On generic and dense chaos for maps induced on hyperspaces

TL;DR

This paper investigates how different types of chaos in a continuous map on a compact metric space are transmitted to the induced map on the hyperspace of closed subsets, establishing conditions for transmission and converse transmission.

Contribution

It proves that various forms of chaos always transmit from a map to its hyperspace induced map, and identifies conditions under which the reverse transmission occurs.

Findings

All four types of chaos transmit from $f$ to $ ilde f$.

Transmission from $ ilde f$ to $f$ holds for generic, generic $ ext{ extepsilon}$-, and dense $ ext{ extepsilon}$-chaos.

Transmission of dense $ ext{ extepsilon}$- and generic $ ext{ extepsilon}$-chaos from $ ilde f$ to $f$ holds for general compact metric spaces.

Abstract

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.