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An Improved Climenhaga-Thompson Criterion for Locally Maximal Sets

Maria Jose Pacifico, Fan Yang, Jiagang Yang, Gongran Yao

TL;DR

The paper strengthens the Climenhaga-Thompson Criterion for equilibrium states of continuous flows on compact locally maximal invariant sets by incorporating weak, non-uniform versions of specification, expansivity, and the Bowen property. It introduces a two-scale partition framework and a pressure-gap mechanism to control orbit segments within Λ and in a surrounding isolating neighborhood, enabling the construction of a unique, ergodic equilibrium state supported on Λ even when shadowing orbits must be pulled from larger sets. The approach leverages Decompositions into (P,G,S), tail (W)-specification, and Bowen-type distortion bounds to derive both lower and upper bounds on partition sums, culminating in a Gibbs-type structure and ergodicity. The results broaden applicability to flows with singularities and locally maximal sets beyond attractors, preserving a robust notion of uniqueness and providing tools for further study of sectional-hyperbolic and related attractors (e.g., Lorenz-type systems).

Abstract

We study the existence and uniqueness of equilibrium states for continuous flows on a compact, locally maximal invariant set under weak, non-uniform versions of specification, expansivity, and the Bowen property, further improving the Climenhaga-Thompson Criterion.

An Improved Climenhaga-Thompson Criterion for Locally Maximal Sets

TL;DR

The paper strengthens the Climenhaga-Thompson Criterion for equilibrium states of continuous flows on compact locally maximal invariant sets by incorporating weak, non-uniform versions of specification, expansivity, and the Bowen property. It introduces a two-scale partition framework and a pressure-gap mechanism to control orbit segments within Λ and in a surrounding isolating neighborhood, enabling the construction of a unique, ergodic equilibrium state supported on Λ even when shadowing orbits must be pulled from larger sets. The approach leverages Decompositions into (P,G,S), tail (W)-specification, and Bowen-type distortion bounds to derive both lower and upper bounds on partition sums, culminating in a Gibbs-type structure and ergodicity. The results broaden applicability to flows with singularities and locally maximal sets beyond attractors, preserving a robust notion of uniqueness and providing tools for further study of sectional-hyperbolic and related attractors (e.g., Lorenz-type systems).

Abstract

We study the existence and uniqueness of equilibrium states for continuous flows on a compact, locally maximal invariant set under weak, non-uniform versions of specification, expansivity, and the Bowen property, further improving the Climenhaga-Thompson Criterion.

Paper Structure

This paper contains 10 sections, 23 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary and Statement of Result
  3. Obstruction to expansivity and almost expansivity
  4. Decomposition of orbit segments
  5. The Bowen property and specification
  6. The Main Result
  7. Strategy of proof
  8. Lower bound on $\Lambda\times{\mathbb R}^+$ and upper bound on ${\mathcal{O}}(U_1)$
  9. Gibbs Property
  10. Finishing the proof of Theorem \ref{['m.1']}

Key Result

Theorem 1

Let $(f_t)_{t\in{\mathbb R}}$ be a continuous flow on a compact metric space $X$. Assume that $\Lambda$ is a compact invariant set of $(f_t)$ that is locally maximal with isolating neighborhood $U$, and $\phi: U\to{\mathbb R}$ a continuous potential function. Suppose that there exist $\varepsilon>0, Also assume there exists ${\mathcal{D}}_1\subset {\mathcal{O}}(U_1)$ which admits a $({\mathcal{P}

Figures (2)

  • Figure 1: Applying the specification
  • Figure 2: The shadowing orbit $(y, s_4+t_0)\in {\mathcal{O}}(U_1)$

Theorems & Definitions (39)

  • Remark 2.1
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • ...and 29 more