Olga Pochinka
B. Pecherskaya 25, Nizhny Novgorod, 603105 Russia
Higher School of Economics
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.

Galkin V. D., Nozdrinova E. V., Pochinka O. V.
Circular Fleitas Scheme for GradientLike Flows on the Surface
2023, vol. 28, no. 6, pp. 865877
Abstract
In this paper, we obtain a classification of gradientlike flows on arbitrary surfaces
by generalizing the circular Fleitas scheme. In 1975 he proved that such a scheme is a complete
invariant of topological equivalence for polar flows on 2 and 3manifolds. In this paper, we
generalize the concept of a circular scheme to arbitrary gradientlike flows on surfaces.We prove
that the isomorphism class of such schemes is a complete invariant of topological equivalence.
We also solve exhaustively the realization problem by describing an abstract circular scheme
and the process of realizing a gradientlike flow on the surface. In addition, we construct an
efficient algorithm for distinguishing the isomorphism of circular schemes.

Medvedev T. V., Nozdrinova E. V., Pochinka O. V.
Components of Stable Isotopy Connectedness of Morse – Smale Diffeomorphisms
2022, vol. 27, no. 1, pp. 7797
Abstract
In 1976 S.Newhouse, J.Palis and F.Takens introduced a stable arc joining two
structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular
stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation
diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique
nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddlenode or a flip
which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms
on manifolds of any dimension which cannot be joined by a stable arc. There naturally
arises the problem of finding an invariant defining the equivalence classes of Morse – Smale
diffeomorphisms with respect to connectedness by a stable arc. In the present review we present
the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic
connectedness and obstructions to existence of stable arcs including the authors’ recent results.

Pochinka O. V., Zinina S. K.
Construction of the Morse – Bott Energy Function for Regular Topological Flows
2021, vol. 26, no. 4, pp. 350369
Abstract
In this paper, we consider regular topological flows on closed $n$manifolds. Such
flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number
of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale
flows, which are closely related to the topology of the supporting manifold. This connection is
provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It
is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds,
on which dynamical systems can be considered only in a continuous category. The existence of
continuous analogs of regular flows on any topological manifolds is an open question, as is the
existence of energy functions for such flows. In this paper, we study the dynamics of regular
topological flows, investigate the topology of the embedding and the asymptotic behavior of
invariant manifolds of fixed points and periodic orbits. The main result is the construction of
the Morse – Bott energy function for such flows, which ensures their close connection with the
topology of the ambient manifold.

Kruglov V., Malyshev D. S., Pochinka O. V., Shubin D. D.
On Topological Classification of Gradientlike Flows on an $n$sphere in the Sense of Topological Conjugacy
2020, vol. 25, no. 6, pp. 716728
Abstract
In this paper, we study gradientlike flows without heteroclinic intersections on an
$n$sphere up to topological conjugacy. We prove that such a flow is completely defined by a
bicolor tree corresponding to a skeleton formed by codimension one separatrices. Moreover, we
show that such a tree is a complete invariant for these flows with respect to the topological
equivalence also. This result implies that for these flows with the same (up to a change
of coordinates) partitions into trajectories, the partitions for elements, composing isotopies
connecting timeone shifts of these flows with the identity map, also coincide. This phenomenon
strongly contrasts with the situation for flows with periodic orbits and connections, where
one class of equivalence contains continuum classes of conjugacy. In addition, we realize every
connected bicolor tree by a gradientlike flow without heteroclinic intersections on the $n$sphere.
In addition, we present a lineartime algorithm on the number of vertices for distinguishing these
trees.

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.
