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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

In this paper, the $C^1$ interior of the set of all diffeomorphisms satisfying
the OE-property is characterized as the set of all diffeomorphisms satisfying Axiom
A and the strong transversality condition. Thus the $C^1$ interior of the set of all
diffeomorphisms satisfying the OE-property is equal to the $C^1$ interior of the set of
all diffeomorphisms satisfying the shadowing property.

DCDS

Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$.
In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting.
Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.

DCDS

Let Int$^1WS(M)$ be the $C^1$-interior of the set of diffeomorphisms of a smooth closed manifold $M$ having the weak shadowing property.
The second author has shown that if $\dim M = 2$ and all of the sources and sinks of a diffeomorphism $f \in$ Int$^1WS(M)$ are trivial, then $f$ is structurally stable.
In this paper, we show that there exist diffeomorphisms $f \in$ Int$^1WS(M)$, $\dim M = 2$, such that $(i)$ $f$ belongs to the $C^1$-interior of diffeomorphisms for which the $C^0$-transversality condition is not satisfied, $(ii)$ $f$ has a saddle connection.
These results are based on the following theorem: if the phase diagram of an $\Omega$-stable diffeomorphism $f$ of a manifold $M$ of arbitrary dimension does not contain chains of length $m > 3$, then $f$ has the weak shadowing property.

DCDS

We study weak and orbital shadowing properties of dynamical systems related
to the following approach: we look for exact trajectories lying in small
neighborhoods of approximate ones (or containing approximate ones in their
small neighborhoods) or for exact trajectories such that the Hausdorff
distances between their closures and closures of approximate trajectories
are small.

These properties are characterized for linear diffeomorphisms. We also study some $C^1$-open sets of diffeomorphisms defined in terms of these properties. It is shown that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms.

These properties are characterized for linear diffeomorphisms. We also study some $C^1$-open sets of diffeomorphisms defined in terms of these properties. It is shown that the $C^1$-interior of the set of diffeomorphisms having the orbital shadowing property coincides with the set of structurally stable diffeomorphisms.

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

Let $f$ be a $C^1$-regular map of a closed $C^{\infty}$ manifold $M$ and $\Lambda$ be a locally maximal closed invariant set of $f$.
We show that $f|_{\Lambda}$ satisfies the $C^1$-stable specification property if and only if $\Lambda$ is a hyperbolic elementary set.
We also prove that there exists a residual subset $\mathcal{R}$ in the space of $C^1$-regular maps endowed with the $C^1$-topology such that for $f \in \mathcal{R}$, $f|_{\Lambda}$ satisfies the specification property if and only if $\Lambda$ is a hyperbolic elementary set.

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