arXiv subjects

O. Pochinka

Publications and source records attributed to O. Pochinka.

9 recordsLinked to original sources

arXiv2026

Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows

Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold $M$ admitting an Anosov flow must have exponential growth, and $M$ must be universally covered by $\mathbb{R}^{3}$. Nevertheless, there are sufficient mechanisms for constructing distinct Anosov flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or playing with hyperbolic building blocks. A central problem in the field has been to determine the number of Anosov flows that can be supported by a single manifold. The question of whether there exists an infinite set of pairwise non-equivalent Anosov flows on a 3-manifold remains open to this day. However, there are several papers proving the existence of a manifold $M_n$ that admits $n$ pairwise inequivalent Anosov flows for any natural number $n$. In all known examples, the manifolds $M_n$ are composed of several geometric pieces. In the present paper, we prove the existence of a countable number of graph manifolds $M_{k,n}$, $k \in \mathbb{N}$ with a single geometric piece, each of which admits $n$ pairwise non-equivalent transitive Anosov flows. All previously known constructions of different flows on the same graph manifold were based on gluing geodesic flows. The nature of the flows constructed in this paper is completely different; they are constructed from a single hyperbolic plug, which is a suspension over a Morse-Smale diffeomorphism on a surface.

math.DS↗

arXiv2025

Topology of 4-manifolds that admit non-singular flows with saddle orbits of the same index

This paper studies regular topological flows $f^t$ defined on closed {topological} manifolds $M^n$. The chain recurrent set of such a flow consists of a finite number of topologically hyperbolic fixed points and periodic orbits. Like their smooth analogs -- Morse-Smale flows -- regular flows possess a continuous Morse-Bott function that decreases outside the chain recurrent set and is constant on the chain components of the flow. This circumstance leads to a close connection between such flows and the topology of the carrying manifold. In particular, the ambient manifold for non-singular flows (regular flows without fixed points), by the Poincare-Hopf formula, has a zero Euler characteristic. The latter property is a criterion for a manifold $M^n$ to admit a non-singular flow in all dimensions except dimension $n=3$. Thus, in higher dimensions, any odd-dimensional manifold admits a non-singular flow, and the list of even-dimensional manifolds is quite broad; at the very least, it includes all manifolds of the form $M^{n-1}\times\mathbb S^1$, where $M^{n-1}$ is any closed $(n-1)$-manifold. A surprising result of the present paper is the proof of the fact that in dimension 4, all this variety of carrying spaces can only be achieved if the flow has saddle orbits of different Morse indices. Specifically, for dimensional reasons, the Morse index of a saddle orbit of a non-singular flow $f^t: M^4 \to M^4$ can only take two values, 1 or 2. We prove that non-singular 4-flows with saddle orbits of the same Morse index exist only on skew or direct products of the 3-sphere and the circle, i.e., $M^4\cong\mathbb S^3\tilde\times\,\mathbb S^1$ or $M^4\cong\mathbb S^3\times\mathbb S^1$.

math.DS↗

arXiv2025

Stable components for gradient-like diffeomorphisms of torus inducing matrix $\begin{pmatrix} -1 & -1\cr 1& 0\end{pmatrix}$

An isotopy between two diffeomorphisms means the existence of an arc connecting them in the space of diffeomorphisms. Among such arcs there are so-called stable arcs, which do not qualitatively change under small perturbations. In the present paper we consider a set of gradient-like diffeomorphisms f of 2-torus whose induced isomorphism given by a matrix $\begin{pmatrix} -1 & -1\cr 1& 0\end{pmatrix}$. We prove that the set of such diffeomorphisms is decomposed into four stable components. Moreover, we establish that two diffeomorphisms under consideration are stably connected if and only if they have the same number of fixed sinks.

math.DS↗

arXiv2023

On the topology of 3-manifolds admitting Morse-Smale diffeomorphisms with four fixed points of pairwise different Morse indices

In the present paper we consider class $G$ of orientation preserving Morse-Smale diffeomorphisms $f$, which defined on closed 3-manifold $M^3$, and whose non-wandering set consist of four fixed points with pairwise different Morse indices. It follows from S. Smale and K. Meyer results that all gradient-like flows with similar properties has Morse energy function with four critical points of pairwise different Morse indices. This implies, that supporting manifold $M^3$ for these flows admits a Heegaard decomposition of genus 1 and hence it is homeomorphic to a lens space $L_{p,q}$. Despite the simple structure of the non-wandering set in class $G$ there exist diffeomorphisms with wild embedded separatrices. According to V. Grines, F. Laudenbach, O. Pochinka results such diffeomorphisms do not possesses an energy function, and question about topology their supporting manifold is open. According to V. Grines, E. Zhuzhoma and V. Medvedev results $M^3$ is homeomorphic to a lens space $L_{p,q}$ in case of tame embedding of closures of one-dimensional separatrices of diffeomorphism $f\in G$. Moreover, the wandering set of $f$ contains at least $p$ non-compact heteroclinic curves. In the present paper similar result was received for arbitrary diffeomorphisms of class $G$. Also we construct diffeomorphisms from $G$ with wild embedding one-dimensional separatrices on every lens space $L_{p,q}$. Such examples were known previously only on the 3-sphere.

math.GT↗

arXiv2021

On classification of periodic maps on the 2-torus

In this paper, following J.Nielsen, we introduce a complete characteristic of orientation preserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of classes of non-homotopic to the identity orientation preserving periodic homeomorphisms on the 2-torus is realized by an algebraic automorphism. Moreover, it is shown that number of such classes is finite. Due to V.Z. Grines and A. Bezdenezhnykh, any gradient like orientation preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. So, results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

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arXiv2020

Existence of an energy function for 3-dimensional chaotic "sink-source" cascades

The paper is a continuation of research in the direction of energy function (a smooth Lyapunov function whose set of critical points coincides with the chain recurrent set of a system) construction for discrete dynamical systems. The authors established the existence of an energy function for any $ A $-diffeomorphism of a three-dimensional closed orientable manifold whose non-wandering set consists of chaotic one-dimensional attractor and repeller.

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arXiv2017

Topological classification of Morse-Smale diffeomorphisms on 3-manifolds

Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a topological classification of Morse-Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.

math.DS↗

arXiv2017

The wild Fox-Artin arc in invariant sets of dynamical systems

The modern qualitative theory of dynamical systems is thoroughly intertwined with the fairly young science of topology. Strange and even bizarre constructions of topology are found sooner or later in dynamics of discrete or continuous dynamical systems. In the present paper we show that the wild Fox-Artin arc naturally emerges in dynamics as an invariant manifold of a fixed point and as a heteroclinic intersection.

math.DS↗

arXiv2015

On heteroclinic separators of magnetic fields in electrically conducting fluids

In this paper we partly solve the problem of existence of separators of a magnetic field in plasma. We single out in plasma a 3-body with a boundary in which the movement of plasma is of special kind which we call an (a-d)-motion. We prove that if the body is the 3-annulus or the "fat" orientable surface with two holes the magnetic field necessarily have a heteroclinic separator. The statement of the problem and the suggested method for its solution lead to some theoretical problems from Dynamical Systems Theory which are of interest of their own.

physics.plasm-ph↗