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Equidistribution of divergent geodesics in negative curvature

Jouni Parkkonen, Frédéric Paulin, Rafael Sayous

TL;DR

The paper studies equidistribution of divergent geodesics in finite-volume negatively curved manifolds and their tree-quotient analogues. By introducing a natural complexity for divergent geodesics and leveraging skinning measures and the Bowen–Margulis framework, it proves that appropriately rescaled Lebesgue measures along divergent geodesic orbits converge to the measure of maximal entropy $m_{\rm BM}$ as complexity grows. It also provides counting asymptotics for divergent geodesics and extends the analysis to geometrically finite tree quotients, where convergence occurs to the Bowen–Margulis measure on $\Gamma\backslash{\cal G}X$ (or its even-time variant). The results are sharp and yield explicit error terms under stronger dynamical assumptions, connecting noncompact negative curvature dynamics with explicit boundary data and arithmetic tree lattices.

Abstract

In the unit tangent bundle of noncompact finite volume negatively curved Riemannian manifolds, we prove the equidistribution towards the measure of maximal entropy for the geodesic flow of the Lebesgue measure along the divergent geodesic flow orbits, as their complexity tends to infinity. We prove the analogous result for geometrically finite tree quotients, where the equidistribution takes place in the quotient space of geodesic lines towards the Bowen-Margulis measure.

Equidistribution of divergent geodesics in negative curvature

TL;DR

The paper studies equidistribution of divergent geodesics in finite-volume negatively curved manifolds and their tree-quotient analogues. By introducing a natural complexity for divergent geodesics and leveraging skinning measures and the Bowen–Margulis framework, it proves that appropriately rescaled Lebesgue measures along divergent geodesic orbits converge to the measure of maximal entropy as complexity grows. It also provides counting asymptotics for divergent geodesics and extends the analysis to geometrically finite tree quotients, where convergence occurs to the Bowen–Margulis measure on (or its even-time variant). The results are sharp and yield explicit error terms under stronger dynamical assumptions, connecting noncompact negative curvature dynamics with explicit boundary data and arithmetic tree lattices.

Abstract

In the unit tangent bundle of noncompact finite volume negatively curved Riemannian manifolds, we prove the equidistribution towards the measure of maximal entropy for the geodesic flow of the Lebesgue measure along the divergent geodesic flow orbits, as their complexity tends to infinity. We prove the analogous result for geometrically finite tree quotients, where the equidistribution takes place in the quotient space of geodesic lines towards the Bowen-Margulis measure.
Paper Structure (6 sections, 7 theorems, 90 equations)

This paper contains 6 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Background on negative curvature
  3. Generalities on divergent geodesics
  4. Divergent geodesics and common perpendiculars
  5. Equidistribution of divergent geodesics
  6. Equidistribution of divergent geodesics in geometrically finite tree quotients

Key Result

Theorem 1.1

Let $M$ be a finite volume complete connected Riemannian good orbifold with dimension at least $2$ and pinched negative curvature at most $-1$. Let ${\cal T}$ be a nonempty subset of $\operatorname{End}(M)^2$. Assume that the Bowen-Margulis measure on $T^1M$ is finite. (1) As $T\rightarrow+\infty$, If, furthermore, $M$ is locally symmetric with exponential decay of correlations, then there exists

Theorems & Definitions (7)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Lemma 4.1
  • Theorem 5.1
  • Theorem 6.1
  • Theorem 6.2