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 Nelin. Dinam.: Year: Volume: Issue: Page: Find

 Nelin. Dinam., 2017, Volume 13, Number 4, Pages 557–571 (Mi nd585)

On the 75th birthday of A.P.Markeev

On hyperbolic attractors and repellers of endomorphisms

V. Z. Grines, E. D. Kurenkov

National Research University Higher School of Economics, ul. Bolshaya Pecherskaya 25/12, Nizhnii Novgorod, 603155, Russia

Abstract: It is well known that the topological classification of dynamical systems with hyperbolic dynamics is significantly defined by dynamics on a nonwandering set. F. Przytycki generalized axiom $A$ for smooth endomorphisms that was previously introduced by S. Smale for diffeomorphisms, and proved the spectral decomposition theorem which claims that the nonwandering set of an $A$-endomorphism is a union of a finite number of basic sets. In the present paper the criterion for a basic set of an $A$-endomorphism to be an attractor is given. Moreover, dynamics on basic sets of codimension one is studied. It is shown that if an attractor is a topological submanifold of codimension one of type $(n-1,1)$, then it is smoothly embedded in the ambient manifold, and the restriction of the endomorphism to this basic set is an expanding endomorphism. If a basic set of type $(n,0)$ is a topological submanifold of codimension one, then it is a repeller, and the restriction of the endomorphism to this basic set is also an expanding endomorphism.

Keywords: endomorphism, axiom $A$, basic set, attractor, repeller

 Funding Agency Grant Number Russian Science Foundation 17-11-01041 HSE Basic Research Program Ò-90

DOI: https://doi.org/10.20537/nd1704008

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Document Type: Article
UDC: 517.938
MSC: 37D20
Accepted:14.11.2017

Citation: V. Z. Grines, E. D. Kurenkov, “On hyperbolic attractors and repellers of endomorphisms”, Nelin. Dinam., 13:4 (2017), 557–571

Citation in format AMSBIB
\Bibitem{GriKur17}
\by V.~Z.~Grines, E.~D.~Kurenkov
\paper On hyperbolic attractors and repellers of endomorphisms
\jour Nelin. Dinam.
\yr 2017
\vol 13
\issue 4
\pages 557--571
\mathnet{http://mi.mathnet.ru/nd585}
\crossref{https://doi.org/10.20537/nd1704008}
\elib{http://elibrary.ru/item.asp?id=30780701}