Chaotic sets of continuous and discontinuous maps

Scrambled sets of shift operators

“…Note that some results of the Li-Yorke chaotic sets for the shift operator is already obtained by Fu et al in [2,3]. Our new contribution here is to characterize Li-Yorke chaotic sets by orbit invariants, Furstenberg families and p-scrambled points.…”

Chaotic sets of shift and weighted shift maps

“…We remark that some results in this paper, such as Lemmas 3.1 and 3.2, Corollary 3.3, and Proposition 3.4, are also valid for a general continuous map f : X → X on a compact metric space X [22]. In Theorem 3.5 and Corollary 3.6 we give the sufficient conditions for S to be a scrambled or Li-Yorke chaotic set of σ .…”

“…(We remark here that the one-dimensional Hausdorff measure H 1 (S) of a maximal scrambled set S ⊆ Σ(N) is zero, if S is H 1 -measurable [39].) It can be shown that a totally chaotic subshift must be a subshift of infinite type [22]. The question is open whether it is possible to give an example of a totally scrambled subshift σ on Σ ⊆ Σ(N), i.e., with the whole space Σ being a scrambled set of σ | Σ .…”

“…There have been extensive studies on scrambled sets of an iterated continuous map on a compact interval. Some results have also been achieved on scrambled sets of continuous or discontinuous self-maps on higher-dimensional cube I n , n ≥ 2 or more general spaces [18][19][20][21][22].…”