Inverse shadowing by continuous methods
S. Yu. Pilyugin
Discrete & Continuous Dynamical Systems - A 2002, 8(1): 29-38 doi: 10.3934/dcds.2002.8.29
We show that a structurally stable diffeomorphism has the inverse shadowingproperty with respect to classes of continuous methods. We also show thatany diffeomorphism belonging to the $C^1$-interior of the set of diffeomorphisms withthe above-mentioned property is structurally stable.
keywords: pseudotrajectories Dynamical systems inverse shadowing structural stability.
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
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

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