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Equilibrium states by synchronization, symbolic extensions, and factors

Katrin Gelfert, Dominik Kwietniak, Yuri Lima

TL;DR

The paper advances the thermodynamic analysis of geodesic flows on compact rank-one surfaces with nonpositive curvature by integrating time-preserving topological factors, synchronization of equilibrium states through time-changes, and symbolic extensions. It establishes a synchronization framework showing that under suitable time-reparametrizations, equilibrium states for hyperbolic potentials correspond to measures of maximal entropy for the time-changed flow, enabling transfer of uniqueness results between extensions and factors. Applying these tools to geodesic flows, it proves that for $q<1$ the scaled geometric potential $q\varphi^{({\rm u})}$ has a unique equilibrium state, with a detailed examination of when hyperbolicity and coexisting states arise for $q=1$ due to regular vs singular dynamics. The overarching contribution is a versatile methodology that preserves and transfers entropy-related properties across time-changes and symbolic or topological factor constructions, yielding a robust approach to uniqueness and structure of equilibrium states in nonuniformly hyperbolic settings and offering insights into expansivity, local product structure, and pseudo-orbit tracing in this geometric context.

Abstract

We combine the two classical topological concepts, time-preserving topological factors and synchronizing time-changes of a continuous flow, and explore some of their thermodynamic consequences. Particular focus is put on equilibrium states and, in particular, measures of maximal entropy, with emphasis on geodesic flows on rank-one surfaces of nonpositive curvature and their time-preserving expansive topological factors for which we investigate the scaled geometric potentials.

Equilibrium states by synchronization, symbolic extensions, and factors

TL;DR

The paper advances the thermodynamic analysis of geodesic flows on compact rank-one surfaces with nonpositive curvature by integrating time-preserving topological factors, synchronization of equilibrium states through time-changes, and symbolic extensions. It establishes a synchronization framework showing that under suitable time-reparametrizations, equilibrium states for hyperbolic potentials correspond to measures of maximal entropy for the time-changed flow, enabling transfer of uniqueness results between extensions and factors. Applying these tools to geodesic flows, it proves that for the scaled geometric potential has a unique equilibrium state, with a detailed examination of when hyperbolicity and coexisting states arise for due to regular vs singular dynamics. The overarching contribution is a versatile methodology that preserves and transfers entropy-related properties across time-changes and symbolic or topological factor constructions, yielding a robust approach to uniqueness and structure of equilibrium states in nonuniformly hyperbolic settings and offering insights into expansivity, local product structure, and pseudo-orbit tracing in this geometric context.

Abstract

We combine the two classical topological concepts, time-preserving topological factors and synchronizing time-changes of a continuous flow, and explore some of their thermodynamic consequences. Particular focus is put on equilibrium states and, in particular, measures of maximal entropy, with emphasis on geodesic flows on rank-one surfaces of nonpositive curvature and their time-preserving expansive topological factors for which we investigate the scaled geometric potentials.

Paper Structure

This paper contains 23 sections, 27 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Topological pressure and entropy
  3. Equilibrium states and measures of maximal entropy
  4. Measures of maximal entropy for (symbolic) extensions
  5. Measures of maximal entropy for factors
  6. Time-changes
  7. Bijection between spaces of (ergodic) invariant measures
  8. Synchronizing time-changes
  9. Time-change in a TMF
  10. Topological dynamics
  11. Expansivity
  12. Mixing and transitivity
  13. Local product structure
  14. Orbit closing property
  15. Pseudo-orbit-tracing and closing properties
  16. ...and 8 more sections

Key Result

Theorem A

Let $M$ be a connected compact rank-one Riemannian surface of nonpositive sectional curvature, and let $G=(g^t)_{t\in\mathbb{R}}$ be its geodesic flow on the unit tangent bundle. For every $q<1$, the potential $q\varphi^{({\rm u})}$ has a unique equilibrium state (with respect to $G$). Moreover, thi

Figures (1)

  • Figure 1: Co-existence of the equilibrium states $\widetilde{m}$ and $\mu_0$ for the potential $q\varphi^{({\rm u})}$ at $q=1$

Theorems & Definitions (63)

  • Theorem A
  • Theorem B: Synchronization of equilibrium states
  • Corollary 1.1
  • Theorem C
  • Theorem D: Pseudo-orbit-tracing and orbit closing property
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 53 more