The Ergodicity and Sensitivity of Nonautonomous Discrete Dynamical Systems

Abstract

Let (E,h1,∞) be a nonautonomous discrete dynamical system (briefly, N.D.D.S.) that is defined by a sequence (hj)j=1∞ of continuous maps hj:E→E over a nontrivial metric space (E,d). This paper defines and discusses some forms of ergodicity and sensitivity for the system (E,h1,∞) by upper density, lower density, density, and a sequence of positive integers. Under some conditions, if the rate of convergence at which (hj)j=1∞ converges to the limit map h is “fast enough” with respect to a sequence of positive integers with a density of one, it is shown that several sensitivity properties for the N.D.D.S. (E,h1,∞) are the same as those properties of the system (E,h). Some sufficient conditions for the N.D.D.S. (E,h1,∞) to have stronger sensitivity properties are also presented. The conditions in our results are less restrictive than those in some existing works, and the conclusions of all the theorems in this paper improve upon those of previous studies. Thus, these results are extensions of the existing ones.

Keywords

• nonautonomous discrete dynamical systems
• ergodicity
• sensitivity