• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.575-591
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• 2018

Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. Roughly, an Anosov family is a two-sided sequence of diffeomorphisms (or non-stationary dynamical system) with similar behavior to an Anosov diffeomorphisms. We show that the set consisting of Anosov families is an open subset of the set consisting of two-sided sequences of diffeomorphisms, which is equipped with the strong topology (or Whitney topology).

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.897-910
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• 2019

In this paper we introduce the notion of "robust special Anosov endomorphisms", and show that Anosov endomorphisms of tori which are not neither an Anosov diffeomorphism nor an expanding map, are not robust special.

• 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 Chungcheong Mathematical Society
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• v.25 no.4
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• pp.739-745
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• 2012

We show that if a symplectic diffeomorphism has the $C^1$-robustly orbital shadowing property, then the diffeomorphism is Anosov.

• Journal of the Chungcheong Mathematical Society
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• v.28 no.1
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• pp.103-108
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• 2015

In this paper we prove that $C^1$-generically, if a diffeomorphism f on a closed $C^{\infty}$ manifold M satisfies weak specification on a locally maximal set ${\Lambda}{\subset}M$ then ${\Lambda}$ is hyperbolic for f. As a corollary we obtain that $C^1$-generically, every diffeomorphism with weak specification is Anosov.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.883-914
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• 2001

We present a survey of research results obtained for infra-nilmanifolds, their fundamental groups and some of their generalizations. This is presented from two different approaches and covers achievements obtained during the past four decades and showing a remarkable amount of mathematical interdisciplinarity. We go more in depth concerning the existence and construction of polynomial structures for these manifolds and groups, a direction where significant progress was made in the past few years. The bounded-degree polynomial structures developed by the authors triggered a number of challenging open problems. Also, their study already has lead to some interesting results concerning e.g. Anosov diffeomorphisms and expanding maps.