Ergodic maximization problem for expanding maps with differentiable observables

TL;DR

This paper proves that for generic smooth functions on expanding maps, the measures that maximize the integral are supported on a single periodic orbit, highlighting a typical behavior in ergodic optimization.

Contribution

It establishes that for a broad class of differentiable functions, the maximizing measures are supported on periodic orbits, extending understanding in ergodic optimization for expanding maps.

Findings

Maximizing measures are supported on a single periodic orbit for generic $C^r$ functions.

The result applies to open and dense subsets of $C^r$ functions.

Discussion on approximation challenges for $C^{ abla}$ functions.

Abstract

We show that for an expanding map, the maximizing measures of a generic (open and dense) $C^r$ ($r\in\mathbb{N}$) differentiable functions are supported on a single periodic orbit. [There is a gap in the discussions. For the $C^{\infty}$ approximation of the Lipschitz functions, we can only control the $C^1$ derivative, but we can not control the $C^r$ derivatives for $r\geq2$. Elegant approximation methods might be needed to solve this problem.]