# Ergodic Maximizing Measures of Non-Generic, Yet Dense Continuous Functions

## TL;DR

This paper investigates the existence and properties of ergodic maximizing measures for continuous functions in dynamical systems, showing that uncountably many such measures can exist under certain conditions.

## Contribution

It demonstrates the abundance of ergodic maximizing measures for dense sets of continuous functions in specific dynamical systems, including subshifts of finite type.

## Key findings

- Existence of uncountably many ergodic maximizing measures for dense sets of functions.
- Maximizing measures can be fully supported with positive entropy in certain systems.
- Results apply to both general continuous maps and topologically mixing subshifts.

## Abstract

Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given "performance" function. For a continuous self-map of a compact metric space and a dense set of continuous performance functions, we show that the existence of uncountably many ergodic maximizing measures. We also show that, for a topologically mixing subshift of finite type and a dense set of continuous functions there exist uncountably many ergodic maximizing measures which are fully supported and have positive entropy.