Transitivity in nonautonomous systems

TL;DR

This paper surveys known results and introduces new findings on how transitivity properties are inherited between sequences of functions and their uniform limits in nonautonomous dynamical systems.

Contribution

It provides a comprehensive survey and new results on conditions for transitivity inheritance in nonautonomous systems converging to a limit function.

Findings

Established new sufficient conditions for transitivity inheritance.

Reviewed existing results on transitivity in nonautonomous systems.

Demonstrated cases where transitivity is preserved or not preserved in limits.

Abstract

Let $(X,d)$ be a metric space and $f_{1,\infty}=\{f_n\}_{i=0}^{\infty}$ be a sequence of continuous maps $f_n:X\rightarrow X$ such that $(f_n)$ converges uniformly to a continuous map $f$. We investigate which conditions ensure that the transitivity of functions $f_n$ or the transitivity of the nonautonomous system $(X,f_{1,\infty})$ is inherited to the limit function $f$ and vice versa. Such problem have been studied for instance by A. Fedeli, A. Le Donne or J. Li who give different sufficient condition for inheriting of transitivity from $f_n$ to $f$. In this paper we give a survey of know result relating to this problem and prove new results concerning to transitivity.