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Gibbs measures for geodesic flow on CAT(-1) spaces

Caleb Dilsavor, Daniel J. Thompson

TL;DR

The paper extends the thermodynamic formalism to geodesic flows on CAT(-1) spaces with discrete group actions by constructing weighted Patterson densities for bounded Bowen potentials and producing a Gibbs measure m_φ on GX_0. It introduces a coarse, non-tempered framework that replaces ill-defined orbit integrals with a supremum weight ar{φ}, yielding a Gibbs quasi-cocycle and a Patterson–Sullivan type density, from which a Gibbs current and state are built. A Ledrappier–Mañé–Otal–Peigné partition is constructed to recover entropy considerations without smooth structure, under a finite upper box-dimension assumption; this enables the proof that finite Gibbs states are equilibrium states, and they prove uniqueness via a robust KL-divergence argument. Collectively, the results generalize the Bowen–Margulis/equilibrium-state theory to CAT(-1) spaces with branching, eliminating previous restrictions such as tempered potentials or curvature-derivative bounds, and establishing a versatile framework for non-compact dynamics with coarse geometric control.

Abstract

For a proper geodesically complete CAT(-1) space equipped with a discrete non-elementary action, and a bounded continuous potential with the Bowen property, we construct weighted quasi-conformal Patterson densities and use them to build a Gibbs measure on the space of geodesic lines. Our construction yields a Gibbs measure with local product structure for any potential in this class, which includes bounded Hölder continuous potentials. Furthermore, if the Gibbs measure is finite, then we prove that it is the unique equilibrium state. In contrast to previous results in this direction, we do not require any condition that the potential must take the same value on two geodesic lines which share a common segment.

Gibbs measures for geodesic flow on CAT(-1) spaces

TL;DR

The paper extends the thermodynamic formalism to geodesic flows on CAT(-1) spaces with discrete group actions by constructing weighted Patterson densities for bounded Bowen potentials and producing a Gibbs measure m_φ on GX_0. It introduces a coarse, non-tempered framework that replaces ill-defined orbit integrals with a supremum weight ar{φ}, yielding a Gibbs quasi-cocycle and a Patterson–Sullivan type density, from which a Gibbs current and state are built. A Ledrappier–Mañé–Otal–Peigné partition is constructed to recover entropy considerations without smooth structure, under a finite upper box-dimension assumption; this enables the proof that finite Gibbs states are equilibrium states, and they prove uniqueness via a robust KL-divergence argument. Collectively, the results generalize the Bowen–Margulis/equilibrium-state theory to CAT(-1) spaces with branching, eliminating previous restrictions such as tempered potentials or curvature-derivative bounds, and establishing a versatile framework for non-compact dynamics with coarse geometric control.

Abstract

For a proper geodesically complete CAT(-1) space equipped with a discrete non-elementary action, and a bounded continuous potential with the Bowen property, we construct weighted quasi-conformal Patterson densities and use them to build a Gibbs measure on the space of geodesic lines. Our construction yields a Gibbs measure with local product structure for any potential in this class, which includes bounded Hölder continuous potentials. Furthermore, if the Gibbs measure is finite, then we prove that it is the unique equilibrium state. In contrast to previous results in this direction, we do not require any condition that the potential must take the same value on two geodesic lines which share a common segment.
Paper Structure (35 sections, 61 theorems, 243 equations, 3 figures)

This paper contains 35 sections, 61 theorems, 243 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous results
  3. Proof ideas and challenges
  4. On the proof of Theorem \ref{['maintheorem']}
  5. On the proof of Theorem \ref{['thmx:op']}
  6. On the proof of Theorem \ref{['thmx:existunique']}
  7. Organization of paper
  8. Preliminaries
  9. $\operatorname{CAT}(-1)$ spaces and geodesic flow
  10. Busemann functions and partitions into dynamical sets
  11. Thermodynamic Formalism and the Gibbs property
  12. Thermodynamic Formalism for geodesic flows on non-compact manifolds
  13. Facts from coarse geometry
  14. Invariance from Quasi-invariance
  15. More on isometries
  16. ...and 20 more sections

Key Result

Theorem 1

Suppose that $\Gamma$ has finite critical exponent. Let $\varphi: GX_0 \to \mathbb{R}$ be a bounded continuous potential function whose lift to $GX$ satisfies the Bowen property. Then there exists a Patterson-Sullivan construction of a flow-invariant Radon measure $m_\varphi$ on $GX_0$ which is full

Figures (3)

  • Figure 3.1: The tree $\mathcal{T}$ used in the proof of (a)
  • Figure 3.2: The tree $\mathcal{T}$ used in the proof of (b)
  • Figure 4.1: The set $B(x,y, R)$ and a local quasi-geodesic

Theorems & Definitions (132)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Paulin, Pollicott and Schapira
  • Definition 2.6
  • Definition 2.7
  • ...and 122 more