Entropy of induced maps of regular curves homeomorphisms

TL;DR

This paper investigates the topological entropy of induced maps on hyperspaces of regular curves, establishing conditions under which the entropy is infinite or zero, and exploring their dynamical properties.

Contribution

It characterizes when the induced systems on hyperspaces have infinite or zero entropy, especially for regular curves, and links entropy to Li-Yorke chaos and periodic subcontinua.

Findings

Induced system $(2^X,2^f)$ has infinite entropy if $X\setminus \Omega(f)$ is not empty.

For regular curves, $(2^X,2^f)$ has infinite entropy iff $X\setminus \Omega(f)$ is not empty.

The entropy of induced systems $(2^X,2^f)$ or $(C(X),C(f))$ is either 0 or infinity.

Abstract

Let $f:X\to X$ be a self homeomorphism of a continuum $X$, we show that the topological entropy of the induced system $(2^X,2^f)$ is infinite provided that $X\setminus \Omega(f)$ is not empty. If furthermore $X$ is a regular curve then it is shown that $(2^X,2^f)$ has infinite topological entropy if and only if $X\setminus \Omega(f)$ is not empty. Moreover we prove for the induced system $(C(X),C(f))$ the equivalence between the following properties: (i) zero topological entropy; (ii) there is no Li-Yorke pair and (iii) for any periodic subcontinnum $A$ of $X$ and any connected component $C$ of $X\setminus \Omega(f)$, $C\subset A$ if $A\cap C\neq \emptyset$. In particular, the topological entropy of either $(2^X,2^f)$ or $(C(X),C(f))$ has only two possible values $0$ or $\infty$. At the end, we give an example of a pointwise periodic rational curve homeomorphism $F:Y\to Y$ with infinite topological entropy induced map $C(F)$.