• Title/Summary/Keyword: shadowing property

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CONTINUOUS SHADOWING AND STABILITY FOR GROUP ACTIONS

  • Kim, Sang Jin
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.53-65
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    • 2019
  • Recently, Chung and Lee [2] introduced the notion of topological stability for a finitely generated group action, and proved a group action version of the Walters's stability theorem. In this paper, we introduce the concepts of continuous shadowing and continuous inverse shadowing of a finitely generated group action on a compact metric space X with respect to various classes of admissible pseudo orbits and study the relationships between topological stability and continuous shadowing and continuous inverse shadowing property of group actions. Moreover, we introduce the notion of structural stability for a finitely generated group action, and we prove that an expansive action on a compact manifold is structurally stable if and only if it is continuous inverse shadowing.

WEAK INVERSE SHADOWING AND Ω-STABILITY

  • Zhang, Yong;Choi, Taeyoung
    • Journal of the Chungcheong Mathematical Society
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    • v.17 no.2
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    • pp.137-145
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    • 2004
  • We give characterization of ${\Omega}$-stable diffeomorphisms via the notions of weak inverse shadowing. More precisely, it is proved that the $C^1$ interior of the set of diffeomorphisms with the weak inverse shadowing property with respect to the class $\mathcal{T}_h$ coincides with the set of ${\Omega}$-stable diffeomorphisms.

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EXPANDING MEASURES FOR HOMEOMORPHISMS WITH EVENTUALLY SHADOWING PROPERTY

  • Dong, Meihua;Lee, Keonhee;Nguyen, Ngocthach
    • Journal of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.935-955
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    • 2020
  • In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.

SPECTRAL DECOMPOSITION OF k-TYPE NONWANDERING SETS FOR ℤ2-ACTIONS

  • Kim, Daejung;Lee, Seunghee
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.387-400
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    • 2014
  • We prove that the set of k-type nonwandering points of a Z2-action T can be decomposed into a disjoint union of closed and T-invariant sets $B_1,{\ldots},B_l$ such that $T|B_i$ is topologically k-type transitive for each $i=1,2,{\ldots},l$, if T is expansive and has the shadowing property.

SHADOWABLE POINTS FOR FINITELY GENERATED GROUP ACTIONS

  • Kim, Sang Jin;Lee, Keonhee
    • Journal of the Chungcheong Mathematical Society
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    • v.31 no.4
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    • pp.411-420
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    • 2018
  • In this paper we introduce the notion of shadowable points for finitely generated group actions on compact metric spaace and prove that the set of shadowable points is invariant and Borel set and if chain recurrent set contained shadowable point set then it coincide with nonwandering set. Moreover an action $T{\in}Act(G, X)$ has the shadowing property if and only if every point is shadowable.

A NOTE ON TOPOLOGICAL PROPERTIES IN MULTI-VALUED DYNAMICAL SYSTEMS

  • Cho, Chihyun;Chu, Hahng-Yun;Kang, No-Weon;Kim, Myoung-Jung
    • Journal of the Chungcheong Mathematical Society
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    • v.35 no.2
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    • pp.171-176
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    • 2022
  • In this article, we investigate the transitivity and chain transitivity on multi-valued dynamical systems. For compact-valued continuous dynamics, we prove that the notion of transitivity is expressed by the notions of the shadowing property and chain transitivity under locally maximal condition.