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Thermodynamic formalism for non-uniform systems with controlled specification and entropy expansiveness

Tianyu Wang, Weisheng Wu

TL;DR

The paper develops a robust thermodynamic framework for dynamical flows with non-uniform structure by extending orbit-decomposition methods to entropy-expansive settings and introducing a weakened weak-controlled specification. It proves that a unique equilibrium state exists when an obstruction to expansiveness is strictly below the topological pressure and when the orbit decomposition satisfies quantifiable distortion and gap conditions, with proofs anchored in precise partition-sum estimates at multiple scales. The theory is then instantiated in two concrete applications: suspension flows with roof functions meeting a weak Walters condition and frame flows on rank-one manifolds of nonpositive curvature under a curvature-bunched regime, yielding uniqueness for a broad class of potentials. The results unify non-uniform hyperbolic phenomena and provide practical tools for proving uniqueness of equilibrium states beyond uniformly hyperbolic systems, through explicit decompositions and pressure-gap arguments for flows.

Abstract

We study thermodynamic formalism of dynamical systems with non-uniform structure. Precisely, we obtain the uniqueness of equilibrium states for a family of non-uniformly expansive flows by generalizing Climenhaga-Thompson's orbit decomposition criteria. In particular, such family includes entropy expansive flows. Meanwhile, the essential part of the decomposition is allowed to satisfy an even weaker version of specification, namely controlled specification, thus also extends the corresponding results by Pavlov. Two applications of our abstract theorems are explored. Firstly, we introduce a notion of regularity condition called weak Walters condition, and study the uniqueness of measure of maximal entropy for a suspension flow with roof function satisfying such condition. Secondly, we investigate topologically transitive frame flows on rank one manifolds of nonpositive curvature, which is a group extension of nonuniformly hyperbolic flows. Under a bunched curvature condition and running a Gauss-Bonnet type of argument, we show the uniqueness of equilibrium states with respect to certain potentials.

Thermodynamic formalism for non-uniform systems with controlled specification and entropy expansiveness

TL;DR

The paper develops a robust thermodynamic framework for dynamical flows with non-uniform structure by extending orbit-decomposition methods to entropy-expansive settings and introducing a weakened weak-controlled specification. It proves that a unique equilibrium state exists when an obstruction to expansiveness is strictly below the topological pressure and when the orbit decomposition satisfies quantifiable distortion and gap conditions, with proofs anchored in precise partition-sum estimates at multiple scales. The theory is then instantiated in two concrete applications: suspension flows with roof functions meeting a weak Walters condition and frame flows on rank-one manifolds of nonpositive curvature under a curvature-bunched regime, yielding uniqueness for a broad class of potentials. The results unify non-uniform hyperbolic phenomena and provide practical tools for proving uniqueness of equilibrium states beyond uniformly hyperbolic systems, through explicit decompositions and pressure-gap arguments for flows.

Abstract

We study thermodynamic formalism of dynamical systems with non-uniform structure. Precisely, we obtain the uniqueness of equilibrium states for a family of non-uniformly expansive flows by generalizing Climenhaga-Thompson's orbit decomposition criteria. In particular, such family includes entropy expansive flows. Meanwhile, the essential part of the decomposition is allowed to satisfy an even weaker version of specification, namely controlled specification, thus also extends the corresponding results by Pavlov. Two applications of our abstract theorems are explored. Firstly, we introduce a notion of regularity condition called weak Walters condition, and study the uniqueness of measure of maximal entropy for a suspension flow with roof function satisfying such condition. Secondly, we investigate topologically transitive frame flows on rank one manifolds of nonpositive curvature, which is a group extension of nonuniformly hyperbolic flows. Under a bunched curvature condition and running a Gauss-Bonnet type of argument, we show the uniqueness of equilibrium states with respect to certain potentials.

Paper Structure

This paper contains 22 sections, 34 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partition sums and topological pressure
  4. Variations and distortions
  5. Specification
  6. Decompositions
  7. Expansiveness
  8. Maximal ergodic theorem and maximal inequality
  9. Proof of Theorem \ref{['thm:weprove']}
  10. Upper bound of partition sums
  11. Lower bound of partition sums
  12. Applications
  13. Suspension flow
  14. Preliminaries
  15. Proof of Theorem \ref{['thm:suspension']}
  16. ...and 7 more sections

Key Result

Theorem A

Let $(X,F)$ be a continuous flow on a compact metric space, and $\varphi:X\to \mathbb{R}$ be a continuous potential. Suppose there exists some $\varepsilon>0$ at which $(X,F)$ is entropy expansive, and there exists $\mathcal{D}\subset X\times\mathbb{R}^+$ which admits a decomposition $(\mathcal{P},\ Then $(X,F,\varphi)$ has a unique equilibrium state.

Theorems & Definitions (67)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 57 more