• Korean Journal of Mathematics
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• v.13 no.1
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• pp.91-101
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• 2005

In this paper, we give the notion of the D-shadowing property, D-inverse shadowing property for dynamical systems. and investigate the density of D-shadowing dynamical systems and the D-inverse shadowing dynamical systems. Moreover we study some relationships between the D-shadowing property and other dynamical properties such as expansivity and topological stability.

• Journal for History of Mathematics
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• v.18 no.1
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• pp.93-102
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• 2005

In this paper, we introduce the history of dynamical systems and the notion of the H-shadowing property. And we study some relationships between the H-shadowing property and other dynamical properties such as expansivity and topological stability.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1421-1431
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• 2021

In this paper we study the preservation of various notions of expansivity in discrete dynamical systems and the induced map for n-fold symmetric products and hyperspaces. Then we give a characterization of a compact metric space admitting hyper N-expansive homeomorphisms via the topological dimension. More precisely, we show that C⁰-generically, any homeomorphism on a compact manifold is not hyper N-expansive for any N ∈ ℕ. Also we give some examples to illustrate our results.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.319-343
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• 2022

We discuss the notions of positive expansivity, chain transitivity, uniform rigidity, chain mixing, weak specification, and pseudo orbital specification in terms of finite open covers for Hausdorff topological spaces and entourages for uniform spaces. We show that the two definitions for each notion are equivalent in compact Hausdorff spaces and further they are equivalent to their standard definitions in compact metric spaces. We show that a homeomorphism on a Hausdorff uniform space has uniform h-shadowing if and only if it has uniform shadowing and its inverse is uniformly equicontinuous. We also show that a Hausdorff positively expansive system with a Hausdorff shadowing property has Hausdorff h-shadowing.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.337-344
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• 2018

We extend Bowen's decomposition theorem to topologically Anosov homeomorphisms on first countable, locally compact, paracompact, Hausdorff spaces which are not necessarily metrizable and not necessarily compact.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.287-300
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• 2020

We introduce and study notions of expansivity, topological stability and persistence for Borel measures with respect to time varying bi-measurable maps on metric spaces. We prove that on Mandelkern locally compact metric spaces expansive persistent measures are topologically stable in the class of all time varying homeomorphisms.