Fluctuations of ergodic sums on periodic orbits under specification

Manfred Denker; Samuel Senti; Xuan Zhang

doi:10.3934/dcds.2020197

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2020, Volume 40, Issue 8: 4665-4687. Doi: 10.3934/dcds.2020197

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Fluctuations of ergodic sums on periodic orbits under specification

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ 21941-909, Brazil
3.
Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP 05508-090, Brazil

Received: April 2019

Published online: August 2020

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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.

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Mathematics Subject Classification: Primary: 37A50; Secondary: 37B99, 60F05.

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