Induced dynamics on the hyperspaces

Abstract

In this paper, we study the dynamics induced by finite commutative relation on the hyperspaces. We prove that the dynamics induced on the hyperspace by a non-trivial commutative family of continuous self maps cannot be transitive and hence cannot exhibit higher degrees of mixing. We also prove that the dynamics induced on the hyperspace by such a collection cannot have dense set of periodic points. We also give example to show that the induced dynamics in this case may or may not be sensitive.

Keywords

• hyperspace
• combined dynamics
• relations
• induced map
• transitivity
• super-transitivity
• dense periodicity.