Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit

TL;DR

This paper proves that within the space of continuous endomorphisms on a compact manifold, those with a maximizing measure supported on a periodic orbit form a dense subset, highlighting the prevalence of such measures.

Contribution

It establishes the density of endomorphisms with a maximizing measure supported on a periodic orbit in the space of all endomorphisms.

Findings

Dense subset of endomorphisms with periodic maximizing measures

Existence of invariant measures supported on periodic orbits

Maximizing measures are prevalent in the considered space

Abstract

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi: M\to\mathbb{R}$ continuous. We prove that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $\mu_{\max}$ supported on a periodic orbit that maximizes the integral of $\phi$ among all $f$ invariant Borel probability measures.