Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval

Abstract

We consider nonautonomous discrete dynamical systems $\{ f_n\}_{n\ge 1}$, where every $f_n$ is a surjective continuous map $[0,1]\to [0,1]$ such that $f_n$ converges uniformly to a map $f$. It is well-known that $f$ has positive topological entropy iff $\{ f_n\}_{n\ge 1}$ has. On the other hand, for systems with zero topological entropy,$\{ f_n\}_{n\ge 1}$ with very complex dynamics can converge even to the identity map.We study the following question: Which properties of the limit function $f$ are inherited by nonautonomous system$\{ f_n\}_{n\ge 1}$?We show that Li-Yorke chaos, distributional chaos DC1 and, for zero entropy maps, infinite $\omega$-limit sets are inherited by nonautonomous systems and, for zero entropy maps, we give a criterion on $f$ under which $\{ f_n\}_{n\ge 1}$ is DC1. More precisely, our main results are: (i) If $f$ is Li-Yorke chaotic then $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and the analogous implication is true for distributional chaos DC1;(ii) If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $\omega$-limit sets;(iii) We introduce new notion of a quasi horseshoe, a generalization of horseshoe. It turns out that $\{f_n\}_{n\ge 1}$ exhibits distributional chaos DC1 if $f$ has a quasi horseshoe. The last result is true for maps defined on arbitrary compact metric spaces.