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Sergiy Maksymenko

Publications and source records attributed to Sergiy Maksymenko.

54 records · Page 3Linked to original sources
arXiv2015

Reparametrizations of vector fields and their shift maps

Let $M$ be a smooth manifold, $F$ be a smooth vector field on $M$, and $F_t$ be the local flow of $F$. Denote by $Sh(F)$ the space of smooth maps $h:M\to M$ of the following form: $h(x) = F_{f(x)}(x)$, where $f:M\to\mathbb{R}$ runs over all smooth functions on $M$ which can be substituted into the flow $F_t$ instead of time. This space often coincides with the identity component of the group of diffeomorphisms preserving orbits of $F$. In this note it is shown that $Sh(F)$ is not changed under reparametrizations and pushforwards of $F$. As an application it is proved that a vector field $F$ without non-closed orbits can be reparametrized to induce a circle action on $M$ if and only if there exists a smooth function $f:M\to (0,+\infty)$ such that for each non-singular point $x$ of $M$, the value $f(x)$ is an integer multiple of the period of $x$ with respect to $F$.

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arXiv2015

Functions on surfaces and incompressible subsurfaces

Let $M$ be a smooth connected compact surface and $P$ be either a real line or a circle. This paper proceeds the study of the stabilizers and orbits of smooth functions on $M$ with respect to the right action of the group of diffeomorphisms of $M$. A large class of smooth maps $f:M\to P$ with isolated singularities is considered and it is shown that the general problem of calculation of the fundamental group of the orbit of $f$ reduces to the case when the Euler characteristic of $M$ is non-negative. For the proof of main result incompressible subsurfaces and cellular automorphisms of surfaces are investigated.

math.GT↗
arXiv2015

Foliations with non-compact leaves on surfaces

We study non-compact surfaces obtained by gluing strips $\mathbb{R}\times(-1,1)$ with at most countably many boundary intervals along some these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.

math.GT↗
arXiv2015

Symmetries of degenerate center singularities of plane vector fields

Let $D$ be a closed unit $2$-disk on the plane centered at the origin $O$, and $F$ be a smooth vector field such that $O$ is a unique singular point of $F$ and all other orbits of $F$ are simple closed curves wrapping once around $O$. Thus topologically $O$ is a "center" singularity. Let also $\mathrm{Diff}(F)$ be the group of all diffeomorphisms of $D$ which preserve orientation and orbits of $F$. In arXiv:0907.0359 the author described the homotopy type of $\mathrm{Diff}(F)$ under assumption that the $1$-jet of $F$ at $O$ is non-degenerate. In this paper degenerate case is considered. Under additional "non-degeneracy assumptions" on $F$ the path components of $\mathrm{Diff}(F)$ with respect to distinct weak topologies are described.

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arXiv2015

Deformations of circle-valued Morse functions on surfaces

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.

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arXiv2015

Image of a shift map along the orbits of a flow

Let $(F_t)$ be a smooth flow on a smooth manifold $M$ and $h:M\to M$ be a smooth orbit preserving map. The following problem is studied: suppose that for every point $z$ of $M$ there exists a germ of a smooth function $f_z$ at $z$ such that near $z$ we have that $h(x)=F_{f_z(x)}(x)$. Can the functions $(f_z)$ be glued together to give a smooth function on all of $M$? This question is closely related to reparametrizations of flows. We describe a large class of flows for which the above problem can be resolved, and show that they have the following property: any smooth flow $(G_t)$ whose orbits coincides with the ones of $(F_t)$ is obtained from $(F_t)$ by smooth reparametrization of time. The proof of our principal statement uses results of D. Hoffman and L. N. Mann about diameters of effective actions of Lie grous of Riemannian manifolds.

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arXiv2015

$\infty$-jets of difeomorphisms preserving orbits of vector fields

Let $F$ be a smooth vector field defined in a neighborhood of the origin in $\mathbb{R}^n$, $F(O)=0$, and let $F_t$ be its local flow. Denote by $E$ the set of germs of diffeomorphisms $h:\mathbb{R}^n \to \mathbb{R}^n$ preserving orbits of $F$ and let $E_{\mathrm{id}}^r$ be the identity component of $E$ with respect to $C^r$-topology. Then every $E_{\mathrm{id}}^{r}$ contains a subset $Sh$ consisting of mappings of the form $F_{f(x)}(x)$, where $f: \mathbb{R}^n \to \mathbb{R}$ is a smooth function. It was proved earlier by the author that if $F$ is a linear vector field, then $Sh=E_{\mathrm{id}}^0$. In this paper we present a class of vector fields for which $Sh$ and $E_{\mathrm{id}}^1$ coincide on the level of $\infty$-jets. We also establish a parameter rigidity of linear vector fields and "reduced" Hamiltonian vector fields of real homogeneous polynomials in two variables.

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arXiv2015

Entropy determination based on the ordinal structure of a dynamical system

The ordinal approach to evaluate time series due to innovative works of Bandt and Pompe has increasingly established itself among other techniques of nonlinear time series analysis. In this paper, we summarize and generalize the theory of determining the Kolmogorov-Sinai entropy of a measure-preserving dynamical system via increasing sequences of order generated partitions of the state space. Our main focus are measuring processes without information loss. Particularly, we consider the question of the minimal necessary number of measurements related to the properties of a given dynamical system.

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arXiv2014

Smooth functions on 2-torus whose Kronrod-Reeb graph contains a cycle

Let $f:M\to \mathbb{R}$ be a Morse function on a connected compact surface $M$, and $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be respectively the stabilizer and the orbit of $f$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$. In a series of papers the first author described the homotopy types of connected components of $\mathcal{S}(f)$ and $\mathcal{O}(f)$ for the cases when $M$ is either a $2$-disk or a cylinder or $χ(M)<0$. Moreover, in two recent papers the authors considered special classes of smooth functions on $2$-torus $T^2$ and shown that the computations of $π_1\mathcal{O}(f)$ for those functions reduces to the cases of $2$-disk and cylinder. In the present paper we consider another class of Morse functions $f:T^2\to\mathbb{R}$ whose KR-graphs have exactly one cycle and prove that for every such function there exists a subsurface $Q\subset T^2$, diffeomorphic with a cylinder, such that $π_1\mathcal{O}(f)$ is expressed via the fundamental group $π_1\mathcal{O}(f|_{Q})$ of the restriction of $f$ to $Q$. This result holds for a larger class of smooth functions $f:T^2\to \mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple factors.

math.GT↗
arXiv2014

Orbits of smooth functions on 2-torus and their homotopy types

Let $f:T^2\to\mathbb{R}$ be a Morse function on $2$-torus $T^2$ such that its Kronrod-Reeb graph $Γ(f)$ has exactly one cycle, i.e. it is homotopy equivalent to $S^1$. Under some additional conditions we describe a homotopy type of the orbit of $f$ with respect to the action of the group of diffeomorphism of $T^2$. This result holds for a larger class of smooth functions $f:T^2\to\mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.

math.GT↗
arXiv2014

Structure of the fundamental groups of orbits of smooth functions on surfaces

Let $M$ be a smooth compact connected surface, $P$ be either the real line $\mathbb{R}$ or the circle $S^1$ and $f:M\to P$ be a Morse map. Denote by $\mathcal{S}(f)$ and $\mathcal{O}(f)$ the corresponding stabilizer and orbit of $f$ with respect to the right action of the group $\mathcal{D}(M)$ of diffeomorphisms of $M$. In a series of papers the author described homotopy types of $\mathcal{S}(f)$ and computed higher homotopy groups of $\mathcal{O}(f)$. The present paper describes the structure of the remained fundamental group $π_1 \mathcal{O}(f)$ for the case when $M$ is orientable and differs from $2$-sphere and $2$-torus. The result holds as well for a larger class of smooth maps $f:M\to P$ having the following property: the germ of $f$ at each of its critical points is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.

math.GT↗
arXiv2014

Homotopy properties of spaces of smooth functions on 2-torus

Let $f:T^2\to\mathbb{R}$ be a Morse function on a 2-torus, $\mathcal{S}(f)$ and $\mathcal{O}(f)$ be its stabilizer and orbit with respect to the right action of the group $\mathcal{D}(T^2)$ of diffeomorphisms of $T^2$, $\mathcal{D}_{\mathrm{id}}(T^2)$ be the identity path component of $\mathcal{D}(T^2)$, and $\mathcal{S}'(f) = \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(T^2)$. We give sufficient conditions under which $$ π_1\mathcal{O}_f(f) \ \cong \ π_1\mathcal{D}(T^2) \times π_0 \mathcal{S}'(f) \ \equiv \ \mathbb{Z}^2 \times π_0 \mathcal{S}'(f).$$ In fact this result holds for a larger class of smooth functions $f:T^2\to\mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to \mathbb{R}$ without multiple factors.

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arXiv2013

Kolmogorov-Sinai entropy via separation properties of order-generated sigma-algebras

In a recent paper, K.Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed from order relations obtained via a given real-valued random vector, which can be interpreted as a collection of observables on the system and is assumed to separate points of it. In the present paper we relax the separation condition in order to generalize the given characterization of Kolmogorov-Sinai entropy, providing a statement on equivalence of sigma-algebras. On its base we show that in the case that a dynamical system is living on an m-dimensional smooth manifold and the underlying measure is Lebesgue absolute continuous, the set of smooth random vectors of dimension n>m with given characterization of Kolmogorov-Sinai entropy is large in a certain sense.

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arXiv2012

Functions with isolated singularities on surfaces, II

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1. For a subset X of M denote by D(M,X) the group of diffeomorphisms of M fixed on X. In this note we consider a special class F of smooth maps f:M\to P with isolated singularities which includes all Morse maps. For each such map f from F we consider certain submanifolds X of M that are "adopted" with f in a natural sense, and study the right action of the group D(M,X) on C^{\infty}(M,P). The main result describes the homotopy types of the connected components of the stabilizers S(f) and orbits O(f) for all maps f from F. It extends previous author results on this topic.

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arXiv2009

Functions with isolated singularities on surfaces

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of stabilizers and orbits of f with respect to the right action of the diffeomorphisms group of M. The present paper extends those calculations to a large class of maps M-->P with degenerate singularities satisfying certain set of axioms.

math.GT↗
arXiv2009

Period functions for $C^0$ and $C^1$ flows

Let $(F_t)$ be a continuous flow on a topological manifold M. For every open $V \subset M$ denote by P(V) the set of all continuous functions $α:V \to R$ such that $F_{α(z)}(z)=z$ for all $z\in V$. Such functions vanish at non-periodic points and their values at periodic points are equal to the corresponding periods (in general not minimal). They can be used for reparametrizations of flows to circle actions. In this paper P(V) is described for connected open subsets V, which extends previous results of the author. It is also shown that there is a difference between the sets P(V) for $C^{0}$ and $C^1$ flows.

math.DS↗
arXiv2009

Symmetries of center singularities of plane vector fields

Let D be a closed unit 2-disk on the plane centered at the origin 0, and F be a smooth vector field on D such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a ``center'' singularity. Let p:D\0-->(0,+\infty) be the function associating to each $z\not=O$ its period with respect to F. This function can be discontinuous at O. Let Diff(F) be the group of all diffeomorphisms of D which preserve orientation and orbits of F. We prove that p smoothly extends to all of D if and only if the 1-jet of F at the origin is a non-degenerate linear map, and that in this case Diff(F) is homotopy equivalent to the circle.

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