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

Abstract

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 $A^{\ast }(y)$, where $(x,y)\in \hat{\Sigma }$. Consider ${\mu }_{\infty ,A^{\ast }}$ a maximizing probability for $A^{\ast }$. We would like to consider the transport problem from ${\mu }_{\infty ,A}$ to ${\mu }_{\infty ,A^{\ast }}$. 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 $(-V,-V^{\ast }$), where we denote the two calibrated sub-actions by $V$ and $V^{\ast }$, respectively, for $A$ and $A^{\ast }$ for ${\mu }_{\infty ,A}$. We analyze the graph property for the optimal plan $\hat{\mu }$.

to appear in Mathematics of Planet Earth, Volume 1 Dynamics, Games and Science. Springer Verlag, Edit. Alberto Pinto et all

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