Vyacheslav Grines
Professor: HSE Campus in Nizhny Novgorod, Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), Department of Funda
mental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod / Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory
of Dynamical Systems and Applications
Born: December 13, 1946 in Isyaslavl', Ukraina.
Positions held:
2015Present: Professor: HSE Campus in Nizhny Novgorod,
Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod),
Department of Fundamental Mathematics;
Chief Research Fellow: HSE Campus in Nizhny Novgorod, Faculty of Informatics,
Mathematics, and Computer Science (HSE Nizhny Novgorod), International Laboratory
of Dynamical Systems and Applications;
20132015: Professor of department of numerical and functional analysis, Lobachevskii
State University, Nizhnii Novgorod;
19772013: Professor of Mathematics, Head of department of mathematics of Nizhny
Novgorod State Agriculture Academy;
19691977: Researcher, Res.Inst. of Appl. Math.&Cybernetics, State University,
N.Novgorod
Scientific degrees:
1976: candidate of physical and mathematical sciences.
1998: doctor of physical and mathematical sciences.
Area of expertise:
Dynamical Systems and Foliations on Manifolds.
Publications:
Barinova M. K., Grines V. Z., Pochinka O. V., Zhuzhoma E. V.
Hyperbolic Attractors Which are Anosov Tori
2024, vol. 29, no. 2, pp. 369375
Abstract
We consider a topologically mixing hyperbolic attractor $\Lambda\subset M^n$ for a diffeomorphism $f:M^n\to M^n$ of a compact orientable $n$manifold $M^n$, $n>3$. Such an attractor $\Lambda$ is called an Anosov torus provided the restriction $f_{\Lambda}$ is conjugate to Anosov algebraic automorphism of $k$dimensional torus $\mathbb T^k$.
We prove that $\Lambda$ is an Anosov torus for two cases:
1) $\dim{\Lambda}=n1$, $\dim{W^u_x}=1$, $x\in\Lambda$;
2) $\dim\,\Lambda=k,\,\dim\, W^u_x=k1,\,x\in\Lambda$, and $\Lambda$ belongs to an $f$invariant closed $k$manifold, $2\leqslant k\leqslant n$, topologically embedded in $M^n$.

Grines V. Z., Pochinka O. V., Chilina E. E.
On Homeomorphisms of ThreeDimensional Manifolds with PseudoAnosov Attractors and Repellers
2024, vol. 29, no. 1, pp. 156173
Abstract
The present paper is devoted to a study of orientationpreserving homeomorphisms
on threedimensional manifolds with a nonwandering set consisting of a finite number of surface
attractors and repellers. The main results of the paper relate to a class of homeomorphisms
for which the restriction of the map to a connected component of the nonwandering set
is topologically conjugate to an orientationpreserving pseudoAnosov homeomorphism. The
ambient $\Omega$conjugacy of a homeomorphism from the class with a locally direct product of a
pseudoAnosov homeomorphism and a rough transformation of the circle is proved. In addition,
we prove that the centralizer of a pseudoAnosov homeomorphisms consists of only pseudo
Anosov and periodic maps.

Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
Classification of Axiom A Diffeomorphisms with Orientable Codimension One Expanding Attractors and Contracting Repellers
2024, vol. 29, no. 1, pp. 143155
Abstract
Let $\mathbb{G}_k^{cod 1}(M^n)$, $k\geqslant 1$, be the set of axiom A diffeomorphisms such that
the nonwandering set of any $f\in\mathbb{G}_k^{cod 1}(M^n)$ consists of $k$ orientable connected codimension one expanding attractors and contracting repellers where $M^n$ is a closed orientable $n$manifold, $n\geqslant 3$. We classify the diffeomorphisms from $\mathbb{G}_k^{cod 1}(M^n)$ up to the global conjugacy on nonwandering sets. In addition, we show that any $f\in\mathbb{G}_k^{cod 1}(M^n)$ is $\Omega$stable and is not structurally stable. One describes the topological structure of a supporting manifold $M^n$.

Grines V. Z., Mints D. I.
On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms
2023, vol. 28, no. 3, pp. 295308
Abstract
In P.D.McSwiggen’s article, it was proposed Derived from Anosov type construction
which leads to a partially hyperbolic diffeomorphism of the 3torus. The nonwandering set
of this diffeomorphism contains a twodimensional attractor which consists of onedimensional
unstable manifolds of its points. The constructed diffeomorphism admits an invariant onedimensional
orientable foliation such that it contains unstable manifolds of points of the
attractor as its leaves. Moreover, this foliation has a global cross section (2torus) and defines
on it a Poincar´e map which is a regular Denjoy type homeomorphism. Such homeomorphisms
are the most natural generalization of Denjoy homeomorphisms of the circle and play an
important role in the description of the dynamics of aforementioned partially hyperbolic
diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps
provides necessary conditions for the topological conjugacy of the restrictions of such partially
hyperbolic diffeomorphisms to their twodimensional attractors. The nonwandering set of
each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism
is, by definition, semiconjugate to the minimal translation of the 2torus. We introduce a
complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is
characterized by the minimal translation, which is semiconjugation of the given regular Denjoy
type homeomorphism, with a distinguished, no more than countable set of orbits.

Grines V. Z., Medvedev V. S., Zhuzhoma E. V.
On the Topological Structure of Manifolds Supporting Axiom A Systems
2022, vol. 27, no. 6, pp. 613628
Abstract
Let $M^n$, $n\geqslant 3$, be a closed orientable $n$manifold and $\mathbb{G}(M^n)$ the set of Adiffeomorp\hisms $f: M^n\to M^n$ whose nonwandering set satisfies the following conditions:
$(1)$ each nontrivial basic set of the nonwandering set is either an orientable codimension one expanding attractor or an orientable codimension one contracting repeller;
$(2)$ the invariant manifolds of isolated saddle periodic points intersect transversally and codimension one separatrices of such points can intersect only onedimensional separatrices of other isolated periodic orbits.
We prove that the ambient manifold $M^n$ is homeomorphic to either the sphere $\mathbb S^n$ or
the connected sum of $k_f \geqslant 0$ copies of the torus
$\mathbb T^n$, $\eta_f\geqslant 0$ copies of $\mathbb S^{n1}\times \mathbb S^1$ and $l_f\geqslant 0$
simply connected manifolds $N^n_1, \dots, N^n_{l_f}$ which are not homeomorphic to the sphere.
Here $k_f\geqslant 0$ is the number of connected components of all nontrivial basic sets, $\eta_{f}=\frac{\kappa_f}{2} k_f+\frac{\nu_f  \mu_f +2}{2},$
$ \kappa_f\geqslant 0$ is the number of bunches of all nontrivial basic sets, $\mu_f\geqslant 0$ is the number of sinks and sources, $\nu_f\geqslant 0$ is the number of isolated saddle periodic points with Morse index
$1$ or $n1$, $0\leqslant l_f\leqslant \lambda_f$, $\lambda_f\geqslant 0$ is the number of all periodic points whose Morse index does not belong to the set $\{0,1,n1,n\}$ of diffeomorphism $f$. Similar statements hold for gradientlike flows on $M^n$. In this case there are no
nontrivial basic sets in the nonwandering set of a flow. As an application, we get sufficient
conditions for the existence of heteroclinic intersections and periodic trajectories for Morse – Smale flows.

Grines V. Z., Gurevich E. Y., Pochinka O. V.
On the Number of Heteroclinic Curves of Diffeomorphisms with Surface Dynamics
2017, vol. 22, no. 2, pp. 122135
Abstract
Separators are fundamental plasma physics objects that play an important role in many astrophysical phenomena. Looking for separators and their number is one of the first steps in studying the topology of the magnetic field in the solar corona. In the language of dynamical
systems, separators are noncompact heteroclinic curves. In this paper we give an exact lower estimation of the number of noncompact heteroclinic curves for a 3diffeomorphism with the socalled “surface dynamics”. Also, we prove that ambient manifolds for such diffeomorphisms are mapping tori.

Grines V. Z., Malyshev D. S., Pochinka O. V., Zinina S. K.
Efficient Algorithms for the Recognition of Topologically Conjugate Gradientlike Diffeomorhisms
2016, vol. 21, no. 2, pp. 189203
Abstract
It is well known that the topological classification of structurally stable flows on surfaces as well as the topological classification of some multidimensional gradientlike systems can be reduced to a combinatorial problem of distinguishing graphs up to isomorphism. The isomorphism problem of general graphs obviously can be solved by a standard enumeration algorithm. However, an efficient algorithm (i. e., polynomial in the number of vertices) has not yet been developed for it, and the problem has not been proved to be intractable (i. e., NPcomplete). We give polynomialtime algorithms for recognition of the corresponding graphs for two gradientlike systems. Moreover, we present efficient algorithms for determining the orientability and the genus of the ambient surface. This result, in particular, sheds light on the classification of configurations that arise from simple, pointsource potentialfield models in efforts to determine the nature of the quietSun magnetic field.

Grines V. Z., Levchenko Y. A., Medvedev V. S., Pochinka O. V.
On the Dynamical Coherence of Structurally Stable 3diffeomorphisms
2014, vol. 19, no. 4, pp. 506512
Abstract
We prove that each structurally stable diffeomorphism $f$ on a closed 3manifold $M^3$ with a twodimensional surface nonwandering set is topologically conjugated to some model dynamically coherent diffeomorphism.

Grines V. Z., Pochinka O. V.
Energy functions for dynamical systems
2010, vol. 15, nos. 23, pp. 185193
Abstract
The paper contains exposition of results devoted to the existence of an energy functions for dynamical systems.

Grines V. Z., Zhuzhoma E. V.
Expanding attractors
2006, vol. 11, no. 2, pp. 225246
Abstract
The article is a survey on local and global structures (including classification results) of expanding attractors of diffeomorphisms $f : M \to M$ of a closed smooth manifold $M$. Beginning with the most familiar expanding attractors (Smale solenoid; DAattractor; Plykin attractor; Robinson–Williams attractors), one reviews the Williams theory, Bothe's classification of onedimensional solenoids in 3manifolds, Grines–Plykin–Zhirov's classification of onedimensional expanding attractors on surfaces, and Grines–Zhuzhoma's classification of codimension one expanding attractors of structurally stable diffeomorphisms. The main theorems are endowed with ideas of proof
