Li-Yorke Chaos for Composition Operators on $L^p$-Spaces

TL;DR

This paper provides simplified characterizations of Li-Yorke chaos specifically for composition operators on $L^p$ spaces, including a clear criterion for weighted shifts, supported by numerous sharp examples.

Contribution

It introduces more straightforward and practical characterizations of Li-Yorke chaos in the context of composition operators on $L^p$ spaces, enhancing understanding and analysis.

Findings

Simplified criteria for Li-Yorke chaos in composition operators

Characterization of Li-Yorke chaotic weighted shifts

Numerous sharp examples demonstrating the results

Abstract

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on $L^p$ spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp.