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ON PERIODIC SHADOWING OF INDUCED HYPERSPACE MAPS
Namjip Koo,Hyunhee Lee,Nyamdavaa Tsegmid 충청수학회 2021 충청수학회지 Vol.34 No.1
In this paper we deal with the preservation of the pe- riodic shadowing property of induced hyperspatial systems. More precisely, we show that an expansive system has the periodic shad- owing property if and only if its induced hyperspatial system has the periodic shadowing property.
TOPOLOGICALLY STABLE POINTS AND UNIFORM LIMITS
Namjip Koo,Hyunhee Lee Korean Mathematical Society 2023 대한수학회지 Vol.60 No.5
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.
A NOTE ON WEAK EXPANSIVE HOMEOMORPHISMS ON A COMPACT METRIC SPACE
Namjip Koo,Gansukh Tumur 충청수학회 2020 충청수학회지 Vol.33 No.1
In this paper we introduce the notion of the expansivity for homeomorphisms on a compact metric space and study some properties concerning weak expansive homeomorphisms. Also, we give some examples to illustrate our results.
A NOTE ON EXPLICIT SOLUTIONS OF CERTAIN IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS
Namjip Koo 충청수학회 2017 충청수학회지 Vol.30 No.1
This paper deals with linear impulsive differential equa-tions involving the Caputo fractional derivative. We provide exact solutions of nonhomogeneous linear impulsive fractional differential equations with constant coefficients by means of the Mittag-Leffler functions.
ON THE DENSITY OF VARIOUS SHADOWING PROPERTIES
Koo, Namjip,Tsegmid, Nyamdavaa Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.3
In this paper we deal with some shadowing properties of discrete dynamical systems on a compact metric space via the density of subdynamical systems. Let $f:X{\rightarrow}X$ be a continuous map of a compact metric space X and A be an f-invariant dense subspace of X. We show that if $f{\mid}_A:A{\rightarrow}A$ has the periodic shadowing property, then f has the periodic shadowing property. Also, we show that f has the finite average shadowing property if and only if $f{\mid}_A$ has the finite average shadowing property.
PRESERVATION OF EXPANSIVITY IN HYPERSPACE DYNAMICAL SYSTEMS
Koo, Namjip,Lee, Hyunhee Korean Mathematical Society 2021 대한수학회지 Vol.58 No.6
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 C<sup>0</sup>-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.
Pointwise Topological Stability
Koo, Namjip,Lee, Keonhee,Morales, C. A. Cambridge University Press 2018 Proceedings of the Edinburgh Mathematical Society Vol.61 No.4
<B>Abstract</B><P>We decompose the topological stability (in the sense of P. Walters) into the corresponding notion for points. Indeed, we define a <I>topologically stable point</I> of a homeomorphism <I>f</I> as a point <I>x</I> such that for any <I>C</I><SUP>0</SUP>-perturbation <I>g</I> of <I>f</I> there is a continuous semiconjugation <I>defined on the <I>g</I>-orbit closure of <I>x</I></I> which tends to the identity as <I>g</I> tends to <I>f</I>. We obtain some properties of the topologically stable points, including preservation under conjugacy, vanishing for minimal homeomorphisms on compact manifolds, the fact that topologically stable chain recurrent points belong to the periodic point closure, and that the chain recurrent set coincides with the closure of the periodic points when all points are topologically stable. Next, we show that the topologically stable points of an expansive homeomorphism of a compact manifold are precisely the shadowable ones. Moreover, an expansive homeomorphism of a compact manifold is topologically stable if and only if every point is topologically stable. Afterwards, we prove that a pointwise recurrent homeomorphism of a compact manifold has no topologically stable points. Finally, we prove that every chain transitive homeomorphism with a topologically stable point of a compact manifold has the pseudo-orbit tracing property. Therefore, a chain transitive expansive homeomorphism of a compact manifold is topologically stable if and only if it has a topologically stable point.</P>
TOPOLOGICAL STABILITY IN HYPERSPACE DYNAMICAL SYSTEMS
Koo, Namjip,Lee, Hyunhee,Tsegmid, Nyamdavaa Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.4
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.
A NOTE ON EXPLICIT SOLUTIONS OF CERTAIN IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS
Koo, Namjip Chungcheong Mathematical Society 2017 충청수학회지 Vol.30 No.1
This paper deals with linear impulsive differential equations involving the Caputo fractional derivative. We provide exact solutions of nonhomogeneous linear impulsive fractional differential equations with constant coefficients by means of the Mittag-Leffler functions.