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### Open Access Journals

DCDS

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

DCDS

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.

DCDS

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.

DCDS

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

DCDS

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.

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