Maximal measures for flows with nonuniform structure

TL;DR

This paper investigates the ergodic optimization of continuous functions for flows with non-uniform hyperbolic structures, focusing on the entropy spectrum of their maximizing measures and extending previous results to broader classes of flows.

Contribution

It extends the orbit decomposition technique to non-uniformly hyperbolic flows, providing a comprehensive description of maximizing measures' entropy spectrum.

Findings

Describes coexistence of maximizing measures with large and small entropy.

Applies results to geodesic and frame flows on nonpositive curvature manifolds.

Extends previous work from symbolic spaces to broad classes of flows.

Abstract

In this paper, we study ergodic optimization of continuous functions for flows by concentrating on the entropy spectrum of their maximizing measures. Precisely, over a wide family of flows with non-uniformly hyperbolic structure, we obtain a picture describing coexistence of continuous functions whose maximizing measures have large and small entropy respectively in $C^0$-topology. Our proof relies on the orbit decomposition technique, originally introduced by Climenhaga and Thompson, for flows with weakened versions of expansiveness and specification property. In particular, our results extend \cite{STY} from non-Markov shift on symbolic spaces to a considerably broad class of continuous flows with nonuniform structure. To illustrate this, we apply our general results to both geodesic flows and frame flows over closed rank one manifolds of nonpositive curvature.