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
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.
endomorphism, axiom $A$, basic set, attractor, repeller
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V. Z. Grines, E. D. Kurenkov, “On hyperbolic attractors and repellers of endomorphisms”, Nelin. Dinam., 13:4 (2017), 557–571
Citation in format AMSBIB
\by V.~Z.~Grines, E.~D.~Kurenkov
\paper On hyperbolic attractors and repellers of endomorphisms
\jour Nelin. Dinam.
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