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Counting and equidistribution of strongly reversible closed geodesics in negative curvature

Jouni Parkkonen, Frédéric Paulin

TL;DR

This work develops a comprehensive framework for counting and equidistributing strongly reversible closed geodesics in negatively curved spaces with 2-torsion, by linking strongly reversible loxodromics to common perpendiculars between fixed sets of involutions. It extends results of Sarnak and Erlandsson–Souto to a thermodynamic formalism setting, providing asymptotics and equidistribution toward the Bowen–Margulis measure with explicit constants given by skinning measures and Gibbs data. The analysis unifies manifold and tree settings, translating geometric counting into Perron–Frobenius-type spectral data and exploiting multiplicity formulas for dihedral subgroups and reversible pairs. The paper also demonstrates concrete applications across real and complex hyperbolic spaces and Bruhat–Tits trees, including real hyperbolic Coxeter groups and Nagao lattices, yielding explicit constants and connecting to classical reciprocity phenomena in modular and Hecke-type groups.

Abstract

Let $M$ be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order $2$. Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in $M$, and prove their equidistribution towards the Bowen-Margulis measure. The result is proved in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with $2$-torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.

Counting and equidistribution of strongly reversible closed geodesics in negative curvature

TL;DR

This work develops a comprehensive framework for counting and equidistributing strongly reversible closed geodesics in negatively curved spaces with 2-torsion, by linking strongly reversible loxodromics to common perpendiculars between fixed sets of involutions. It extends results of Sarnak and Erlandsson–Souto to a thermodynamic formalism setting, providing asymptotics and equidistribution toward the Bowen–Margulis measure with explicit constants given by skinning measures and Gibbs data. The analysis unifies manifold and tree settings, translating geometric counting into Perron–Frobenius-type spectral data and exploiting multiplicity formulas for dihedral subgroups and reversible pairs. The paper also demonstrates concrete applications across real and complex hyperbolic spaces and Bruhat–Tits trees, including real hyperbolic Coxeter groups and Nagao lattices, yielding explicit constants and connecting to classical reciprocity phenomena in modular and Hecke-type groups.

Abstract

Let be a pinched negatively curved Riemannian orbifold, whose fundamental group has torsion of order . Generalizing results of Sarnak and Erlandsson-Souto for constant curvature oriented surfaces, and with very different techniques, we give an asymptotic counting result on the number of strongly reversible periodic orbits of the geodesic flow in , and prove their equidistribution towards the Bowen-Margulis measure. The result is proved in the more general setting with weights coming from thermodynamic formalism, and also in the analogous setting of graphs of groups with -torsion. We give new examples in real hyperbolic Coxeter groups, complex hyperbolic orbifolds and graphs of groups.
Paper Structure (10 sections, 17 theorems, 99 equations)

This paper contains 10 sections, 17 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Strongly reversible elements and common perpendiculars
  3. Multisets of strongly reversible loxodromic elements
  4. Dynamics of the geodesic flow
  5. Counting strongly reversible closed geodesics
  6. Equidistribution of strongly reversible closed geodesics
  7. Examples
  8. Examples in real hyperbolic spaces
  9. Examples in complex hyperbolic space
  10. Examples in trees

Key Result

Theorem 1.1

Assume that the Bowen-Margulis measure $m_{\rm BM}$ is finite and mixing for the geodesic flow on $T^1M$, and that the skinning measures $\sigma^+_{I}$ and $\sigma^-_{J}$ are finite nonzero. (1) As $T\rightarrow+\infty$, we have (2) When $J=I$, the sum (with multiplicities) of the Lebesgue measures along the $\{I,I\}$-reversible closed geodesics of length at most $T$, normalized to be a probabili

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 3.1
  • ...and 16 more