N-Convergence and Chaotic Properties of Non-autonomous Discrete Systems

Let f(1,infinity) := (f(n))(infinity)(n=1) be a non-autonomous dynamical system on a compact metric space X. For a given N? N we consider Nth iterate f ([N]similar to)(1,infinity) of the system (i.e. f ([N]similar to)(1,infinity) = (f(N) (N (n-1)+1))(infinity) (n =1), where f(i)(n) = f(i)+((n-1))o similar to. . . o f(i) and f(1)(0) = id(X).) We also investigate N- convergent non-autonomous systems this is weaker notion than uniform convergence. In this setting we generalize results regarding different types of chaos. Particularly we prov(1) f(1,infinity) is distributionally chaotic of type 1 if and only if f ([N]similar to)(1,infinity) is also.(2) f(1,infinity) is distributionally chaotic of type 2 if and only if f ([N]similar to)(1,infinity)is also.(3) f(1,infinity) is distributionally chaotic of type 221 if and only if f ([N]similar to)(1,infinity) is also.(4) f(1,infinity) is P-chaotic if and only if f ([N]similar to)(1,infinity) is also, where P-chaos denotes one of the following properties: Li-Yorke chaos, dense chaos, densed-chaos, generic chaos, generic d-chaos, Li-Yorke sensitivity and spatio-temporal chaos.(5) f(1,)infinity is sensitive (resp. ergodically sensitive) if and only if f ([N]similar to)(1,infinity) is also.We also discuss and partly solve a problem given by [Xinxing Wu, Peiyong Zhu, Chaos in a class of non-autonomous discrete systems. Applied Mathematics Letters 26 (2013) 431-436]. Furthermore, we present two examples which show that conditions of N-convergence and continuity in some results cannot be removed.