Sufficient Conditions under Which a Transitive System Is Chaotic

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×Y, T ×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.