Typical Uniqueness in Ergodic Optimization

TL;DR

This paper proves that in ergodic optimization, the set of potential functions with a unique maximizing measure is large, being residual and prevalent, with non-uniqueness only on a small set of hypersurfaces.

Contribution

It establishes that typically, the maximizing measure is unique for a broad class of potential functions in ergodic optimization, strengthening previous genericity results.

Findings

The set of functions with unique maximizing measure is residual.

Non-uniqueness occurs only on a countable collection of hypersurfaces.

The result applies to any topological dynamical system and separable Banach space of continuous functions.

Abstract

For ergodic optimization on any topological dynamical system, with real-valued potential function $f$ belonging to any separable Banach space $B$ of continuous functions, we show that the $f$-maximizing measure is typically unique, in the strong sense that a countable collection of hypersurfaces contains the exceptional set of those $f\in B$ with non-unique maximizing measure. This strengthens previous results asserting that the uniqueness set is both residual and prevalent.