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arXiv · 2501.05801

On recurrence and entropy in hyperspace of continua in dimension one

Domagoj Jelić·Piotr Oprocha
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Abstract

We show that if $G$ is a topological graph, and $f$ is continuous map, then the induced map $\tilde{f}$ acting on the hyperspace $C(G)$ of all connected subsets of $G$ by natural formula $\tilde{f}(C)=f(C)$ carries the same entropy as $f$. This is well known that it does not hold on the larger hyperspace of all compact subsets. Also negative examples were given for the hyperspace $C(X)$ on some continua $X$, including dendrites. Our work extends previous positive results obtained first for much simpler case of compact interval by completely different tools.

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Domagoj Jelić, Piotr Oprocha. 2025-01-10. On recurrence and entropy in hyperspace of continua in dimension one. https://doi.org/10.4064/fm235-4-2023

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