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arXiv · 1608.06269

On generic and dense chaos for maps induced on hyperspaces

Michaela Mlíchová·Marta Štefánková
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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.

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Michaela Mlíchová, Marta Štefánková. 2016-08-22. On generic and dense chaos for maps induced on hyperspaces. https://arxiv.org/abs/1608.06269

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