Hyperbolicity of $C^1$-stably expansive homoclinic classes
Keonhee Lee Manseob Lee
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 expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
keywords: shadowing chain component homoclinic class chain recurrent germ expansive Axiom A. $C^1$- persistently expansive hyperbolic $C^1$-stably expansive
Attractors under discretizations with variable stepsize
Barnabas M. Garay Keonhee Lee
The standard upper and lower semicontinuity results for discretized attractors [22], [13], [5] are generalized for discretizations with variable stepsize. Several examples demonstrate that the limiting behaviour depends crucially on the stepsize sequence. For stepsize sequences suitably chosen, convergence to the exact attractor in the Hausdorff metric is proven. Connections to pullback attractors in cocycle dynamics are pointed out.
keywords: variable stepsize discretization. Attractor
Various shadowing properties and their equivalence
Keonhee Lee Kazuhiro Sakai
In this paper, various shadowing properties are considered for expansive homeomorphisms. More precisely, we show that the continuous shadowing property, the Lipschitz shadowing property, the limit shadowing property and the strong shadowing property are all equivalent to the (usual) shadowing property for expansive homeomorphisms on compact metric spaces.
keywords: strong shadowing limit shadowing Pseudo-orbit Lipschitz shadowing continuous shadowing expansive. shadowing
$C^1$-stable shadowing diffeomorphisms
Keonhee Lee Kazumine Moriyasu Kazuhiro Sakai
Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold. In this paper, we define the notion of the $C^1$-stable shadowing property for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $R(f)$ of $f$ has the $C^1$-stable shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and $(ii)$ for the chain component $C_f(p)$ of $f$ containing a hyperbolic periodic point $p$, $C_f(p)$ has the $C^1$-stable shadowing property if and only if $C_f(p)$ is the hyperbolic homoclinic class of $p$.
keywords: hyperbolic set $C^1$-stable shadowing property chain component Axiom A. chain recurrent set shadowing pseudo-orbit homoclinic class
Topological stability and spectral decomposition for homeomorphisms on noncompact spaces
Keonhee Lee Ngoc-Thach Nguyen Yinong Yang

In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for homeomorphisms on noncompact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Then we extend the Walters's stability theorem and Smale's spectral decomposition theorem to homeomorphisms on locally compact metric spaces.

keywords: Expansiveness shadowing property spectral decomposition topological stability

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