Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms

TL;DR

This paper investigates the dynamics of pointwise periodic homeomorphisms on hyperspaces, revealing conditions for chaos, rigidity, and the structure of minimal sets, including the presence of adding machines and the absence of Devaney chaos.

Contribution

It establishes new rigidity results for induced hyperspace maps and characterizes conditions under which chaos occurs or is absent, including examples of chaotic systems from pointwise periodic homeomorphisms.

Findings

Induced maps of distal homeomorphisms have zero topological entropy.

Set of almost periodic points equals the set of uniformly recurrent points for $2^f$.

Hyperspace systems are either equicontinuous or exhibit Li-Yorke and $oldsymbol{ ext{ω}}$-chaos.

Abstract

In this paper, we first prove that the topological entropy of induced map of any distal homeomorphism of a compact metric space is null. Then we consider induced map $2^f$ of an arbitrary pointwise periodic homeomorphism $f:X\to X$ of a compact metric space $X$, we show that the set of almost periodic points coincides with the set of uniformly recurrent points, i.e. $AP(2^f)=UR(2^f)$. Furthermore, we prove that inside any infinite $\omega$-limit set $\omega_{2^f}(A)$ there is a unique minimal set and this minimal set is an adding machine. As a consequence, $(2^X,2^f)$ has no Devaney chaotic subsystems. In contrast to these rigidity properties, we obtain some results with chaotic flavor. In fact, we prove the following dichotomy, the hyperspace system $(2^X,2^f)$ is either equicontinuous or choatic with respect to Li-Yorke chaos and $\omega$-chaos. It is shown that the later case occurs if and only if $R(2^f)\setminus AP(2^f)\neq\emptyset$. This enables us to provide simple examples of pointwise periodic homeomorphisms with chaotic induced systems.