The dynamics of nonautonomous dynamical systems with the large deviations theorem

Abstract

Let $(X,\mathcal{F})$ 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 $(X,\mathcal{F})$ is ergodicaly sensitive if the pair $(\mathcal{F},\mu )$ satisfies the large deviations theorem and $(X,\mathcal{F})$ is topologically strongly ergodic; $(X,\mathcal{F})$ is topologically ergodic if the pair $(\mathcal{F},\mu )$ satisfies the large deviations theorem in a sequence of positive integers; μ is expansive if the pair $(\mathcal{F},\mu )$ satisfies the large deviations theorem and $(X,\mathcal{F})$ is topologically strongly ergodic and each measurable set with positive measure with respect to μ has a nonempty interior.

Keywords:

• Large deviations theorem
• topological ergodicity
• ergodic sensitivity
• measure expansiveness

2010 Mathematics Subject Classifications:

• 54H20
• 37D45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by the National Natural Science Foundation of China (Grant No. 12061043).