Chaos for endomorphisms of completely metrizable groups and linear operators on Fr\'echet spaces

TL;DR

This paper explores various forms of chaos in continuous endomorphisms of metrizable groups and linear operators on Fréchet spaces, providing characterizations and examples to deepen understanding of chaotic dynamics.

Contribution

It introduces a unified approach to different chaos types in topological groups and extends chaos analysis to linear operators on Fréchet spaces, improving existing results.

Findings

Characterization of Li-Yorke, mean Li-Yorke, and distributional chaos via semi-irregular points

Examples of inner automorphisms illustrating chaos properties

Enhanced results on chaos in linear operators on Fréchet spaces

Abstract

Using some techniques from topological dynamics, we give a uniform treatment of Li-Yorke chaos, mean Li-Yorke chaos and distributional chaos for continuous endomorphisms of completely metrizable groups, and characterize three kinds of chaos (resp. extreme chaos) in terms of the existence of the so-called semi-irregular points (resp. irregular points). We exhibit some examples of inner automorphisms of Polish groups to illustrate the results. We also apply our results to the chaos theory of continuous linear operators on Fr\'echet spaces, which improves some results in the literature.