• 제목/요약/키워드: generic diffeomorphism

검색결과 17건 처리시간 0.018초

EVENTUAL SHADOWING FOR CHAIN TRANSITIVE SETS OF C1 GENERIC DYNAMICAL SYSTEMS

  • Lee, Manseob
    • 대한수학회지
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    • 제58권5호
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    • pp.1059-1079
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    • 2021
  • We show that given any chain transitive set of a C1 generic diffeomorphism f, if a diffeomorphism f has the eventual shadowing property on the locally maximal chain transitive set, then it is hyperbolic. Moreover, given any chain transitive set of a C1 generic vector field X, if a vector field X has the eventual shadowing property on the locally maximal chain transitive set, then the chain transitive set does not contain a singular point and it is hyperbolic. We apply our results to conservative systems (volume-preserving diffeomorphisms and divergence-free vector fields).

HYPERBOLIC STRUCTURE OF POINTWISE INVERSE PSEUDO-ORBIT TRACING PROPERTY FOR C1 DIFFEOMORPHISMS

  • Manseob Lee
    • 대한수학회논문집
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    • 제38권1호
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    • pp.243-256
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    • 2023
  • We deal with a type of inverse pseudo-orbit tracing property with respect to the class of continuous methods, as suggested and studied by Pilyugin [54]. In this paper, we consider a continuous method induced through the diffeomorphism of a compact smooth manifold, and using the concept, we proved the following: (i) If a diffeomorphism f of a compact smooth manifold M has the robustly pointwise inverse pseudoorbit tracing property, f is structurally stable. (ii) For a C1 generic diffeomorphism f of a compact smooth manifold M, if f has the pointwise inverse pseudo-orbit tracing property, f is structurally stable. (iii) If a diffeomorphism f has the robustly pointwise inverse pseudo-orbit tracing property around a transitive set Λ, then Λ is hyperbolic for f. Finally, (iv) for C1 generically, if a diffeomorphism f has the pointwise inverse pseudo-orbit tracing property around a locally maximal transitive set Λ, then Λ is hyperbolic for f. In addition, we investigate cases of volume preserving diffeomorphisms.

DYNAMICAL SYSTEMS WITH SPECIFICATION

  • Lee, Keonhee;Tajbakhsh, Khosro
    • 충청수학회지
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    • 제28권1호
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    • pp.103-108
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    • 2015
  • In this paper we prove that $C^1$-generically, if a diffeomorphism f on a closed $C^{\infty}$ manifold M satisfies weak specification on a locally maximal set ${\Lambda}{\subset}M$ then ${\Lambda}$ is hyperbolic for f. As a corollary we obtain that $C^1$-generically, every diffeomorphism with weak specification is Anosov.

HYPERBOLICITY OF CHAIN TRANSITIVE SETS WITH LIMIT SHADOWING

  • Fakhari, Abbas;Lee, Seunghee;Tajbakhsh, Khosro
    • 대한수학회보
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    • 제51권5호
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    • pp.1259-1267
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    • 2014
  • In this paper we show that any chain transitive set of a diffeomorphism on a compact $C^{\infty}$-manifold which is $C^1$-stably limit shadowable is hyperbolic. Moreover, it is proved that a locally maximal chain transitive set of a $C^1$-generic diffeomorphism is hyperbolic if and only if it is limit shadowable.