• Communications of the Korean Mathematical Society
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• v.32 no.1
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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.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.657-661
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• 2016

Let $f:X{\rightarrow}X$ be a continuous surjection of a compact metric space and let ($X_f,{\tilde{f}}$) be the inverse limit of a continuous surjection f on X. We show that for a continuous surjective map f, if f has the asymptotic average, the average shadowing, the ergodic shadowing property then ${\tilde{f}}$ is topologically transitive.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.4
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• pp.461-466
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• 2018

$Let\;f:X{\rightarrow}X$ be a continuous surjection of a compact metric space X and let ${\sigma}_f:X_f{\rightarrow}X_f$ be the shift map on the inverse limit space $X_f$ constructed by f. We show that if a continuous surjective map f has some shadowing properties: the asymptotic average shadowing property, the average shadowing property, the two side limit shadowing property, then ${\sigma}_f$ also has the same properties.

• Journal of the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.165-170
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• 2019

$C^1$ generically, if a diffeomorphism $f:M{\rightarrow}M$ of a closed smooth Riemannian manifold M has the asymptotic average shadowing property or the average shadowing property then f has a decomposition property.

• 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.

• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.355-361
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• 2006

We prove that if a continuous surjective map f on a compact metric space X has the average shadowing property, then every point x is chain recurrent. We also show that if a homeomorphism f has more than two fixed points on $S^1$, then f does not satisfy the average shadowing property. Moreover, we construct a homeomorphism on a circle which satisfies the shadowing property but not the average shadowing property. This shows that the converse of the theorem 1.1 in [6] is not true.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.4
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• pp.651-655
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• 2016

In this paper, we show that if a quasi-Anosov diffeomorphism has the various types of shadowing property then it is Anosov.

• Journal of the Korean Mathematical Society
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• v.58 no.1
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• pp.207-218
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• 2021

In this paper, the notion of two-sided limit shadowing property is considered for a positively expansive open map. More precisely, let f be a positively expansive open map of a compact metric space X. It is proved that if f is topologically mixing, then it has the two-sided limit shadowing property. As a corollary, we have that if X is connected, then the notions of the two-sided limit shadowing property and the average-shadowing property are equivalent.

• 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.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.