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On recurrence and entropy in the hyperspace of continua in dimension one

Volume 263 / 2023

Domagoj Jelić, Piotr Oprocha Fundamenta Mathematicae 263 (2023), 23-50 MSC: Primary 37E25; Secondary 54F16. DOI: 10.4064/fm235-4-2023 Published online: 19 July 2023


We show that if $G$ is a topological graph, and $f\colon G\to G$ is a continuous map, then the induced map $\tilde {f}$ defined on the hyperspace $C(G)$ of all connected subsets of $G$ by the natural formula $\tilde {f}(C)=f(C)$ carries the same entropy as $f$. It is well known that this 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 the much simpler case of a compact interval by completely different tools.


  • Domagoj JelićFaculty of Science
    University of Split
    21000 Split, Croatia
  • Piotr OprochaFaculty of Applied Mathematics
    AGH University of Science and Technology
    30-059 Kraków, Poland
    Centre of Excellence IT4Innovations
    Institute for Research and Applications of Fuzzy Modeling
    University of Ostrava
    701 03 Ostrava 1, Czech Republic

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