• Journal of the Korean Mathematical Society
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• v.58 no.5
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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).

• Journal of the Chungcheong Mathematical Society
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• v.30 no.2
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• pp.177-181
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• 2017

Let $f:M{\rightarrow}M$ be a diffeomorphism of a compact smooth manifold M. In this paper, we show that $C^1$ generically, if a chain transitive set ${\Lambda}$ is locally maximal then it admits a dominated splitting. Moreover, $C^1$ generically if a chain transitive set ${\Lambda}$ of f is locally maximal then it has zero entropy.

• Bulletin of the Korean Mathematical Society
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• v.49 no.2
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• pp.263-270
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• 2012

We prove that a locally maximal chain transitive set of a $C^1$-generic diffeomorphism is hyperbolic if and only if it is shadowable.

• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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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.

• Communications of the Korean Mathematical Society
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• v.25 no.1
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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.

• The Pure and Applied Mathematics
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• v.18 no.4
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• pp.285-291
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• 2011

In this paper, we show that if a transitive set ${\Lambda}$ is $C^1$-stably expansive, then ${\Lambda}$ admits a dominated splitting.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.821-829
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• 2013

Let f be a diffeomorphism of a closed $C^{\infty}$ manifold M, and let ${\Lambda}{\subset}M$ be a closed f-invariant set. We show that if $f{\mid}_{\Lambda}$ is robustly chain transitive with shadowing, then ${\Lambda}$ is the hyperbolic homoclinic class.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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• pp.27-33
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• 2012

Let $f$ be a difeomorphism of a closed $n$ -dimensional smooth manifold M, and $p$ be a hyperbolic periodic point of $f$. Let ${\Lambda}(p)$ be a closed set which containing $p$. In this paper, we show that (i) if $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$, then ${\Lambda}(p)$ is the chain component which contains $p$. (ii) suppose $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$. Then if $f|_{\Lambda(p)}$ has the $C^{1}$-stably shadowing property then it is hyperbolic.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.91-101
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• 2022

We study some properties of nonwandering set Ω(𝜙) and chain recurrent set CR(𝜙) for an expansive flow which has the POTP on a compact TVS-cone metric spaces. Moreover we shall prove a spectral decomposition theorem for an expansive flow which has the POTP on TVS-cone metric spaces.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.617-623
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• 2010

Let p be a hyperbolic periodic point of f, and let ${\Lambda}(p)$ be a closed set which containing p. In this paper, we show that $C^1$-generically, if $f{\mid}_{{\Lambda}(p)}$ has the $C^1$-stably asymptotic average shadowing property, then it admits a dominated splitting.