Fluctuations of ergodic sums on periodic orbits under specification

TL;DR

This paper investigates the fluctuations of ergodic sums on periodic orbits using specifications, establishing central limit theorems and showing convergence to mixtures of normal distributions in systems with a unique measure of maximal entropy.

Contribution

It introduces new CLT results for ergodic sums on periodic points under both global and local specifications, and demonstrates convergence to normal mixtures in certain systems.

Findings

Established Lindeberg-type CLTs for ergodic sums

Proved weak convergence to mixtures of normal distributions

Highlighted the importance of decomposing variances into global and local sources

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.