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# On hyperbolic attractors and repellers of endomorphisms

Received 08 November 2017; accepted 11 November 2017

2017, Vol. 13, No. 4, pp.  557–571

Author(s): Grines V. Z., Kurenkov E. D.

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
Citation: Grines V. Z., Kurenkov E. D., On hyperbolic attractors and repellers of endomorphisms, Rus. J. Nonlin. Dyn., 2017, Vol. 13, No. 4, pp.  557–571
DOI:10.20537/nd1704008