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Maximal measures for flows with nonuniform structure

Qiao Liu, Tianyu Wang, Weisheng Wu

TL;DR

The paper addresses ergodic optimization for continuous flows with nonuniform hyperbolicity, focusing on the entropy spectrum of maximizing measures and the existence/uniqueness of equilibrium states. It extends symbolic non-Markov results to flows via an orbit-decomposition framework with weakened expansiveness and specification, and establishes dense $G_{\delta}$-type results for low-entropy maximizers while proving the persistence and abundance of high-entropy maximizers through ground-state constructions and convex-analytic perturbations. The authors apply the theory to geodesic and frame flows on rank-one manifolds of nonpositive curvature, deriving generic entropy-control results and detailed equilibrium-state behavior, including full support and continuity properties. The work broadens the landscape of ergodic optimization beyond uniformly hyperbolic or Markov systems, highlighting the rich structure of maximizing measures and their thermodynamic limits in nonuniformly hyperbolic dynamics, with potential implications for geometric flows and statistical properties of flows in curved spaces.

Abstract

In this paper, we study ergodic optimization of continuous functions for flows by concentrating on the entropy spectrum of their maximizing measures. Precisely, over a wide family of flows with non-uniformly hyperbolic structure, we obtain a picture describing coexistence of continuous functions whose maximizing measures have large and small entropy respectively in $C^0$-topology. Our proof relies on the orbit decomposition technique, originally introduced by Climenhaga and Thompson, for flows with weakened versions of expansiveness and specification property. In particular, our results extend \cite{STY} from non-Markov shift on symbolic spaces to a considerably broad class of continuous flows with nonuniform structure. To illustrate this, we apply our general results to both geodesic flows and frame flows over closed rank one manifolds of nonpositive curvature.

Maximal measures for flows with nonuniform structure

TL;DR

The paper addresses ergodic optimization for continuous flows with nonuniform hyperbolicity, focusing on the entropy spectrum of maximizing measures and the existence/uniqueness of equilibrium states. It extends symbolic non-Markov results to flows via an orbit-decomposition framework with weakened expansiveness and specification, and establishes dense -type results for low-entropy maximizers while proving the persistence and abundance of high-entropy maximizers through ground-state constructions and convex-analytic perturbations. The authors apply the theory to geodesic and frame flows on rank-one manifolds of nonpositive curvature, deriving generic entropy-control results and detailed equilibrium-state behavior, including full support and continuity properties. The work broadens the landscape of ergodic optimization beyond uniformly hyperbolic or Markov systems, highlighting the rich structure of maximizing measures and their thermodynamic limits in nonuniformly hyperbolic dynamics, with potential implications for geometric flows and statistical properties of flows in curved spaces.

Abstract

In this paper, we study ergodic optimization of continuous functions for flows by concentrating on the entropy spectrum of their maximizing measures. Precisely, over a wide family of flows with non-uniformly hyperbolic structure, we obtain a picture describing coexistence of continuous functions whose maximizing measures have large and small entropy respectively in -topology. Our proof relies on the orbit decomposition technique, originally introduced by Climenhaga and Thompson, for flows with weakened versions of expansiveness and specification property. In particular, our results extend \cite{STY} from non-Markov shift on symbolic spaces to a considerably broad class of continuous flows with nonuniform structure. To illustrate this, we apply our general results to both geodesic flows and frame flows over closed rank one manifolds of nonpositive curvature.
Paper Structure (23 sections, 33 theorems, 156 equations)

This paper contains 23 sections, 33 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. General results
  3. Application to geodesic flows
  4. Some history
  5. Organization of the paper
  6. Preliminaries
  7. Backgrounds on thermodynamical formalism
  8. Pressure and orbit decomposition
  9. Distortions, specification and expansiveness
  10. Uniqueness of equilibrium states
  11. Lemmas that come in handy
  12. Limit behavior of equilibrium states
  13. Functional analysis
  14. Some generic results for flows
  15. Proofs for Theorem \ref{['thm:main']}, \ref{['thm:pathelogic']}, and \ref{['thm:groundstates']}
  16. ...and 8 more sections

Key Result

Theorem A

Let $(X,F)$ be a fixed-points free continuous flow on a compact metric space. Suppose that $(X,F)$ has expansive type of E1 or E2, and $X\times\mathbb{R}^+$ admits a decomposition $(\mathcal{P},\mathcal{G},\mathcal{S})$ such that $\mathcal{G}$ satisfies weak controlled specification at all scale $\e Then $R_{H}$ is dense $G_\delta$ in $C(X,\mathbb{R})$ for all $H\in [H^{\perp},h)$.

Theorems & Definitions (59)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Theorem G
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 49 more