Constrained Ergodic optimization for generic continuous functions

TL;DR

This paper extends ergodic optimization results to constrained settings, showing that for generic continuous functions, maximizing invariant measures under constraints have zero entropy, highlighting a fundamental property of such systems.

Contribution

It introduces a new result in constrained ergodic optimization, generalizing previous findings to systems with constraints and establishing zero entropy for generic maximizers.

Findings

Maximizing measures under constraints have zero entropy for generic functions.

The result generalizes classical ergodic optimization to constrained systems.

Provides a theoretical foundation for constrained ergodic optimization.

Abstract

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function $f$ every invariant probability measure that maximizes the space average of $f$ must have zero entropy. We establish the analogical result in the context of constraint ergodic optimization, which is introduced by Garibaldi and Lopes (2007).