• 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.25 no.4
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• pp.725-730
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• 2012

Let $f$ be a diffeomorphism of a closed $C^{\infty}$ manifold M. We show that $C^1$-generically, if $f$ has the shadowing property on a robustly transitive set, then it is hyperbolic.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.581-587
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• 2013

Let ${\Lambda}$ be a robustly transitive set of a diffeomorphism $f$ on a closed $C^{\infty}$ manifold. In this paper, we characterize hyperbolicity of ${\Lambda}$ in $C^1$-generic sense.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.1
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• pp.45-52
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• 2013

We show that a nontrivial transitive set is $C^1$-stably limit shadowable if and only if the transitive set is hyperbolic.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.1
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• pp.65-71
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• 2010

Let $\Lambda$ be a transitive set for f. In this paper, we show that if a f-invariant set $\Lambda$ has the $C^{1}$-stably shadowing property, then $\Lambda$ admits a dominated splitting.

• 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.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.613-619
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• 2012

In this paper, we prove that locally maximal transitive set of a $C^1$-generic diffeomorphism is hyperbolic if and only if it is limit weak shadowable.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.389-394
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• 2016

Let $f:M{\rightarrow}M$ be a diffeomorphism on a closed $C^{\infty}\;d({\geq}2)$ dimensional manifold M. For $C^1$-generic f, if a diffeomorphism f satisfies the local star condition on a transitive set, then it is hyperbolic.

• 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.

• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.323-330
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• 2014

The notion of transitive filters is introduced in lattice implication algebras. A necessary and sufficient condition is derived for every filter to become a transitive filter. Some sufficient conditions are also derived for a filter to become a transitive filter. The concept of absorbent filters is introduced and their properties are studied. A set of equivalent conditions is obtained for a filter to become an absorbent filter.