Shadowing Property on Finitely Generated Group Actions

Barzanouni, Ali

doi:10.1080/1726037x.2014.922253

Cited by 4 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, where m > 1. One can follow the same steps as in Proposition 2.2 [3] to show that Φ has shadowing. It is also easy to check that Φ is expansive.…”

Section: Stability Theoremsmentioning

confidence: 97%

Stability Theorems for Group Actions on Uniform Spaces

Das,

Das

2018

Preprint

View full text Add to dashboard Cite

We extend the notions of topological stability, shadowing and persistence from homeomorphisms to finitely generated group actions on uniform spaces and prove that an expansive action with either shadowing or persistence is topologically stable. Using the concept of null set of a Borel measure µ, we introduce the notions of µ-expansivity, µ-topological stability, µ-shadowing and µpersistence for finitely generated group actions on uniform spaces and show that a µ-expansive action with either µ-shadowing or µ-persistence is µ-topologically stable.

show abstract

“…, where m > 1. One can follow the same steps as in Proposition 2.2 [3] to show that Φ has shadowing. It is also easy to check that Φ is expansive.…”

Section: Stability Theoremsmentioning

confidence: 97%

Stability Theorems for Group Actions on Uniform Spaces

Das,

Das

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, shadowing property for a countable group is defined in [4] and [13]. We used the definition of shadowing property for a countable group in [4] and [13], which generalize the definition for finitely generated group in [2] and [16]. For S ⊂ G and δ > 0, (S, δ)-pseudo-orbit of (X, G, Φ) is a sequence {x g } g∈G in X satisfying that d(Φ s (x g ), x sg ) < δ holds for any g ∈ G and s ∈ S. For ǫ > 0, a point z ∈ X is said to ǫ-shadow a sequence {x g } g∈G in X if d(Φ g (z), x g ) < ǫ for any g ∈ G. For S ⊂ G, we say Φ has S-shadowing property if for any ǫ > 0, there exists δ > 0 such that each (S, δ)-pseudo-orbit can be ǫ-shadowed by some point in X.…”

Section: Introductionmentioning

confidence: 99%

Shadowing and mixing on systems of countable group actions

Lin¹,

Chen²,

Zhou³

2020

Preprint

View full text Add to dashboard Cite

Let (X, G, Φ) be a dynamical system, where X is a compact Hausdorff space, and G is a countable discrete group. We investigated shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property, fix some finite subset S ⊂ G. We proved that if X is totally disconnected and Φ has S-shadowing property, then (X, G, Φ) is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that X is a metric space (may be not totally disconnected), we proved that if Φ has S-shadowing property, then (X, G, Φ) is a factor of an inverse limit of a sequence of shifts of finite type by a factor map which almost lifts pseudo-orbit for S.On the other hand, let property P be one of the following properties: transitivity, minimal, totally transitivity, weakly mixing, mixing, and specification property. We proved that if X is totally disconnected, then Φ has property P if and only if (X, G, Φ) is conjugate to an inverse limit of an inverse system that consists of subshifts with property P which satisfies Mittag-Leffler condition. Also, for the case of metric space (may be not totally disconnected), if property P is not minimal or specification property, we proved that Φ has property P if and only if (X, G, Φ) is a factor of an inverse limit of a sequence of subshifts with property P which satisfies Mittag-Leffler condition.

show abstract

“…Recently, in 2014, Osipov and Tikhomirov [12], introduced the notion of shadowing property for finitely generated group actions. Barzanouni in [3] used the notion of shadowing property to study chain recurrent sets whereas Chung and Lee in [5] showed that expansive action which has shadowing property are always topologically stable. Hurder in [8] discusses the dynamics of expansive actions on the unit circle S 1 .…”

Section: Introductionmentioning

confidence: 99%