Fluctuations of ergodic sums on periodic orbits under specification

Abstract

We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal entropy, we show weak convergence of ergodic sums to a mixture of normal distributions. Our results suggest decomposing the variances of ergodic sums according to global and local sources.

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

“…The classical Central Limit Theorem can be obtained quite robustly using a variety of techniques. We do not attempt to survey the literature here, but we recommend recent papers by [9,1,8,5,17,14] for an excellent paper trail. One might expect the classical CLT to hold in the setting of this paper, but none of these proof techniques are currently known to apply.…”

“…[11]. We will formulate our results using a dynamical version of the Lindeberg condition on periodic orbits, following Denker, Senti and Zhang [8].…”

“…It can be obtained quite easily from the classical Lindeberg CLT [11]. We give a short formal proof based on verifying Denker-Senti-Zhang's hypotheses [8,Proposition 3.3].…”

“…Roughly, the Lindeberg condition guarantees that the variance of a single random variable is negligible in comparison to the sum of all the variances. This idea was recently explored by Denker, Senti and Zhang [8] in the setting of maps with the specification property. They showed that a Lindeberg condition on the sequence of periodic orbit measures was equivalent to a Central Limit Theorem in the limit.…”

“…The technique is an extension of Denker, Senti and Zhang [8]. The idea is to build ǫ-independent collections of regular closed geodesics whose growth rate is the topological entropy.…”