On the Large Deviations Theorem of Weaker Types

Abstract

In this paper, we introduce the concepts of the large deviations theorem of weaker types, i.e. type I, type I[Formula: see text], type II, type II[Formula: see text], type III, and type III[Formula: see text], and present a systematic study of the ergodic and chaotic properties of dynamical systems satisfying the large deviations 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[Formula: see text] is ergodic, which is equivalent to the large deviations theorem of type II[Formula: see text] 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 [Formula: see text]-system.