Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms

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.