Difference between Devaney chaos associated with two systems

TL;DR

This paper investigates the relationship between Devaney chaos in a base dynamical system and its induced hyperspace system, revealing that chaos in the hyperspace does not necessarily imply chaos in the base system, and provides an equivalent condition for periodic density.

Contribution

It clarifies the non-implication of Devaney chaos from hyperspace to base systems and offers an equivalent condition for periodic density in hyperspace.

Findings

Devaney chaos in hyperspace does not imply chaos in the base system.

The implication of chaos is not valid even under strengthened conditions.

An equivalent condition for periodic density in hyperspace is established.

Abstract

We discuss the relation between Devaney chaos in the base system and Devaney chaos in its induced hyperspace system. We show that the latter need not imply the former. We also argue that this implication is not true even in the strengthened condition. Additionally we give an equivalent condition for the periodically density in the hyperspace system.