Olga Pochinka
B. Pecherskaya 25, Nizhny Novgorod, 603105 Russia
Higher School of Economics
Publications:
Osenkov E. M., Pochinka O. V.
Morse – Smale 3Diffeomorphisms with Saddles of the Same Unstable Manifold Dimension
2024, Vol. 20, no. 1, pp. 167178
Abstract
In this paper, we consider a class of Morse – Smale diffeomorphisms defined on a closed
3manifold (not necessarily orientable) under the assumption that all their saddle points have
the same dimension of the unstable manifolds. The simplest example of such diffeomorphisms is
the wellknown “sourcesink” or “north pole – south pole” diffeomorphism, whose nonwandering
set consists of exactly one source and one sink. As Reeb showed back in 1946, such systems can
only be realized on the sphere. We generalize his result, namely, we show that diffeomorphisms
from the considered class also can be defined only on the 3sphere.

Pochinka O. V., Shubin D. D.
Topology of Ambient 3Manifolds of NonSingular Flows with Twisted Saddle Orbit
2023, Vol. 19, no. 3, pp. 371381
Abstract
In the present paper, nonsingular Morse – Smale flows on closed orientable 3manifolds are
considered under the assumption that among the periodic orbits of the flow there is only one
saddle and that it is twisted. An exhaustive description of the topology of such manifolds is
obtained. Namely, it is established that any manifold admitting such flows is either a lens space
or a connected sum of a lens space with a projective space, or Seifert manifolds with a base
homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the
result obtained refutes the claim that, among prime manifolds, the flows considered admit only
lens spaces.

Baranov D. A., Grines V. Z., Pochinka O. V., Chilina E. E.
On a Classification of Periodic Maps on the 2Torus
2023, Vol. 19, no. 1, pp. 91110
Abstract
In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving
periodic maps on the twodimensional torus. All admissible complete characteristics
were found and realized. In particular, each of the classes of orientationpreserving periodic
homeomorphisms on the 2torus that are nonhomotopic to the identity is realized by an algebraic
automorphism. Moreover, it is shown that the number of such classes is finite. According to
V. Z. Grines and A.Bezdenezhnykh, any gradientlike orientationpreserving diffeomorphism of
an orientable surface is represented as a superposition of the time1 map of a gradientlike flow
and some periodic homeomorphism. Thus, the results of this work are directly related to the
complete topological classification of gradientlike diffeomorphisms on surfaces.

Barinova M. K., Gogulina E. Y., Pochinka O. V.
Omegaclassification of Surface Diffeomorphisms Realizing Smale Diagrams
2021, Vol. 17, no. 3, pp. 321334
Abstract
The present paper gives a partial answer to Smale's question
which diagrams can correspond to $(A,B)$diffeomorphisms.
Model diffeomorphisms of the twodimensional torus derived
by ``Smale surgery'' are considered, and necessary and
sufficient conditions for their topological conjugacy are
found. Also, a class $G$ of $(A,B)$diffeomorphisms on surfaces which are the connected
sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse
diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$conjugated are constructed. Moreover, a subset $G_{*}^{} \subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$conjugacy is singled out.

Pochinka O. V., Nozdrinova E. V.
Stable Arcs Connecting Polar Cascades on a Torus
2021, Vol. 17, no. 1, pp. 2337
Abstract
The problem of the existence of an arc with at most countable (finite) number of bifurcations
connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the
list of fifty Palis – Pugh problems at number 33. In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection). In the article, the components of the stable isotopic connection of polar gradientlike diffeomorphisms on a twodimensional torus are found under the assumption that all nonwandering points are fixed and have a positive orientation type. 
Grines V. Z., Kruglov E. V., Pochinka O. V.
The Topological Classification of Diffeomorphisms of the TwoDimensional Torus with an Orientable Attractor
2020, Vol. 16, no. 4, pp. 595606
Abstract
This paper is devoted to the topological classification of structurally stable diffeomorphisms
of the twodimensional torus whose nonwandering set consists of an orientable onedimensional
attractor and finitely many isolated source and saddle periodic points, under the assumption
that the closure of the union of the stable manifolds of isolated periodic points consists of simple
pairwise nonintersecting arcs. The classification of onedimensional basis sets on surfaces has
been exhaustively obtained in papers by V. Grines. He also obtained a classification of some
classes of structurally stable diffeomorphisms of surfaces using combined algebrageometric invariants.
In this paper, we distinguish a class of diffeomorphisms that admit purely algebraic
differentiating invariants.

Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.
On a Class of Isotopic Connectivity of Gradientlike Maps of the 2sphere with Saddles of Negative Orientation Type
2019, Vol. 15, no. 2, pp. 199211
Abstract
We consider the class $G$ of gradientlike orientationpreserving diffeomorphisms of the 2sphere with saddles of negative orientation type. We show that the for every diffeomorphism $f\in G$ every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) $G=G_1\cup G_2\cup G_3$ where a diffeomorphism $f_1\in G_1$ has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism $f_2\in G_2$ has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism $f_3\in G_3$ is topologically conjugate to a diffeomorphism $f_1^{1}$. The main result is the proof that every diffeomorphism $f\in G$ can be connected to the ``sourcesink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of perioddoubling bifurcations.

Pochinka O. V., Loginova A. S., Nozdrinova E. V.
OneDimensional ReactionDiffusion Equations and Simple SourceSink Arcs on a Circle
2018, Vol. 14, no. 3, pp. 325330
Abstract
This article presents a number of models that arise in physics, biology, chemistry, etc.,
described by a onedimensional reactiondiffusion equation. The local dynamics of such models
for various values of the parameters is described by a rough transformation of the circle. Accordingly,
the control of such dynamics reduces to the consideration of a continuous family of
maps of the circle. In this connection, the question of the possibility of joining two maps of the
circle by an arc without bifurcation points naturally arises. In this paper it is shown that any
orientationpreserving sourcesink diffeomorphism on a circle is joined by such an arc. Note that
such a result is not true for multidimensional spheres.

Pochinka O. V., Kruglov E. V., Dolgonosova A.
Scenario of reconnection in the solar corona with a simple discretization
2017, Vol. 13, No. 4, pp. 573–578
Abstract
In this paper, one of the possible scenarios for the creation of heteroclinic separators in the solar corona is described and realized. This reconnection scenario connects the magnetic field with two zero points of different signs, the fan surfaces of which do not intersect, with a magnetic field with two zero points which are connected by two heteroclinic separators. The method of proof is to create a model of the magnetic field produced by the plasma in the solar corona and to study it using the methods of dynamical systems theory. Namely, in the space of vector fields on the sphere $S^3$ with two sources, two sinks and two saddles, we construct a simple arc with two saddlenode bifurcation points that connects the system without heteroclinic curves to a system with two heteroclinic curves. The discretization of this arc is also a simple arc in the space of diffeomorphisms. The results are new.

Grines V. Z., Levchenko Y. A., Pochinka O. V.
On topological classification of diffeomorphisms on 3manifolds with twodimensional surface attractors and repellers
2014, Vol. 10, No. 1, pp. 1733
Abstract
We consider a class of diffeomorphisms on 3manifolds which satisfy S. Smale’s axiom A such that their nonwandering set consists of twodimensional 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 Ωconjugated with a model diffeomorphism of mapping torus. Under certain assumptions on asymptotic properties of twodimensional 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.

Pochinka O. V.
Necessary and sufficient conditions for topological classification of Morse–Smale cascades on 3manifolds
2011, Vol. 7, No. 2, pp. 227238
Abstract
In this paper class $MS(M^3)$ of Morse–Smale diffeomorphisms (cascades) given on connected closed orientable 3manifolds are considered. For a diffeomorphism $f \in MS(M^3)$ it is introduced a notion scheme $S_f$, which contains an information on the periodic data of the cascade and a topology of embedding of the sepsrstrices of the saddle points. It is established that necessary and sufficient condition for topological conjugacy of diffeomorphisms $f$, $f’ \in MS(M^3)$ is the equivalence of the schemes $S_f$, $S_f’$.

Mitryakova T. M., Pochinka O. V.
To a question on classification of diffeomorphisms of surfaces with a finite number of moduli of topological conjugacy
2010, Vol. 6, No. 1, pp. 91105
Abstract
In this paper diffeomorphisms on orientable surfaces are considered, whose nonwandering set consists of a finite number of hyperbolic fixed points and the wandering set contains a finite number of heteroclinic orbits of transversal and nontransversal intersections. We investigate substantial class of diffeomorphisms for which it is found complete topological invariant — a scheme consisting of a set of geometrical objects equipped by numerical parametres (moduli of topological conjugacy).
