Sensitivity and Transitivity of Systems Satisfying the Large Deviations Theorem in a Sequence

Abstract

Let [Formula: see text] be a continuous selfmap of a nontrivial compact metric space [Formula: see text]. This paper derives the large deviations theorem in a sequence. It is shown that if [Formula: see text] satisfies the large deviations theorem in a sequence then it is an E-system which implies that [Formula: see text] is syndetically transitive, and that if the pair [Formula: see text] satisfies the large deviations theorem in a sequence and [Formula: see text] is an infinite space, then [Formula: see text] is syndetically sensitive, which implies that [Formula: see text] is ergodically sensitive. Moreover, it is proved that if the pair [Formula: see text] satisfies the large deviations theorem in a syndetic sequence, then [Formula: see text] is an ergodic measure with supp[Formula: see text]. Our main results extend and improve the corresponding results in the literature.