• 충청수학회지
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• 제25권4호
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• pp.725-730
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• 2012

Let $f$ be a diffeomorphism of a closed $C^{\infty}$ manifold M. We show that $C^1$-generically, if $f$ has the shadowing property on a robustly transitive set, then it is hyperbolic.

• 대한수학회지
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• 제56권1호
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• pp.53-65
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• 2019

Recently, Chung and Lee [2] introduced the notion of topological stability for a finitely generated group action, and proved a group action version of the Walters's stability theorem. In this paper, we introduce the concepts of continuous shadowing and continuous inverse shadowing of a finitely generated group action on a compact metric space X with respect to various classes of admissible pseudo orbits and study the relationships between topological stability and continuous shadowing and continuous inverse shadowing property of group actions. Moreover, we introduce the notion of structural stability for a finitely generated group action, and we prove that an expansive action on a compact manifold is structurally stable if and only if it is continuous inverse shadowing.

• 충청수학회지
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• 제26권2호
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• pp.309-313
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• 2013

In this paper, we show that for a linear dynamical system $f(x)=Ax$ of $\mathbb{C}^n$, $f$ has the limit shadowing property if and only if the matrix A is hyperbolic.

• 충청수학회지
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• 제17권2호
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• pp.137-145
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• 2004

We give characterization of ${\Omega}$-stable diffeomorphisms via the notions of weak inverse shadowing. More precisely, it is proved that the $C^1$ interior of the set of diffeomorphisms with the weak inverse shadowing property with respect to the class $\mathcal{T}_h$ coincides with the set of ${\Omega}$-stable diffeomorphisms.

• 대한수학회지
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• 제57권4호
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• pp.935-955
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• 2020

In this paper we present a measurable version of the Smale's spectral decomposition theorem for homeomorphisms on compact metric spaces. More precisely, we prove that if a homeomorphism f on a compact metric space X is invariantly measure expanding on its chain recurrent set CR(f) and has the eventually shadowing property on CR(f), then f has the spectral decomposition. Moreover we show that f is invariantly measure expanding on X if and only if its restriction on CR(f) is invariantly measure expanding. Using this, we characterize the measure expanding diffeomorphisms on compact smooth manifolds via the notion of Ω-stability.

• 대한수학회보
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• 제51권2호
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• pp.387-400
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• 2014

We prove that the set of k-type nonwandering points of a Z2-action T can be decomposed into a disjoint union of closed and T-invariant sets $B_1,{\ldots},B_l$ such that $T|B_i$ is topologically k-type transitive for each $i=1,2,{\ldots},l$, if T is expansive and has the shadowing property.

• 충청수학회지
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• 제29권1호
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• pp.171-176
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• 2016

In this article, we study various notions on multi-valued dynamical systems. We first investigate important tools to express the systems, and prove that the notion of chain recurrence is equivalent to the notion of nonwandering set on compact metric spaces.

• 충청수학회지
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• 제28권2호
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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.

• 충청수학회지
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• 제31권4호
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• pp.411-420
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• 2018

In this paper we introduce the notion of shadowable points for finitely generated group actions on compact metric spaace and prove that the set of shadowable points is invariant and Borel set and if chain recurrent set contained shadowable point set then it coincide with nonwandering set. Moreover an action $T{\in}Act(G, X)$ has the shadowing property if and only if every point is shadowable.

• 충청수학회지
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• 제35권2호
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• pp.171-176
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• 2022

In this article, we investigate the transitivity and chain transitivity on multi-valued dynamical systems. For compact-valued continuous dynamics, we prove that the notion of transitivity is expressed by the notions of the shadowing property and chain transitivity under locally maximal condition.