Ergodic Optimization with Linear Constraints

Abstract

Let $T:X\to X$ be a continuous map of a compact metrizable space, and let $\ddot{I}•:X\to \mathbb{R}$ be a continuous function. The ergodic optimization problem is to maximize the integral $\int \ddot{I}•d\hat{I}¼$ as $\hat{I}¼$ ranges over all $T$-invariant Borel probability measures on $X$. In this paper we consider a constrained version of the ergodic optimization problem. Given a `constraint set' $\mathcal{C}\subset C(X)$, let $M_{\mathcal{C}}(X,T)$ be the set of $T$-invariant Borel probability measures $\hat{I}¼$ on $X$ such that $\int gd\hat{I}¼=0$ for all $g\in \mathcal{C}$. We investigate the problem of maximizing the integral $\int \ddot{I}•d\hat{I}¼$ over the constrained set $M_{\mathcal{C}}(X,T)$. We address basic properties of this optimization problem, beginning with nonemptiness of $M_{\mathcal{C}}(X,T)$ and existence of optimal solutions. Additionally, we establish the generic and prevalent uniqueness of optimal measures, we provide a realization result, and we give a characterization of the dual problem. This framework provides a common generalization of several previously considered optimization problems in dynamical systems and optimal transport.

27 pages, no figure