Joint typical periodic optimization

TL;DR

This paper proves a generalized conjecture showing that for various dynamical systems and function spaces, most map-function pairs have a unique periodic orbit as their maximizing invariant measure.

Contribution

It establishes that in a broad setting, the typical behavior is the uniqueness of periodic maximizing measures for a large class of systems and functions.

Findings

Open dense subset of map-function pairs with unique periodic maximizing measure.

Applicable to Lipschitz expanding maps, Anosov diffeomorphisms, and beta-transformations.

Supports the generalized Yuan--Hunt--Ma e Conjecture.

Abstract

We prove a generalised Yuan--Hunt--Ma\~n\'e Conjecture: if $\mathcal{F}$ is the Banach space of $\alpha$-H\"older functions, and $\mathcal{T}$ is either a space of Lipschitz expanding maps, or of Anosov diffeomorphisms, or the family of beta-transformations on the interval, there is an open dense subset of $\mathcal{T}\times\mathcal{F}$ consisting of map-function pairs whose maximizing invariant measure is unique and supported on a periodic orbit.