The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos

Wang, Lidong; Liao, Gongfu; Chen, Zhizhi; Duan, Xiaodong

doi:10.4064/ap82-3-7

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Cited by 3 publications

(2 citation statements)

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“…As an immediate consequence we obtain that È xy () ¼ 0. A similar technique of proving distributional chaos via semiconjugacy was first introduced in [29] (see also [30]). …”

Section: P Oprochamentioning

confidence: 96%

Invariant scrambled sets and distributional chaos

Oprocha

2009

Dynamical Systems

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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.

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“…As an immediate consequence we obtain that È xy () ¼ 0. A similar technique of proving distributional chaos via semiconjugacy was first introduced in [29] (see also [30]). …”

Section: P Oprochamentioning

confidence: 96%

Invariant scrambled sets and distributional chaos

Oprocha

2009

Dynamical Systems

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“…Wang et al [15] proved that a dynamical system having a regular shift-invariant set is distributionally chaotic and posed the following question:…”

Section: Introductionmentioning

confidence: 99%

Shifts, rotations and distributional chaos

Xiang

Liang

2019

Adv Differ Equ

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Let R r 0 , R r 1 : S 1 − → S 1 be rotations on the unit circle S 1 and define f : Σ 2 × S 1 − → Σ 2 × S 1 as f (x, t) = (σ (x), R r x 1 (t)), for x = x 1 x 2 • • • ∈ Σ 2 := {0, 1} N , t ∈ S 1 , where σ : Σ 2 − → Σ 2 is the shift, and r 0 and r 1 are rotational angles. It is first proved that the system (Σ 2 × S 1 , f) exhibits maximal distributional chaos for any r 0 , r 1 ∈ R (no assumption of r 0 , r 1 ∈ R \ Q), generalizing Theorem 1 in Wu and Chen (Topol. Appl. 162:91-99, 2014). It is also obtained that (Σ 2 × S 1 , f) is cofinitely sensitive and (M 1 ,M 1)-sensitive and that (Σ 2 × S 1 , f) is densely chaotic if and only if r 1-r 0 ∈ R \ Q.

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Non-weakly almost periodic recurrent points and distributionally scrambled sets on Σ2×S1

Chen

2014

Topology and its Applications

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The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos

Cited by 3 publications

References 9 publications

Invariant scrambled sets and distributional chaos

Invariant scrambled sets and distributional chaos

Shifts, rotations and distributional chaos

Non-weakly almost periodic recurrent points and distributionally scrambled sets on Σ2×S1

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