Sufficient conditions under which a transitive system is chaotic

Abstract

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 × 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 fact. 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.

Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

“…Clearly, the '2D Li-Yorke chaos' case, which is known as 'strong Li-Yorke chaos' for cascades on compact metric spaces in [2], is already conceptually stronger than the usual Li-Yorke chaos introduced in §1.2.…”

“…Now the following result completes the proof of Proposition 2.8. (2), which means that Devaney chaos implies sensitive at the same time. It should be mentioned that the non-minimality of (T, X, π) is essential for our conclusion.…”

“…There are systematical studies on chaos for Zor Z + -action dynamical systems on compact metric spaces; see, e.g., [28,2,33] and references therein. In this paper, we will study Devaney chaos, Li-Yorke chaos, and multi-dimensional Li-Yorke chaos of flows or semiflows (T, X, π) on uniform spaces X with general topological phase group or semigroup T .…”

A dynamical version of the Kuratowski–Mycielski theorem and invariant chaotic sets

“…For any n ∈ N, the n-fold product system of (X, f ) is denoted by (X n , f (n) ), where f (n) = f × f × · · · × f (n-times). A dynamical system (X, f ) is called weakly mixing if the product system (X 2 , f (2) ) is transitive, and strongly mixing if for every two non-empty open subsets U and V of X there is an N ∈ N such that U ∩ f −n (V) ∅ for all n ≥ N. It is clear that strong mixing implies weak mixing, which in turn implies transitivity.…”

“…A subset K ⊆ X is called a uniformly chaotic set if there are Cantor sets C 1 ⊆ C 2 ⊆ · · · such that (1) K = ∞ i=1 C i is a recurrent subset of X and also a proximal subset of X; (2) for each i = 1, 2, · · · , C i is uniformly recurrent; (3) for each i = 1, 2, · · · , C i is uniformly proximal. It is shown in [2] that for a non-trivial transitive system (X, f ), if there exists some subsystem (Y, f ) such that (X × Y, f × f ) is transitive, then there exists a dense uniformly chaotic set. In particular, if there exists a fixed point, then there exists a dense uniformly chaotic set.…”