Asymptotic average shadowing property on compact metric spaces

Honary, Bahman; Bahabadi, Alireza Zamani

doi:10.1016/j.na.2007.08.058

Cited by 16 publications

(11 citation statements)

References 8 publications

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“…But the set of β for which the β-shift has the Spe-Pro has zero Lebesgue measure [5,24]. [11].) Let X be a compact metric space with more than one element and, let a ∈ X be arbitrary.…”

Section: Alm-spe-promentioning

confidence: 99%

“…For example, Honary and Bahabadi [11] showed that if for a continuous map f on a compact metric space X, the chain recurrent set R(f ) has more than one chain component, then f does not satisfy the asymptotic average shadowing property. They also showed that [12] if f is either a conservative average shadowing stable or a conservative asymptotic average shadowing stable, then f admits a dominated splitting.…”

Section: Alm-spe-promentioning

confidence: 99%

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Ergodic properties of systems with asymptotic average shadowing property

Dong

Tian

Yuan

2015

Journal of Mathematical Analysis and Applications

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In this paper, we explore a topological system f : M → M with asymptotic average shadowing property and extend Sigmund's results from Bowen's specification case. We show that every non-empty, compact and connected subset V ⊆ M inv (f ) coincides with some V f (y). Moreover, we show that the setAs consequences, we have several corollaries. One is that every invariant measure has generic points. Another is that the set consisting of those points for which the Birkhoff ergodic average does not exist (called irregular set) is either dense in Δ max (residual provided that Δ max = M ) or empty. In particular, we give an uncountable division of irregular set and obtain a refined characterization which can be as a substantial generalization of [18] with a different method. Remark that for systems with asymptotic average shadowing property, this article studies Birkhoff ergodic average from topological viewpoint but it is still unknown how about the theory from the perspective of dimension theory.

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“…But the set of β for which the β-shift has the Spe-Pro has zero Lebesgue measure [5,24]. [11].) Let X be a compact metric space with more than one element and, let a ∈ X be arbitrary.…”

Section: Alm-spe-promentioning

confidence: 99%

Section: Alm-spe-promentioning

confidence: 99%

Ergodic properties of systems with asymptotic average shadowing property

Dong

Tian

Yuan

2015

Journal of Mathematical Analysis and Applications

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show abstract

“…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.…”

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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“…But the converse is not true. To see this we use an example of [12] to introduce a map which is shadowable but does not have ergodic shadowing property.…”

Section: })mentioning

confidence: 99%