Devaney's chaos or 2-scattering implies Li–Yorke's chaos

Huang, Wen; Ye, Xiangdong

doi:10.1016/s0166-8641(01)00025-6

Cited by 266 publications

(159 citation statements)

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“…Relevant references are [GM89], [GW93], [AAB96], [Ak97], [AG01], [HY02], [Ak03], [AAG08]. The latter is perhaps a good starting point for a beginner.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

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Sufficient conditions under which a transitive system is chaotic

Akin

Glasner

Huang

et al. 2009

Ergod. Th. Dynam. Sys.

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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.

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“…Relevant references are [GM89], [GW93], [AAB96], [Ak97], [AG01], [HY02], [Ak03], [AAG08]. The latter is perhaps a good starting point for a beginner.…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…In the rest of this section we will obtain some applications of the above criterion. First, we need to recall some definitions (see [BHM02,HY02]). …”

Section: Theorem 31 (A Criterion For Chaos) Let (X T ) Be a Transimentioning

confidence: 99%

Sufficient conditions under which a transitive system is chaotic

Akin

Glasner

Huang

et al. 2009

Ergod. Th. Dynam. Sys.

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“…Devaney chaos implies Li-Yorke chaos [19,21,23] and positive topological entropy implies Li-Yorke chaos [5]. If X is the unit interval we have that positive topological entropy is equivalent to Devaney chaos on a subsystem.…”

Section: Li-yorke Chaosmentioning

confidence: 99%

Chaotic behaviour of the map x ↦ ω(x, f)

D’Aniello

Steele

2014

Open Mathematics

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Let K(2 N ) be the class of compact subsets of the Cantor space 2 N , furnished with the Hausdorff metric. Let ∈ C (2 N ). We study the map ω : 2 N → K(2 N ) defined as ω ( ) = ω( ), the ω-limit set of under . Unlike the case of -dimensional manifolds, ≥ 1, we show that ω is continuous for the generic self-map of the Cantor space, even though the set of functions for which ω is everywhere discontinuous on a subsystem is dense in C (2 N ). The relationships between the continuity of ω and some forms of chaos are investigated. MSC:37B99, 54H20, 37B10

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“…It was shown by Banks et al [1] that (D1) + (D2) implies (D3) for any aperiodic system. Devaney chaos is a stronger version of chaos than Li-Yorke chaos, which was given by Huang and Ye [8] and Mai [12].…”

Section: Introductionmentioning

confidence: 99%