Induced Dynamics on the Hyperspaces

TL;DR

This paper investigates the dynamics induced by finite commutative relations, showing such systems cannot be transitive or mixing, and exploring properties like periodic points and sensitivity in the induced hyperspace dynamics.

Contribution

It establishes that dynamics generated by finite commutative collections lack transitivity and mixing, and analyzes properties of the induced hyperspace systems, providing new insights into their behavior.

Findings

Induced systems cannot be transitive or super-transitive.

Such systems cannot exhibit higher degrees of mixing.

Under certain conditions, they lack dense periodic points.

Abstract

In this paper, we study the dynamics induced by finite commutative relation. We prove that the dynamics generated by such a non-trivial collection cannot be transitive/super-transitive and hence cannot exhibit higher degrees of mixing. As a consequence we establish that the dynamics induced by such a collection on the hyperspace endowed with any admissible hit and miss topology cannot be transitive and hence cannot exhibit any form of mixing. We also prove that if the system is generated by such a commutative collection, under suitable conditions the induced system cannot have dense set of periodic points. In the end we give example to show that the induced dynamics in this case may or may not be sensitive.