Adversarial ergodic optimization

TL;DR

This paper explores a new ergodic optimization problem inspired by information theory, analyzing the extremal behavior of time averages over fibers in topological dynamical systems.

Contribution

It introduces an ergodic optimization framework involving factor maps and continuous functions, connecting dynamical systems with information-theoretic concepts.

Findings

Characterization of extremal values in the ergodic optimization problem

Connections established between dynamical systems and information theory

Potential applications to understanding complex system behaviors

Abstract

In this article, we introduce an ergodic optimization problem inspired by information theory, which can be presented informally as follows: given a factor map $\pi : (X, T) \to (Y, S)$ of topological dynamical systems, and a continuous function $f \in C(X)$, what can be said about the extrema $$\sup_{y \in Y} \inf_{x \in \pi^{-1} \{y\}} \lim_{k \to \infty} \frac{1}{k} \sum_{j = 0}^{k - 1} f \left( T^j x \right) ? $$