(Differencialnie Uravnenia i Protsesy Upravlenia)

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Русская версия

**Sergei Yurievich Piliygin**

Faculty of Mathematics and Mechanics,

St.Petersburg State University

Universitetsky prospekt, 28,

198504, Peterhof, St. Petersburg, Russia

Professor

Doctor in physical and mathematical science, professor

In many papers devoted to connections between the theory of shadowing of pseudotrajectories of dynamical systems and the theory of structural stability, the authors apply the Mane theorem on characterization of structural stability in terms of the analytic strong transversality condition. The original proof of this theorem contains a proof of the implication "the analytic strong transversality condition implies the density of periodic points in the nonwandering set", which is based on a nontrivial theory of hyperbolic limit sets. In this short note, we show that the proof of this implication can be essentially simplified in the case of a diffeomorphism having the shadowing property. Bibl. 17 titles

- Pilyugin S. Yu.
*Shadowing in Dynamical Systems*. Lecture Notes in Math., vol. 1706, Springer-Verlag, 1999. xvi+271 p - Palmer K.
*Shadowing in Dynamical Systems. Theory and Applications*. Kluwer, 2000. xiv+299 p - Pilyugin S. Yu. [Theory of shadowing of pseudotrajectories in dynamical systems].
*Differencial'nie uravnenia i processy upravlenia*, 2011, no. 4, pp. 96-112 (In Russ. )**Available at**http://www.math.spbu.ru/diffjournal/pdf/pilyugin2.pdf - Robinson C. Stability theorems and hyperbolicity in dynamical systems.
*Rocky Mount. J. of Math.*, 1977: 425-437 - Morimoto A. [The method of pseudo-orbit tracing and stability of dynamical systems].
*Sem . Note Tokyo Univ.*, 1979, vol. 30 - Sawada K. Extended f-orbits are approximated by orbits.
*Nagoya Math. J.*, 1980, vol. 79: 33-45 - Pilyugin S. Yu. Variational shadowing.
*Discrete Continuous Dynam. Systems, Ser. B.*, 2010, vol. 14: 733-737 - Anosov D. V. Ob odnom klasse invariantnykh mnozhestv gladkikh dinamicheskikh sistem [On a class of invariant sets of smooth dynamical systems].
*Trudy 5 Mezhd. Konf. po Nelin. Koleb.*[Proc. 5th Intern. Conf. on Nonlin. Oscill. ], Kiev, 1970, pp. 39-45 (in Russian) - Bowen R.
*Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms.*Lecture Notes in Math., vol. 470, Springer-Verlag, 1975. i+108 p - Pilyugin S. Yu., Tikhomirov S. B. Lipschitz shadowing implies structural stability.
*Nonlinearity*, 2010; vol. 23, pp. 2509-2515 - Mane R.
*Characterizations of AS diffeomorphisms.*Lecture Notes in Math., vol. 597, Springer-Verlag, 1977, pp. 389-394 - Pliss V. A. Ogranichennye resheniya neodnorodnykh lineinykh sistem differentsial'nykh uravnenii [Bounded solutions of nonhomogeneous linear systems of differential equations].
*Problemy Asimptoticheskoi Teorii Nelineinykh Kolebanii*[Problems of the Asymptotic Theory of Nonlinear Oscillations], Kiev, 1977, pp. 168-173 (In Russian) - Tikhomirov S. B. Holder shadowing on finite intervals.
*Ergodic Theory and Dynamical Systems*(to appear). Preprint: arXiv:1106. 4053v4, 2011 - Todorov D. Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory.
*Discrete Continuous Dynam. Systems, Ser. A*, 2013; vol. 33, pp 4187 - 4205 - Pilyugin S. Yu.
*Vvedenie v grubye sistemy differentsial'nykh uravnenii*[Introduction to structurally stable systems of differential equations]. Leningrad Univ. Publ., 1988 (in Russ). 159 p - Newhouse S. Hyperbolic limit sets.
*Trans. Amer. Math. Soc.*, 1972; vol. 167, pp. 125-150 - Pilyugin S. Yu.
*Prostranstva dinamicheskikh sistem*[Spaces of dynamical systems]. Regular and Chaotic Dynamics, Izhevsk, 2008, 270 p. (In Russian)