Flexibility of statistical properties for smooth systems satisfying the central limit theorem

TL;DR

This paper introduces new classes of smooth dynamical systems that satisfy the Central Limit Theorem while exhibiting diverse statistical properties such as zero entropy, various mixing conditions, and non-Bernoulli behavior, expanding understanding of statistical behaviors in dynamical systems.

Contribution

It provides explicit examples of smooth systems satisfying the CLT with a range of statistical properties, highlighting the flexibility of these properties in such systems.

Findings

Constructed systems with zero entropy satisfying CLT

Examples of systems with weak but not strong mixing

Systems with polynomial mixing not K or Bernoulli

Abstract

In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties: (1) zero entropy; (2) weak but not strong mixing; (3) (polynomially) mixing but not $K$; (4) $K$ but not Bernoulli; (5) non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index $1$.