Marta Štefánková | Searcharxiv

arXiv2016

On generic and dense chaos for maps induced on hyperspaces

A continuous map $f$ on a compact metric space $X$ induces in a natural way the map $\tilde f$ on the hyperspace $\mathcal K(X)$ of all closed non-empty subsets of $X$. We study the question of transmission of chaos between $f$ and $\tilde f$. We deal with generic, generic $\varepsilon$-, dense and dense $\varepsilon$-chaos for interval maps. We prove that all four types of chaos transmit from $f$ to $\tilde f$, while the converse transmission from $\tilde f$ to $f$ is true for generic, generic $\varepsilon$- and dense $\varepsilon$-chaos. Moreover, the transmission of dense $\varepsilon$- and generic $\varepsilon$-chaos from $\tilde f$ to $f$ is true for maps on general compact metric spaces.

math.DS↗

arXiv2013

Inheriting of chaos in nonautonomous dynamical systems

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 $f$. We show, among others, that if $f$ is chaotic in the sense of Li and Yorke then the nonautonomous system $ \{ f_n\}_{n\ge 1}$ is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If $f$ has zero topological entropy then the nonautonomous system inherits its infinite $ω$-limit sets.

math.DS↗