Li-Yorke chaos for maps on G-Spaces

TL;DR

This paper extends the concept of Li-Yorke chaos to G-spaces, establishing conditions under which maps exhibit this form of chaos and exploring its properties and implications.

Contribution

It introduces the definition of G-Li-Yorke chaos, proves its properties, and provides sufficient conditions for chaos in G-spaces, advancing the understanding of chaos in this setting.

Findings

G-Li-Yorke chaos is iterable for f

Li-Yorke chaos implies G-Li-Yorke chaos, but not vice versa

G-transitivity and fixed points imply G-Li-Yorke chaos

Abstract

We introduce the definition of Li-Yorke chaos for the map f on G-spaces, and show G-Li-Yorke chaos is iterable for f. Li-Yorke chaos implies G-Li-Yorke chaos, while the converse is not true. Then we give a sufficient condition for f to be chaotic in the sense of G-Li-Yorke. Also, we prove that if f is G-transitive and there exists a common fixed point for f and all of the maps in G, then f is chaotic in the sense of G-Li-Yorke.