On recurrence and entropy in hyperspace of continua in dimension one

TL;DR

This paper proves that for topological graphs, the entropy of a continuous map is preserved when induced on the hyperspace of connected subsets, extending known results from simpler cases to more complex continua.

Contribution

It establishes entropy preservation for induced maps on the hyperspace of connected subsets of topological graphs, expanding previous results beyond intervals.

Findings

Entropy is preserved under induced maps on $C(G)$ for topological graphs.

Counterexamples exist for larger hyperspaces and more complex continua.

The work extends results from intervals to more general one-dimensional continua.

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.