DCDS
Various shadowing properties and their equivalence
Keonhee Lee Kazuhiro Sakai
Discrete & Continuous Dynamical Systems - A 2005, 13(2): 533-540 doi: 10.3934/dcds.2005.13.533
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
The oe-property of diffeomorphisms
Kazuhiro Sakai
Discrete & Continuous Dynamical Systems - A 1998, 4(3): 581-591 doi: 10.3934/dcds.1998.4.581
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.
keywords: pseudo-orbits Axiom A and strong transversality condition structurally stable. Extended orbits shadowing property
DCDS
$C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings
Shaobo Gan Kazuhiro Sakai Lan Wen
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 205-216 doi: 10.3934/dcds.2010.27.205
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.
keywords: dominated splitting partially hyperbolic. shadowing Weak shadowing pseudo-orbit homoclinic class chain component chain recurrent set
DCDS
Transversality properties and $C^1$-open sets of diffeomorphisms with weak shadowing
S. Yu. Pilyugin Kazuhiro Sakai O. A. Tarakanov
Discrete & Continuous Dynamical Systems - A 2006, 16(4): 871-882 doi: 10.3934/dcds.2006.16.871
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.
keywords: transversality condition. Weak shadowing property shadowing property Axiom A no-cycle condition
DCDS
Orbital and weak shadowing properties
S. Yu. Pilyugin A. A. Rodionova Kazuhiro Sakai
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 287-308 doi: 10.3934/dcds.2003.9.287
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.
keywords: pseudotrajectories shadowing structural stability. Axiom A
DCDS
$C^1$-stable shadowing diffeomorphisms
Keonhee Lee Kazumine Moriyasu Kazuhiro Sakai
Discrete & Continuous Dynamical Systems - A 2008, 22(3): 683-697 doi: 10.3934/dcds.2008.22.683
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
Regular maps with the specification property
Kazumine Moriyasu Kazuhiro Sakai Kenichiro Yamamoto
Discrete & Continuous Dynamical Systems - A 2013, 33(7): 2991-3009 doi: 10.3934/dcds.2013.33.2991
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.
keywords: $C^1$-topology. Specification property regular maps Axiom A elementary sets $C^1$-stable specification property

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