Li-Yorke chaos for invertible mappings on non-compact spaces

TL;DR

This paper demonstrates that Li-Yorke chaos in invertible mappings does not necessarily imply chaos in their inverses and that such chaos is not preserved under topological conjugacy, through specific examples.

Contribution

It provides the first examples showing non-preservation of Li-Yorke chaos under inversion and conjugacy for invertible mappings on non-compact spaces.

Findings

Li-Yorke chaos is not preserved under inversion.

Li-Yorke chaos is not preserved under topological conjugacy.

Examples include an invertible bounded linear operator and a homeomorphism.

Abstract

In this paper, we give two examples to show that an invertible mapping is Li-Yorke chaotic does not imply its inverse being Li-Yorke chaotic, in which one is an invertible bounded linear operator on an infinite dimensional Hilbert space and the other is a homeomorphism on the unit open disk. Moreover, we use the last example to prove that Li-Yorke chaos is not preserved under topological conjugacy.