• 대한수학회지
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• 제60권5호
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• pp.987-998
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• 2023

It is shown that every continuum-wise expansive C¹ generic vector field X on a compact connected smooth manifold M satisfies Axiom A and has no cycles, and every continuum-wise expansive homoclinic class of a C¹ generic vector field X on a compact connected smooth manifold M is hyperbolic. Moreover, every continuum-wise expansive C¹ generic divergence-free vector field X on a compact connected smooth manifold M is Anosov.

• 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).

• 대한수학회보
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• 제49권2호
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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.

• 대한수학회논문집
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• 제27권3호
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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.

• 대한수학회지
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• 제61권3호
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• pp.495-513
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• 2024

In the paper, we show that for a generic C¹ vector field X on a closed three dimensional manifold M, any isolated transitive set of X is singular hyperbolic. It is a partial answer of the conjecture in [13].

• 대한수학회논문집
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• 제28권3호
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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 applied mathematics & informatics
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• 제18권1_2호
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• pp.585-589
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• 2005

In [1], the structure of C(2,2) is determined as the polynomial ring in 5 variables. In this work, we show that C(2,3) is a free module over the subring of 9 variables. We explicitly give a presentation of C(2, 3) as free module over the polynomial ring.

• 충청수학회지
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• 제28권3호
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• pp.391-396
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• 2015

$C^1$-generically, if a transitive diffeomorphism f is periodically expansive, then it is hyperbolic.

• 대한수학회보
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• 제50권3호
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• pp.727-730
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• 2013

We prove an inequality between topological entropy and asymptotical growth of periodic orbits for $C^1$ generic diffeomorphisms.

• 대한수학회논문집
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• 제31권2호
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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.