The dynamics of nonautonomous dynamical systems with the large deviations theorem

Let be a nonautonomous dynamical system and μ be a Borel measure on X with a full support. The purpose of this paper is to introduce the concepts of large deviations theorem and expansive measure for nonautonomous discrete systems and investigate the dynamics of nonautonomous dynamical systems with the large deviations theorem. Actually, it is proved that is ergodicaly sensitive if the pair satisfies the large deviations theorem and is topologically strongly ergodic; is topologically ergodic if the pair satisfies the large deviations theorem in a sequence of positive integers; μ is expansive if the pair satisfies the large deviations theorem and is topologically strongly ergodic and each measurable set with positive measure with respect to μ has a nonempty interior.