• 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.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.133-146
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• 2021

For pseudo-free and recurrent self-similar groups, we show that continuous orbit equivalence of inverse semigroup partial actions implies continuous orbit equivalence of group actions. Conversely, if group actions are continuous orbit equivalent, and the induced homeomorphism commutes with the shift maps on their groupoids, we obtain continuous orbit equivalence of inverse semigroup partial actions.

• 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.

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1299-1307
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• 2020

We show that the Liechti-Strenner's example for the closed nonorientable surface in [13] minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in [13] minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial p(x) of the action induced on the first cohomology nonpositive or all but the first coefficient of p(x)(x ± 1)², p(x)(x² ± 1), or p(x)(x² ± x + 1) nonpositive.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.915-932
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• 2010

The goal of this paper is to study extension problems of several continuities in computer topology. To be specific, for a set $X\;{\subset}\;Z^n$ take a subspace (X, $T_n^X$) induced from the Khalimsky nD space ($Z^n$, $T^n$). Considering (X, $T_n^X$) with one of the k-adjacency relations of $Z^n$, we call it a computer topological space (or a space if not confused) denoted by $X_{n,k}$. In addition, we introduce several kinds of k-retracts of $X_{n,k}$, investigate their properties related to several continuities and homeomorphisms in computer topology and study extension problems of these continuities in relation with these k-retracts.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.411-421
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• 2004

A digital $(k_0,\;k_1)$-homotopy is induced from digital $(k_0,\;k_1)$-continuity with the n kinds of $k_i$-adjacency relations in ${\mathbb{Z}}^n$, $i{\in}\{0,\;1\}$. The k-fundamental group, ${\pi}^k_1(X,\;x_0)$, is derived from the pointed digital k-homotopy, $k{\in}\{3^n-1(n{\geq}2),\;3^n-{\sum}^{r-2}_{k=0}C^n_k2^{n-k}-1(2{\leq}r{\leq}n-1(n{\geq}3)),\;2n(n{\geq}1)\}$. In this paper two kinds of digital 8-pseudotori stemmed from the minimal simple closed 4-curve and the minimal simple closed 8-curve with 8-contractibility or without 8-contractibility, e.g., $DT_8$ and $DT^{\prime}_8$, are introduced and their digital topological properties are studied by the calculation of the k-fundamental groups, $k{\in}\{8,\;32,\;64,\;80\}$.