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On ergodic optimization for unimodal maps

Bing Gao, Rui Gao

TL;DR

The paper advances ergodic optimization for one-dimensional dynamics by proving that Lipschitz typical maximizing measures for a broad class of unimodal maps are supported on periodic orbits. It extends the typically periodic optimization paradigm from uniformly expanding systems to piecewise expanding unimodal and Collet–Eckmann/S-unimodal maps under natural ω-limit assumptions, using a subordination framework that reduces to the expanding case. The approach combines a Mor07-type subordination bound, a reducibility argument, and Contreras-type results to obtain generic Lipschitz TPO, with a parallel circle-map result. This work broadens the non-uniformly expanding settings where the typically periodic optimization phenomenon holds, offering a robust method to identify periodic maximizing structures in one-dimensional dynamics.

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.

On ergodic optimization for unimodal maps

TL;DR

The paper advances ergodic optimization for one-dimensional dynamics by proving that Lipschitz typical maximizing measures for a broad class of unimodal maps are supported on periodic orbits. It extends the typically periodic optimization paradigm from uniformly expanding systems to piecewise expanding unimodal and Collet–Eckmann/S-unimodal maps under natural ω-limit assumptions, using a subordination framework that reduces to the expanding case. The approach combines a Mor07-type subordination bound, a reducibility argument, and Contreras-type results to obtain generic Lipschitz TPO, with a parallel circle-map result. This work broadens the non-uniformly expanding settings where the typically periodic optimization phenomenon holds, offering a robust method to identify periodic maximizing structures in one-dimensional dynamics.

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.
Paper Structure (18 sections, 21 theorems, 49 equations)

This paper contains 18 sections, 21 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Statement of results
  4. Comments on the proof
  5. Conventions and notations
  6. Preliminaries
  7. On Lipschitz typically periodic optimization
  8. On the subordination principle in ergodic optimization
  9. On unimodal maps
  10. Proof of the Main results
  11. Establishing subordination
  12. Proof of Theorem \ref{['thm:unimodal']}
  13. Proof of Theorem \ref{['thm:exclu']}
  14. On non-uniformly expanding circle covering maps
  15. Proof of Theorem \ref{['thm:circle']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $T:X\to X$ be either a piecewise expanding unimodal map or an S-unimodal map satisfying the Collet-Eckmann condition. Let $\omega(c)$ denote the $\omega$-limit set of the turning point $c$, and suppose that at least one of the following assumptions is satisfied: Then Lipschitz TPO holds for $(X,T)$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Remark
  • Theorem 2.1: Con16HLMXZ19
  • Lemma 2.2: YH96BZ15
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 36 more