Fluctuations of time averages around closed geodesics in non-positive curvature

TL;DR

This paper proves an asymptotic Central Limit Theorem for measures derived from closed geodesics on non-positive curvature manifolds, extending previous results to non-uniform and continuous-time settings.

Contribution

It extends the asymptotic CLT for periodic orbits to non-uniform, continuous-time geodesic flows with new techniques and broader conditions.

Findings

Established asymptotic CLT for measures from closed geodesics

Extended techniques to non-uniform, continuous-time flows

Results apply to weighted periodic orbit measures and equilibrium states

Abstract

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the Bowen-Margulis-Knieper measure of maximal entropy. The technique expands on ideas of Denker, Senti and Zhang, who proved this type of asymptotic Lindeberg Central Limit Theorem on periodic orbits for expansive maps with the specification property. We extend these techniques from the uniform to the non-uniform setting, and from discrete-time to continuous-time. We consider H\"older observables subject only to the Lindeberg condition and a weak positive variance condition. If we assume a natural strengthened positive variance condition, the Lindeberg condition is always satisfied. Our results extend to dynamical arrays of H\"older observables, and to weighted periodic orbit measures which converge to a unique equilibrium state.