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 $\hat{ \mu}$.
Categories: math.DS cond-mat.stat-mech math-ph math.MP math.OC math.PR