On ergodic optimization for unimodal maps

TL;DR

This paper demonstrates that for most non-uniformly expanding unimodal maps, the optimal invariant measure for a typical Lipschitz function is supported on a periodic orbit, revealing a generic property of such dynamical systems.

Contribution

It establishes that in non-uniformly expanding unimodal maps, the maximizing measure for generic Lipschitz functions is typically supported on a periodic orbit, a new insight into ergodic optimization.

Findings

Maximizing measure is supported on a periodic orbit for typical maps.

Results apply to a broad class of non-uniformly expanding unimodal maps.

Supports the conjecture that generic Lipschitz functions have simple maximizing measures.

Abstract

In this article, we show that for a typical non-uniformly expanding unimodal map, the unique maximizing measure of a generic Lipschitz function is supported on a periodic orbit.