Sufficient conditions under which a transitive system is chaotic

TL;DR

This paper establishes conditions under which a topologically transitive system exhibits strong Li-Yorke chaos, introduces the concept of uniform chaos, and explores its properties in minimal systems.

Contribution

It provides a new sufficient condition for strong chaos in transitive systems and links uniform chaos with minimal systems and PI extensions.

Findings

Transitivity of a subsystem implies strong Li-Yorke chaos.

Many known chaos conditions are corollaries of the main result.

Uniform chaos is preserved under extensions in minimal systems.

Abstract

Let (X,T) be a topologically transitive dynamical system. We show that if there is a subsystem (Y,T) of (X,T) such that (X\times Y, T\times T) is transitive, then (X,T) is strongly chaotic in the sense of Li and Yorke. We then show that many of the known sufficient conditions in the literature, as well as a few new results, are corollaries of this statement. In fact, the kind of chaotic behavior we deduce in these results is a much stronger variant of Li-Yorke chaos which we call uniform chaos. For minimal systems we show, among other results, that uniform chaos is preserved by extensions and that a minimal system which is not uniformly chaotic is PI.