Kato's chaos of multiple mappings and its continuous self-maps

TL;DR

This paper explores the properties of multiple mappings and their continuous self-maps, focusing on chaos, sensitivity, and accessibility, and establishes conditions under which these properties are preserved or do not imply each other.

Contribution

It introduces new definitions of sensitivity, accessibility, and Kato's chaos for multiple mappings from a set-valued perspective and analyzes their relationships.

Findings

Multiple mappings and their continuous self-maps do not imply each other in sensitivity and accessibility.

Provides sufficient conditions for multiple mappings to be sensitive, accessible, and Kato's chaotic.

Shows that these properties are preserved under topological conjugation.

Abstract

In 2016, Hou and Wang introduced the concept of multiple mappings based on iterated function system, which is an important branch of fractal theory. In this paper, we introduce the definitions of sensitivity, accessibility, and Kato's chaos of multiple mappings from a set-valued perspective. We show that multiple mappings and its continuous self-maps do not imply each other in terms of sensitivity and accessibility. While a sufficient condition for multiple mappings to be sensitive, accessible and Kato's chaotic is provided, respectively. And the sensitivity, accessibility, and Kato's chaos of multiple mappings are preserved under topological conjugation.