Geodesic
On hyperbolic attractors and repellers of endomorphisms
Russian journal of nonlinear dynamics, Tome 13 (2017) no. 4, pp. 557-571

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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.
Mots-clés : endomorphism, axiom $A$
Keywords: basic set, attractor, repeller. 
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     author = {V. Z. Grines and E. D. Kurenkov},
     title = {On hyperbolic attractors and repellers of endomorphisms},
     journal = {Russian journal of nonlinear dynamics},
     pages = {557--571},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2017_13_4_a7/}
}
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V. Z. Grines; E. D. Kurenkov. On hyperbolic attractors and repellers of endomorphisms. Russian journal of nonlinear dynamics, Tome 13 (2017) no. 4, pp. 557-571. http://geodesic.mathdoc.fr/item/ND_2017_13_4_a7/