• Journal of the Chungcheong Mathematical Society
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• v.35 no.3
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• pp.255-267
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• 2022

Uniform space is a generalization of metric space. The main purpose of this paper is to extend several results contained in [5, 6] which have for an expansive homeomorphism with the pseudo orbit tracing property(POTP in short) on a compact metric space (X, d) for an expansive homeomorphism with the POTP on a compact uniform space (X, 𝒰). we characterize stable and unstable sets, sink and source and saddle, recurrent points for an expansive homeomorphism which has the POTP on a compact uniform space (X, 𝒰).

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.569-581
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• 2020

In this paper, we introduce the new general concept of usual expansiveness which is called "positively weak measure expansiveness" and study the basic properties of positively weak measure expansive C¹-differentiable maps on a compact smooth manifold M. And we prove that the following theorems. (1) Let 𝓟𝓦𝓔 be the set of all positively weak measure expansive differentiable maps of M. Denote by int(𝓟𝓦𝓔) is a C¹-interior of 𝓟𝓦𝓔. f ∈ int(𝓟𝓦𝓔) if and only if f is expanding. (2) For C¹-generic f ∈ C¹ (M), f is positively weak measure-expansive if and only if f is expanding.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.1
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• pp.173-179
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• 2020

Let X be a compact space and Λ a finite index set. We deal with dynamical properties of iterated function systems on X. For an iterated function system 𝓕 on X, we prove that 𝓕 is c-expansive if and only if 𝓕^k is also c-expansive for each k ∈ ℕ. Furthermore we prove that the c-expansiveness of 𝓕 is equivalent to the original expansiveness of the shift map of it.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.3
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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.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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• pp.461-464
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• 2019

In this article, we consider that if a continuous map f : X → X of a compact metric space X is Li-York chaotic then it is positively weak measure expansive.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.77-83
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• 2022

In this paper we investigate the topological stability of induced homeomorphisms on a hyperspace. More precisely, we show that an expansive induced homeomorphism on a hyperspace is topologically stable. We also give examples and a diagram about implications to illustrate our results.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.1
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• pp.85-90
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• 2022

In this article, we consider the notion of expansiveness on compact metric spaces for symbolically point of view. And we show that a homeomorphism is symbolically countably expansive if and only if it is symbolically measure expansive. Moreover, we prove that a homeomorphism is symbolically N-expansive if and only if it is symbolically measure N-expanding.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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• pp.287-305
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• 2015

Hyperbolicity is a core of dynamics. Shadowness and expansiveness for homeomorphisms have been studied by J. Om-bach([3], [4], [5]). We study the hyperbolicity (i.e., expansivity and the shadowing property) and the Anosov relation for a closed relation.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1309-1318
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• 2020

In this paper we extend the concept of topological stability from continuous maps to the corresponding induced maps and prove that a continuous map is topologically stable if and only if its induced map also is topologically stable.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.4
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• pp.509-524
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• 2019

Let Act(G, X) be the set of all continuous actions of a finitely generated group G on a compact metric space X. In this paper, we study the concepts of topologically stable points and continuous shadowable points of a group action T ∈ Act(G, X). We show that if T is expansive then the set of continuous shadowable points is contained in the set of topologically stable points.