• 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.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.4
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• pp.329-333
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• 2022

In this paper we investigate some properties concerning the set of shadowable points for homeomorphisms. Then we show that the almost shadowing property is preserved by a topological conjugacy between homeomorphisms. Also, we give an example to illustrate our results.

• 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.24 no.4
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• pp.631-638
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• 2011

In this paper, we prove that for $C^1$ generically, if every hyperbolic periodic point in a chain component is uniformly far away from being nonhyperbolic, and it is $C^1$-robustly average shadowable, then the chain component is hyperbolic.

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1043-1055
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• 2023

In this paper we study a pointwise version of Walters topological stability in the class of homeomorphisms on a compact metric space. We also show that if a sequence of homeomorphisms on a compact metric space is uniformly expansive with the uniform shadowing property, then the limit is expansive with the shadowing property and so topologically stable. Furthermore, we give examples to illustrate our results.

• Communications of the Korean Mathematical Society
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• v.27 no.2
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• pp.369-375
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• 2012

In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let $M$ be a compact manifold with a volume form ${\omega}$ and $f:M{\rightarrow}M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves ${\omega}$. In this paper, we do not assume $f$ is $\mathcal{C}^1$-generic. We prove that $f$ satisfies the chain transitivity and we also show that, on $M$, the $\mathcal{C}^1$-stable shadowability is equivalent to the hyperbolicity.