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MSC2010 37D20, 37G70

Many-dimensional solenoid invariant saddle-type sets

E. V. Zhuzhoma1, N. V. Isaenkova2, V. S. Medvedev3

AnnotationIn the paper we construct some example of smooth diffeomorphism of closed manifold. This diffeomorphism has one-dimensional (in topological sense) basic set with stable invariant manifold of arbitrary nonzero dimension and the unstable invariant manifold of arbitrary dimension not less than two. The basic set has a saddle type, i.e. is neither attractor nor repeller. In addition, it follows from the construction that the diffeomorphism has a positive entropy and is conservative (i.e. its jacobian equals one) in some neighborhood of the one-dimensional solenoidal basic set. The construction represented in this paper allows to construct a diffeomorphism with the properties stated above on the manifold that is diffeomorphic to the prime product of the circle and the sphere of codimension one.
Keywordsdiscrete dynamical system, basic set, solenoid, separator, topological entropy.

1Evgeny V. Zhuzhoma, Professor of Department of Fundamental Mathematics, National Research University <> (25/12 B. Pecherskaya st., Nizhny Novgorod 603005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-8682-7591, zhuzhoma@mail.ru

2Nataliya V. Isaenkova, Professor of Department of Mathematics, Computer Science and Information Technology, Nizhny Novgorod Academy of the Ministry of the Interior of the Russian Federation  (3 Ankudinovskoye Sh., Nizhny Novgorod 603950, Russia), Ph.D. (Physics and Mathematics), ORCID: http:// orcid.org/0000-0003-4880-3526, nisaenkova@mail.ru

3Vyacheslav S. Medvedev, Researcher TAPRADESS laboratory, National Research University <> (25/12 B. Pecherskaya, Nizhny Novgorod 603005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0001-6369-0000, vmedvedev@hse.ru

Citation: E. V. Zhuzhoma, N. V. Isaenkova, V. S. Medvedev, "[Many-dimensional solenoid invariant saddle-type sets]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:1 (2018) 23–29 (In Russian)

DOI 10.15507/2079-6900.20.201801.23-29