• 충청수학회지
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• 제24권4호
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• pp.735-743
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• 2011

Let f be a diffeomorphism of a closed n-dimensional smooth manifold. In this paper, we introduce the notion of $C^1$-stably periodic shadowing property for a closed f-invariant set, and prove that for a transitive set ${\Lambda}$, if f has the $C^1$-stably periodic shadowing property on ${\Lambda}$, then ${\Lambda}$ admits a dominated splitting.

• 대한수학회지
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• 제58권5호
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• pp.1059-1079
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• 2021

We show that given any chain transitive set of a C¹ generic diffeomorphism f, if a diffeomorphism f has the eventual shadowing property on the locally maximal chain transitive set, then it is hyperbolic. Moreover, given any chain transitive set of a C¹ generic vector field X, if a vector field X has the eventual shadowing property on the locally maximal chain transitive set, then the chain transitive set does not contain a singular point and it is hyperbolic. We apply our results to conservative systems (volume-preserving diffeomorphisms and divergence-free vector fields).

• 충청수학회지
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• 제37권2호
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• pp.75-80
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• 2024

In this paper, we study some dynamics of the uniform limits of sequences in dynamical systems on a noncompact metric space. We show that if a sequence of homeomorphisms on a noncompact metric space has the uniform ergodic shadowing property, then the uniform limit also has the ergodic shadowing property. Then we apply this result to nonwandering maps.

• 충청수학회지
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• 제20권4호
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• pp.577-585
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• 2007

We introduce the inverse shadowing property of geometric Lorenz flows and prove that the geometric Lorenz flows do not have the inverse shadowing property.

• 대한수학회논문집
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• 제32권1호
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• pp.201-214
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• 2017

In this paper, we introduce and study notions of average chain transitivity, average chain mixing, total average chain transitivity and almost average shadowing property. We also discuss their interrelations.

• 대한수학회지
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• 제59권5호
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• pp.987-996
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• 2022

We introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.

• 충청수학회지
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• 제23권4호
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• pp.617-623
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• 2010

Let p be a hyperbolic periodic point of f, and let ${\Lambda}(p)$ be a closed set which containing p. In this paper, we show that $C^1$-generically, if $f{\mid}_{{\Lambda}(p)}$ has the $C^1$-stably asymptotic average shadowing property, then it admits a dominated splitting.

• 충청수학회지
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• 제25권1호
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• pp.27-33
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• 2012

Let $f$ be a difeomorphism of a closed $n$ -dimensional smooth manifold M, and $p$ be a hyperbolic periodic point of $f$. Let ${\Lambda}(p)$ be a closed set which containing $p$. In this paper, we show that (i) if $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$, then ${\Lambda}(p)$ is the chain component which contains $p$. (ii) suppose $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$. Then if $f|_{\Lambda(p)}$ has the $C^{1}$-stably shadowing property then it is hyperbolic.

• 한국수학사학회지
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• 제18권1호
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• pp.93-102
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• 2005

본 논문에서는 역학계의 역사와 H-추적 성질(H-shadowing property)의 개념을 소개한다. 또한 콤팩트 거리 공간에서 H-추적 성질과, 확정성, 위상적 안정성 등 다른 역학적 성질 사이의 관계를 연구하였다.

• 대한수학회지
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• 제61권2호
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• pp.395-408
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• 2024

There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a C² strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.