Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

TL;DR

This paper demonstrates that for various complex dynamical systems beyond uniform hyperbolicity, generic functions have unique maximizing measures with zero entropy, some with full support, revealing new ergodic optimization properties.

Contribution

It extends ergodic optimization results to systems like singular hyperbolic attractors and surface diffeomorphisms beyond classical hyperbolic cases.

Findings

Unique maximizing measures with zero entropy for generic functions.

Existence of maximizing measures with full support in some systems.

Applicability to singular hyperbolic attractors and certain surface diffeomorphisms.

Abstract

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems including singular hyperbolic attractors, $C^\infty$ surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.