Counting and equidistribution of strongly reversible closed geodesics in negative curvature

TL;DR

This paper establishes an asymptotic count and equidistribution of strongly reversible closed geodesics in negatively curved orbifolds with torsion, extending previous results to more general settings and new examples.

Contribution

It generalizes counting and equidistribution results for reversible geodesics to orbifolds with torsion and applies thermodynamic formalism, introducing new examples in hyperbolic and graph group settings.

Findings

Asymptotic counting of strongly reversible geodesics established.

Proves equidistribution of these geodesics towards the Bowen-Margulis measure.

Extends results to orbifolds with torsion and graphs of 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.