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Dynamical systems with Lipschitz inverse shadowing properties. (English. Russian original) Zbl 1256.37009

Vestn. St. Petersbg. Univ., Math. 44, No. 3, 208-213 (2011); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2011, No. 3, 48-54(2011).
Summary: In this paper, the notion of the Lipschitz inverse shadowing property with respect to two classes of \(d\)-methods that generate pseudotrajectories of dynamical systems is introduced. It is shown that if a diffeomorphism of a Euclidean space has the Lipschitz inverse shadowing property on the trajectory of an individual point, then the Mañé analytic strong transversality condition must be satisfied at this point. This result is used in the proof of the main theorem: a diffeomorphism of a smooth closed manifold that has the Lipschitz inverse shadowing property is structurally stable.

MSC:

37C50 Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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References:

[1] S. Yu. Pilyugin, Shadowing in Dynamical Systems (Springer, Berlin, 1999).
[2] K. Palmer, Shadowing in Dynamical Systems. Theory and Applications (Klüwer, Dordrecht, 2000). · Zbl 0997.37001
[3] R. M. Corless and S. Yu. Pilyugin, ”Approximate and Real Trajectories for Generic Dynamical Systems,” J. Math. Anal. Appl. 189, 409–423 (1995). · Zbl 0821.58036 · doi:10.1006/jmaa.1995.1027
[4] S. Yu. Pilyugin, ”Inverse Shadowing by Continuous Methods,” Discrete Contin. Dyn. Syst. 8, 29–38 (2002). · Zbl 0998.37006 · doi:10.3934/dcds.2002.8.29
[5] P. E. Kloeden and J. Ombach, ”Hyperbolic Homeomorphisms and Bishadowing,” Ann. Polon. Math. 65, 171–177 (1997). · Zbl 0877.58044 · doi:10.4064/ap-65-2-171-177
[6] S. Yu. Pilyugin and S. B. Tikhomirov, ”Lipschitz Shadowing Implies Structural Stability,” Nonlinearity 23, 2509–2515 (2010). · Zbl 1206.37012 · doi:10.1088/0951-7715/23/10/009
[7] S. Yu. Pilyugin, Spaces of Dynamical Systems (Regular and Chaotic Dynamics, Izhevsk, 2008) [in Russian].
[8] R. Mãné, ”Characterizations of AS diffeomorphisms,” in Geometry and Topology, Lect. Notes in Math. Vol. 597 (Springer, Berlin, 1977), pp. 389–394.
[9] V. A. Pliss, ”Bounded Solutions of Inhomogeneous Linear Systems of Differential Equations,” in Problems in Asymptotic Theory of Nonlinear Oscillations (Naukova Dumka, Kiev, 1977), pp. 168–173.
[10] S. Yu. Pilyugin, ”Generalizations of the Notion of Hyperbolicity,” J. Diff. Eqns. Appl. 12, 271–282 (2006). · Zbl 1136.37322 · doi:10.1080/10236190500489350
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