hat{\Sigma} \to \mathbb{R}$, it is known that can we get the dual potential $A^*(y)$, where $(x,y)\in \hat{\Sigma}$. Consider $\mu_{\infty, A^*}$ a maximizing probability for $A^*$. We would like to consider the transport problem from $\mu_{\infty,A}$ to $\mu_{\infty,A^*}$. In this case, it is natural to consider the cost function $c(x,y) = I(x) - W(x,y) +\gamma$, where $I$ is the deviation function. The pair of functions for the Kantorovich Transport dual Problem are $(-V,-V^*$), where we denote the two calibrated sub-actions by $V$ and $V^*$, respectively, for $A$ and $A^*$ for $\mu_{\infty,A}$. We analyze the graph property for the optimal plan $\hat{\mu}$.