Invariant scrambled sets and distributional chaos

Abstract

In their famous paper 'Period three implies chaos', Li and Yorke started a study of a very important phenomena in dynamical systems (known presently under the name Li-Yorke chaos). Recently, it was proved by Du that an interval map f is turbulent if and only if there is an invariant scrambled set for f. We extend this approach and prove that exactly the same characterization is valid for distributional chaos.

Omega-limit sets and invariant chaos in dimension one

“…This is no longer true when we add the assumption of invariance of a scrambled set. This was proved in [13] for interval maps and here we generalize the results for the case of graph maps.…”

Distributionally chaotic translation semigroups

“…This concept was generalized in [2,22]. We also refer to [14,[18][19][20] for some recent papers dealing with distributional chaos, and to [7,17] for distributional chaos in the linear infinite-dimensional setting, which is the matter of this paper.…”

Syndetically proximal pairs

“…This theorem has been used by many authors (see for instance [5,15]) to show that an interval map with positive topological entropy has dynamically complicated subsets such as scrambled sets, by first establishing the corresponding phenomena in the shift dynamical system and then doing a transfer process via the map φ. Now, this transfer process from the shift dynamical system to the interval setting is not always easy, and the authors generally go through some delicate arguments since φ may not be a homeomorphism.…”

“…The existence of invariant scrambled sets was considered in [5,15]. Using Theorem 7, Theorem 9 and Proposition 4(i), we obtain that for any interval map f with h( f ) > 0, there is an uncountable syndetically scrambled set invariant under some power of f .…”