  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.
