On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property

Niu, Y.; Su, Shoubao

doi:10.1016/j.chaos.2011.03.008

Cited by 8 publications

(5 citation statements)

References 11 publications

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“…As an application, it is shown that for a continuous semi-flow θ on a compact metric space E with the AASP, if the almost periodic points of θ are dense in E then θ is syndetically sensitive. Consequently, by our results obtained in this paper we can easily see that the semi-flow versions of Theorem 3.1 in [40] and Proposition 1, Theorem 1, and Corollary 1 in [39] are true. Also, Theorem 3.5 is stronger than the semi-flow version of Theorem 3.1 in [40], Lemma 3.4 is stronger than the semi-flow versions of Proposition 3.3 from [40] in the case that f = g, and Lemma 3.2 is the semi-flow versions of Lemma 3.2 from [40] in the case that f = g. Moreover, inspired by [18] and [20] we show that for any semi-flow θ ∈ C 0 (R + × E, E) on a compact metric space (E, d), it has the AASP if and only if so does its inverse limit ( E, θ), and if and only if so does its lifting continuous semi-flow ( E, θ).…”

Section: Introductionmentioning

confidence: 73%

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Li¹,

Lu²,

Yu³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, for a continuous semi-flow θ on a compact metric space E with the asymptotic average-shadowing property (AASP), we show that if the almost periodic points of θ are dense in E then θ is multi-sensitive and syndetically sensitive. Also, we show that if θ is a Lyapunov stable semi-flow with the AASP, then the space E is trivial. Consequently, a Lyapunov stable semi-flow with the AASP is minimal. Furthermore, we prove that for a syndetically transitive continuous semi-flow on a compact metric space, sensitivity is equivalent to syndetical sensitivity. As an application, we show that for a continuous semi-flow θ on a compact metric space E with the AASP, if the almost periodic points of ϕ are dense in E then θ is syndetically sensitive. Moreover, we prove that for any continuous semi-flow θ on a compact metric space, it has the AASP if and only if so does its inverse limit ( E, θ), and if only if so does its lifting continuous semi-flow ( E, θ). Also, an example which contains two numerical experiments is given. Our results extend some corresponding and existing ones.

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Section: Introductionmentioning

confidence: 73%

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Li¹,

Lu²,

Yu³

et al. 2017

J. Nonlinear Sci. Appl.

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“…In [8,10], the authors claimed that ϕ has the asymptotic average shadowing property. But, in [12] Niu and Su explained that x = 0 is a distal point for ϕ and consequently this result is false. Now, we consider a similar map that has the asymptotic average shadowing property.…”

Section: Example 34mentioning

confidence: 83%

“…Niu and Su proved that, if f is point distal then f does not have the asymptotic average shadowing property [12].…”

Section: Example 34mentioning

confidence: 99%

Parameterized IFS with the Asymptotic Average Shadowing Property

Nia

2015

Qual. Theory Dyn. Syst.

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In this paper, we generalize the notion of the asymptotic average shadowing property to parameterized IFS's and prove some related theorems on this notion. Specially, it is proved that every uniformly contracting IFS has the asymptotic average shadowing property. As an important result, we show that if a continuous surjective IFS F on a compact metric space has the asymptotic average shadowing property then F is chain transitive. Moreover, we give some examples to illustrate our approach and compare the asymptotic average shadowing property for IFS with the asymptotic average shadowing property in discrete dynamical systems. For example, we will show that there is an IFS F which has the asymptotic average shadowing property but does not satisfy the shadowing property.

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“…5,19 AASP has been studied by many authors. 3,4,7 Recently, distributional chaos in the system with specification property has been fully investigated. 2,8,9,13,16 The latest result is that a system with specification property must display dense uniformly distributional chaos, and exhibit dense invariant uniformly distributional chaos provided it has a fixed point.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic average shadowing property, almost specification property and distributional chaos

Wang

Lei

et al. 2016

Mod. Phys. Lett. B

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It is proved that a nontrivial compact dynamical system with asymptotic average shadowing property (AASP) displays uniformly distributional chaos or distributional chaos in a sequence. Moreover, distributional chaos in a system with AASP can be uniform and dense in the measure center, that is, there is an uncountable uniformly distributionally scrambled set consisting of such points that the orbit closure of every point contains the measure center. As a corollary, the similar results hold for the system with almost specification property.

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On strong ergodicity and chaoticity of systems with the asymptotic average shadowing property

Cited by 8 publications

References 11 publications

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Multi-sensitivity, syndetical sensitivity and the asymptotic average- shadowing property for continuous semi-flows

Parameterized IFS with the Asymptotic Average Shadowing Property

Asymptotic average shadowing property, almost specification property and distributional chaos

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