Devaney Chaos on a Set-valued Map and Its Inverse Limit

TL;DR

This paper investigates the relationship between set-valued maps and their inverse limits, focusing on properties like chaos, transitivity, and periodic points, revealing conditions under which these properties are equivalent or imply each other.

Contribution

It establishes new connections between set-valued maps and inverse limits regarding chaos, periodic points, and transitivity, clarifying when properties are preserved or imply each other.

Findings

Density of periodic points is equivalent between a map and its inverse limit.

Sensitivity does not necessarily transfer between a map and its inverse limit.

Transitivity and Devaney chaos of inverse limits imply the same for the original map.

Abstract

We study relationships between a set-valued map and its inverse limits about the notion of periodic point set, transitivity, sensitivity and Devaney chaos. Density of periodic point set of a set-valued map and its inverse limits implies each other. Sensitivity of a set-valued map and its inverse limits does not imply each other. Transitivity and Devaney chaos of generalized inverse limits implies the corresponding property of a set-valued map.