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2013
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# Vyacheslav Grines

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

## Publications:

 Grines V. Z., Kurenkov E. D. On hyperbolic attractors and repellers of endomorphisms 2017, Vol. 13, No. 4, pp.  557–571 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 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
 Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K. Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma 2014, Vol. 10, No. 4, pp.  427-438 Abstract We obtain properties of three-dimensional phase space and dynamics of Morse–Smale diffeomorphism that led to existence of at least one heteroclinical curve in non-wandering set of the diffeomorphism. We apply this result to solve a problem of existence of separators in magnetic field of plasma. Keywords: Morse – Smale cascades, heteroclinic curves, mapping torus, locally trivial bundle, separators of magnetic field Citation: Grines V. Z., Gurevich E. Y., Zhuzhoma E. V., Zinina S. K.,  Heteroclinic curves of Morse – Smale cascades and separators in magnetic field of plasma, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 4, pp.  427-438 DOI:10.20537/nd1404003
 Grines V. Z., Levchenko Y. A., Pochinka O. V. On topological classification of diffeomorphisms on 3-manifolds with two-dimensional surface attractors and repellers 2014, Vol. 10, No. 1, pp.  17-33 Abstract We consider a class of diffeomorphisms on 3-manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of two-dimensional surface basic sets. Interrelation between dynamics of such diffeomorphism and topology of the ambient manifold is studied. Also we establish that each considered diffeomorphism is Ω-conjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of two-dimensional invariant manifolds of points from the basic sets, we obtain necessary and sufficient conditions of topological conjugacy of structurally stable diffeomorphisms from the considered class. Keywords: diffeomorphism, basic set, topological conjugacy, attractor, repeller Citation: Grines V. Z., Levchenko Y. A., Pochinka O. V.,  On topological classification of diffeomorphisms on 3-manifolds with two-dimensional surface attractors and repellers, Rus. J. Nonlin. Dyn., 2014, Vol. 10, No. 1, pp.  17-33 DOI:10.20537/nd1401002