• Bulletin of the Korean Mathematical Society
• /
• v.32 no.1
• /
• pp.93-102
• /
• 1995

In [6], we introduce a hyperbolic homeomorphism on a compact metrizable space and show that a hyperbolic homeomorphism is topologically stable. The purpose of this paper is to study a necessary and sufficient condition for a homeomorphism to be hyperbolic. We get the following theorem.

• Korean Journal of Mathematics
• /
• v.6 no.1
• /
• pp.117-125
• /
• 1998

In this paper, we investigate some conditions which are equivalent to fuzzy $r$-homeomorphisms and give some characterizing theorems for fuzzy $r$-semicontinuous, $r$-semiopen and $r$-semiclosed maps.

• Journal of the Chungcheong Mathematical Society
• /
• v.11 no.1
• /
• pp.73-85
• /
• 1998

This paper deals with the topological stability of continuous maps. First, the notion of local expansion is given and we show that local expansions of compact metric spaces have the shadowing property. Also, we prove that if a continuous surjective map f is a local homeomorphism and local expansion, then f is topologically stable in the class of continuous surjective maps. Finally, we find homeomorphisms which are not topologically stable.

• Honam Mathematical Journal
• /
• v.33 no.2
• /
• pp.231-246
• /
• 2011

Let $X_{n,k}$ be a Khalimsky topological n dimensional subspace with digital k-connectivity. In relation to the classification of spaces $X_{n,k}$, by comparing several kinds of continuities and homeomorphisms, the paper proposes a category which is suitable for studying the spaces $X_{n,k}$.

• Journal of the Chungcheong Mathematical Society
• /
• v.15 no.1
• /
• pp.61-73
• /
• 2002

We consider the inverse shadowing property of a dynamical system which is an "inverse" form of the shadowing property of the system. In particular, we show that every Morse-Smale system f on a compact smooth manifold has the inverse shadowing property with respect to the class $\mathcal{T}_h(f)$ of continuous methods generated by homeomorphisms, but the system f does not have the inverse\mathrm{T} shadowing property with respect to the class $\mathcal{T}_c(f)$ of continuous methods.

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

• The Pure and Applied Mathematics
• /
• v.2 no.2
• /
• pp.111-114
• /
• 1995

A C$\^$$\infty$/ manifold is a pair (M, C) where a) M is a Hausdorff topological space such that every point $\chi$$\in$M has a neighborhood homeomorphic to an open subset of R$^n$. b) C is a collection of these homeomorphisms whose domains cover M. If ø, $\psi$ $\in$ C then ø o $\psi$$\^$-1/ is C$\^$$\infty$/. c) C is maximal with respect to (b).(omitted)

• Bulletin of the Korean Mathematical Society
• /
• v.43 no.1
• /
• pp.43-52
• /
• 2006

We study the genericity of the first weak inverse shadowing property and the second weak inverse shadowing property in the space of homeomorphisms on a compact metric space, and show that every shift homeomorphism does not have the first weak inverse shadowing property but it has the second weak inverse shadowing property.

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

• Communications of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.1329-1334
• /
• 2019

We define the concept of a generator for a ${\mathbb{Z}}^k$-action T and show that T is expansive if and only it has a generator. Further, we prove several properties of a ${\mathbb{Z}}^k$-action including that the least upper bound of the set of expansive constants is not an expansive constant.