Ergodic Optimization of Super-continuous Functions in the Shift

TL;DR

This paper investigates ergodic optimization for super-continuous functions in the shift space, showing that a dense subset of these functions are optimized by measures supported on periodic orbits, extending previous results to non-separable spaces.

Contribution

It extends ergodic optimization results to super-continuous functions in non-separable spaces, demonstrating that measures supported on periodic orbits are dense in this class.

Findings

A dense subset of super-continuous functions are optimized by periodic orbit measures.

The set of functions optimized by periodic measures contains an open subset in the super-continuous space.

Extends ergodic optimization theory to non-separable function spaces.

Abstract

Ergodic Optimization is the process of finding invariant probability measures that maximize the integral of a given function. It has been conjectured that "most" functions are optimized by measures supported on a periodic orbit, and it has been proved in several separable spaces that an open and dense subset of functions is optimized by measures supported on a periodic orbit. We add to these positive results by presenting a non-separable space, the class of super-continuous functions, where the set of functions optimized by periodic orbit measures contains an open subset dense in super-continuous functions.