Paper
Dynamics on Hyperspace of Pointwise Periodic Homeomorphisms
Authors
Issam Naghmouchi
Abstract
In this paper, we consider the dynamics of induced map of a given pointwise periodic homeomorphism of a compact metric space . First, we show that the topological entropy of is zero, i.e. and that the set of almost periodic points coincides with the set of uniformly recurrent points, i.e. . Furthermore, we prove that inside any infinite -limit set there is a unique minimal set and this minimal set is an adding machine. As a consequence, 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 is either equicontinuous or choatic with respect to Li-Yorke chaos and -chaos. It is shown that the later case occurs if and only if . This enables us to provide simple examples of pointwise periodic homeomorphisms with chaotic induced systems.