ul. Bolshaya Pecherskaya 25/12, Nizhnii Novgorod, 603155, Russia
National Research University Higher School of Economics
Grines V. Z., Kurenkov E. D.
On hyperbolic attractors and repellers of endomorphisms
2017, Vol. 13, No. 4, pp. 557–571
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.
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
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.
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
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.