• Journal of the Chungcheong Mathematical Society
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• v.28 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.48 no.3
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• pp.565-583
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• 2011

In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.629-633
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• 2011

In this paper we study some properties related to hyperbolicity and tautness of model domains.

• Honam Mathematical Journal
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• v.37 no.3
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• pp.353-360
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• 2015

In this paper we study some properties related to the Lemperty hyperbolicity of certain Hartogs type domains. Our results for Lempert hyperbolcity are more generalized than proved in [3].

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.237-247
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• 2007

In this paper, we establish a necessary condition in terms of curvature for the Kobayashi hyperbolicity of a class of almost complex Finsler manifolds. For an almost complex Finsler manifold with the condition (R), so-called Rizza manifold, we show that there exists a unique connection compatible with the metric and the almost complex structure which has the horizontal torsion in a special form. With this connection, we define a holomorphic sectional curvature. Then we show that this holomorphic sectional curvature of an almost complex submanifold is not greater than that of the ambient manifold. This fact, in turn, implies that a Rizza manifold is hyperbolic if its holomorphic sectional curvature is bounded above by -1.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.409-422
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• 2003

Let $F: \mathbb{C}^k\;{\rightarrow}\;\mathbb{C}^k$ be a dynamical system and let $\{x_n\}_{n{\geq}0}$ denote an orbit of F. We study the relation between $\{x_n\}$ and pseudoorbits $\{y_n}, y_0=x_0.\;Here\;y_{n+1}=F(y_n)+s_n.$ In general $y_n$ might diverge away from $x_n.$ Our main problem is whether there exists arbitrarily small $t_n$ so that if $\tilde{y}_{n+1}=F(\tilde{y}_n)+s_n+t_n,$ then $\tilde{y}_n$ remains close to $x_n.$ This leads naturally to the concept of sustainable orbits, and their existence seems to be closely related to the concept of hyperbolicity, although they are not in general equivalent.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1305-1319
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• 2015

We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.167-175
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• 2018

In this paper we show that two notions of the average shadowing property and the hyperbolicity are equivalent in linear dynamical systems on the vector space ${\mathbb{C}}^n$.

• Journal of the Korean Mathematical Society
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• v.27 no.1
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• pp.33-45
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• 1990

• Journal of the Korean Mathematical Society
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• v.16 no.2
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• pp.125-129
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• 1980