On the large deviations theorem of weaker types

TL;DR

This paper introduces various weaker large deviations theorems, studies their implications for dynamical systems' ergodic and chaotic properties, and establishes new connections between these theorems and system classifications.

Contribution

It systematically defines weaker large deviations theorems and explores their implications for ergodic and chaotic properties of dynamical systems, including new equivalences and classifications.

Findings

Systems satisfying type I' are ergodic.

Type III systems are E-systems.

Systems satisfying the central limit theorem do not exist.

Abstract

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e., type I, type I', type II, type II', type III, and type III', and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large eviations theorem of various types. Some characteristics of the ergodic measure are obtained and then applied to prove that every dynamical system satisfying the large deviations theorem of type I' is ergodic, which is equivalent to the large deviations theorem of type II' in this regard, and that every uniquely ergodic dynamical system restricted on its support satisfies the large deviations theorem. Moreover, we prove that every dynamical system satisfying the large deviations theorem of type III is an $E$-system. Finally, we show that a dynamical system satisfying the central limit theorem, introduced in [Y. Niu, Y. Wang, Statist. Probab. Lett., {\bf 80} (2010), 1180--1184], does not exist.