Joint typical periodic optimization

Abstract

We prove a generalised Yuan--Hunt--Mañé Conjecture: if $\mathcal{F}$ is the Banach space of $\hat{I}\pm$-Hölder 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.

101 pages. In this version we introduce the Joint TPO problem, significantly extending the first version, with a different focus. Some of the material on beta-transformations is based on, and supersedes, the results presented in the first version