• 제목/요약/키워드: shadowing property

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SOME PROPERTIES OF THE STRONG CHAIN RECURRENT SET

  • Fakhari, Abbas;Ghane, Fatomeh Helen;Sarizadeh, Aliasghar
    • 대한수학회논문집
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    • 제25권1호
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    • pp.97-104
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    • 2010
  • The article is devoted to exhibit some general properties of strong chain recurrent set and strong chain transitive components for a continuous map f on a compact metric space X. We investigate the relation between the weak shadowing property and strong chain transitivity. It is shown that a continuous map f from a compact metric space X onto itself with the average shadowing property is strong chain transitive.

TAME DIFFEOMORPHISMS WITH C1-STABLE PROPERTIES

  • Lee, Manseob
    • 충청수학회지
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    • 제21권4호
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    • pp.519-525
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    • 2008
  • Let f be a diffeomorphisms of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper, we prove that if generically, f is tame diffeomorphims then the following conditions are equivalent: (i) f is ${\Omega}$-stable, (ii) f has the $C^1$-stable shadowing property (iii) f has the $C^1$-stable inverse shadowing property.

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DYNAMICAL STABILITY AND SHADOWING PROPERTY OF CONTINUOUS MAPS

  • Koo, Ki-Shik;Ryu, Hyun Sook
    • 충청수학회지
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    • 제11권1호
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    • pp.73-85
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    • 1998
  • This paper deals with the topological stability of continuous maps. First, the notion of local expansion is given and we show that local expansions of compact metric spaces have the shadowing property. Also, we prove that if a continuous surjective map f is a local homeomorphism and local expansion, then f is topologically stable in the class of continuous surjective maps. Finally, we find homeomorphisms which are not topologically stable.

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GENERIC DIFFEOMORPHISM WITH SHADOWING PROPERTY ON TRANSITIVE SETS

  • Lee, Manseob;Kang, Bowon;Oh, Jumi
    • 충청수학회지
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    • 제25권4호
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    • pp.643-653
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    • 2012
  • Let $f\;:\;M\;{\rightarrow}\;M$ be a diffeomorphism on a closed $C^{\infty}$ manifold. Let $\Lambda$ be a transitive set. In this paper, we show that (i) $C^1$-generically, $f$ has the shadowing property on a locally maximal $\Lambda$ if and only if $\Lambda$ is hyperbolic, (ii) f has the $C^1$-stably shadowing property on $\Lambda$ if and only if $\Lambda$ is hyperbolic.

VARIOUS SHADOWING PROPERTIES FOR INVERSE LIMIT SYSTEMS

  • Lee, Manseob
    • 충청수학회지
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    • 제29권4호
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    • pp.657-661
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    • 2016
  • Let $f:X{\rightarrow}X$ be a continuous surjection of a compact metric space and let ($X_f,{\tilde{f}}$) be the inverse limit of a continuous surjection f on X. We show that for a continuous surjective map f, if f has the asymptotic average, the average shadowing, the ergodic shadowing property then ${\tilde{f}}$ is topologically transitive.

VARIOUS INVERSE SHADOWING IN LINEAR DYNAMICAL SYSTEMS

  • Choi, Tae-Young;Lee, Keon-Hee
    • 대한수학회논문집
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    • 제21권3호
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    • pp.515-526
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    • 2006
  • In this paper, we give a characterization of hyperbolic linear dynamical systems via the notions of various inverse shadowing. More precisely it is proved that for a linear dynamical system f(x)=Ax of ${\mathbb{C}^n}$, f has the ${\tau}_h$ inverse(${\tau}_h-orbital$ inverse or ${\tau}_h-weak$ inverse) shadowing property if and only if the matrix A is hyperbolic.

C1-STABLE INVERSE SHADOWING CHAIN COMPONENTS FOR GENERIC DIFFEOMORPHISMS

  • Lee, Man-Seob
    • 대한수학회논문집
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    • 제24권1호
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    • pp.127-144
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    • 2009
  • Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

STRUCTURAL STABILITY OF VECTOR FIELDS WITH ORBITAL INVERSE SHADOWING

  • Lee, Keon-Hee;Lee, Zoon-Hee;Zhang, Yong
    • 대한수학회지
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    • 제45권6호
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    • pp.1505-1521
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    • 2008
  • In this paper, we give a characterization of the structurally stable vector fields via the notion of orbital inverse shadowing. More precisely, it is proved that the $C^1$ interior of the set of $C^1$ vector fields with the orbital inverse shadowing property coincides with the set of structurally stable vector fields. This fact improves the main result obtained by K. Moriyasu et al. in [15].