• Journal of the Chungcheong Mathematical Society
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• v.34 no.1
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• pp.55-60
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• 2021

In this paper we deal with the preservation of the periodic shadowing property of induced hyperspatial systems. More precisely, we show that an expansive system has the periodic shadowing property if and only if its induced hyperspatial system has the periodic shadowing property.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.981-989
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• 2019

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.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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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.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.4
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• pp.519-525
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• 2008

Let f be a diffeomorphisms of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper, we prove that if generically, f is tame diffeomorphims then the following conditions are equivalent: (i) f is ${\Omega}$-stable, (ii) f has the $C^1$-stable shadowing property (iii) f has the $C^1$-stable inverse shadowing property.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.1
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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.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.4
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• pp.631-638
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• 2011

In this paper, we prove that for $C^1$ generically, if every hyperbolic periodic point in a chain component is uniformly far away from being nonhyperbolic, and it is $C^1$-robustly average shadowable, then the chain component is hyperbolic.

• Communications of the Korean Mathematical Society
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• v.24 no.1
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• pp.127-144
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• 2009

Let f be a diffeomorphism of a compact $C^{\infty}$ manifold, and let p be a hyperbolic periodic point of f. In this paper we introduce the notion of $C^1$-stable inverse shadowing for a closed f-invariant set, and prove that (i) the chain recurrent set $\cal{R}(f)$ of f has $C^1$-stable inverse shadowing property if and only if f satisfies both Axiom A and no-cycle condition, (ii) $C^1$-generically, the chain component $C_f(p)$ of f associated to p is hyperbolic if and only if $C_f(p)$ has the $C^1$-stable inverse shadowing property.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.195-205
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• 2024

In this paper, we consider the set of periodic shadowable points for homeomorphisms of a compact metric space, and we prove that this set satisfies some properties such as invariance and being a G_δ set. Then we investigate implication relations related to sets consisting of shadowable points, periodic shadowable points and uniformly expansive points, respectively. Assume that the set of periodic points and the set of periodic shadowable points of a homeomorphism on a compact metric space are dense in X. Then we show that a homeomorphism has the periodic shadowing property if and only if so is the restricted map to the set of periodic shadowable points. We also give some examples related to our results.

• Korean Journal of Materials Research
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• v.29 no.11
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• pp.709-714
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• 2019

In order to observe the microstructure and morphology of porous titanium -oxide thin film, deposition is performed under a higher Ar gas pressure than is used in the general titanium thin film production method. Black titanium thin film is deposited on stainless steel wire and Cu thin plate at a pressure of about 12 Pa, but lustrous thin film is deposited at lower pressure. The black titanium thin film has a larger apparent thickness than that of the glossy thin film. As a result of scanning electron microscope observation, it is seen that the black thin film has an extremely porous structure and consists of a separated column with periodic step differences on the sides. In this configuration, due to the shadowing effect, the nuclei formed on the substrate periodically grow to form a step. The surface area of the black thin film on the Cu thin plate changes with the bias potential. It has been found that the bias of the small negative is effective in increasing the surface area of the black titanium thin film. These results suggest that porous titanium-oxide thin film can be fabricated by applying the appropriate oxidation process to black titanium thin film composed of separated columns.