Ergodic Optimization Restricted On Certain Subsets Of Invariant Measures

TL;DR

This paper investigates ergodic optimization within certain invariant measure subsets for specific dynamical systems, establishing generic uniqueness and ergodicity of measures with prescribed entropy.

Contribution

It proves that for generic continuous functions, the ergodic optimization measure is unique, ergodic, has full support, and a specified entropy in these systems.

Findings

Unique ergodic measures with prescribed entropy are generic.

Results extend to suspension flows and Lorenz attractors.

Ergodic optimization measures have full support and are ergodic.

Abstract

In this article, we pay attention to transitive dynamical systems having the shadowing property and the entropy functions are upper semicontinuous. As for these dynamical systems, when we consider ergodic optimization restricted on the subset of invariant measures whose metric entropy are equal or greater than a given constant, we prove that for generic real continuous functions the ergodic optimization measure is unique, ergodic, full support and have metric entropy equal to the given constant. Similar results also hold for suspension flows over transitive subshift of finite type, Cr (r\geq 2)- generic geometric Lorenz attractors and C1-generic singular hyperbolic attractors.