# Sergiy MaksymenkoInstitute of Mathematics of National AcademIy of Sciences of Ukraine

Sergiy Maksymenko

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96

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Introduction

## Publications

Publications (96)

Let M be a smooth manifold and F a Morse-Bott foliation with a compact critical manifold B.
Denote by D(F) the group of diffeomorphisms of M leaving invariant each leaf of F.
Under certain assumptions on F it is shown that the computation of the homotopy type of D(F) reduces to three rather independent groups: the group of diffeomorphisms of B, the...

We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in\mathbb{N}\cup\{\infty\}$ let $H$ be the group of homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of $h$ to $\mathbb{R}\setminus0$ is a $\mathcal{C}^{...

Let \({\mathcal {F}}\) be a Morse–Bott foliation on the solid torus \(T=S^1\times D^2\) into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space \(L_{p,q}\) with a Morse–Bott foliation \({\mathcal {F}}_{p,q}\) obtained from \({\mathcal {F}}\)...

Let G and H be two groups acting on path connected topological spaces X and Y respectively. Assume that H is finite of order m and the quotient maps p:X→X/G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\odds...

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. The...

This note devoted to Volodymyr Vasylyovych Sharko (25.09.1949-07.10.2014)

Let \(T= S^1\times D^2\) be the solid torus, \(\mathcal {F}\) the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle \(S^1\times 0\), which is the central circle of the torus T, and \(\mathcal {D}(\mathcal {F},\partial T)\) the group of diffeomorphisms of T fixed on \(\partial T\) and leaving each leaf of the fol...

Given a compact surface $M$, consider the natural right action of the group of diffeomorphisms $\mathcal{D}(M)$ of $M$ on $\mathcal{C}^{\infty}(M,\mathbb{R})$ given by $(f,h)\mapsto f\circ h$ for $f\in \mathcal{C}^{\infty}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$. Denote by $\mathcal{F}(M)$ the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisti...

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. The...

Programme of the 7th international online conference "Algebraic and Geometric Methods of Analysis" AGMA 2023 (May 29 - June 1, 2023, Odesa, Ukraine) held under the aegis of the Ministry of Education and Science of Ukraine, the Odesa National University of Technology, and the International Geometry Center. Source: https://imath.kiev.ua/~topology/con...

Let $\mathcal{F}$ be a Morse-Bott foliation on the solid torus $T = S^1\times D^2$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$. A diffeomorphism $h:T \to T$ is called foliated (resp. leaf preserving) if for each leaf $\omega\in\mathcal{F}$ its image $h(\omega)$ is also leaf of $\mathcal{F}$ (resp. $h(\omega)=\omega$...

The paper contains an application of van Kampen theorem for groupoids to computation of homotopy types of certain class of non-compact foliated surfaces obtained by at most countably many strips \(\mathbb {R}\times (0,1)\) with boundary intervals in \(\mathbb {R}\times \{\pm 1\}\) along some of those intervals.

Let $T= S^1\times D^2$ be the solid torus, $\mathcal{F}$ the \textit{singluar} foliation on $T$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$, which is the central circle of the torus $T$, and $\mathcal{D}(\mathcal{F},\partial T)$ the group of diffeomorphisms of $T$ fixed on $\partial T$ and leaving each leaf of the f...

Let F be a foliation with a "singular" submanifold B on a smooth manifold M and p : E → B be a regular neighborhood of B in M. Under certain "homogeneity" assumptions on F near B we prove that every leaf preserving diffeomorphism h of M is isotopic via a leaf preserving isotopy to a diffeomorphism which coincides with some vector bundle morphism of...

Programme of the international online conference "Algebraic and Geometric Methods of Analysis" AGMA 2022 (May 24-27, 2022, Odesa, Ukraine) held under the aegis of the Ministry of Education and Science of Ukraine, the Odesa National Academy of Food Technologies and the International Geometry Center.
Source: https://www.imath.kiev.ua/~topology/conf/a...

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals which gives a foliation $\...

On May 25-28, 2021 held an International online conference "Algebraic and geometric methods of analysis" dedicated to the memory of an outstanding mathematician, the Corresponding member of National Academy of Sciences of Ukraine Yuriy Yuriyovych Trokhymchuk.

Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R×(0,1) with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines R×t, t∊(0,1), and boundary intervals which gives a foliation Δ on all of Z. Denote by H(Z,Δ) the group...

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant m...

Programme of the international online conference "Algebraic and Geometric Methods of Analysis" AGMA 2021 (May 25-28, 2021, Odesa, Ukraine) held under the aegis of the Ministry of Education and Science of Ukraine, the Odesa National Academy of Food Technologies and the International Geometry Center.
Source: https://www.imath.kiev.ua/~topology/conf/a...

The paper contains a review on recent progress in the deformational properties of smooth maps from compact surfaces $M$ to a one-dimensional manifold $P$. It covers description of homotopy types of stabilizers and orbits of a large class of smooth functions on surfaces obtained by the author, E. Kudryavtseva, B. Feshchenko, I. Kuznietsova, Yu. Soro...

The paper contains an application of van Kampen theorem for groupoids to computation of homotopy types of certain class of non-compact foliated surfaces obtained by at most countably many strips $\mathbb{R}\times(0,1)$ with boundary intervals in $\mathbb{R}\times\{\pm1\}$ along some of those intervals.

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$...

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular component of each level set of $f$ and reverses its orientation, then $h^2$ is isotopic to the identity map of $M$...

Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It...

Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset. It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence....

Let $\Delta$ be a foliation on a topological manifold $X$, $Y$ be the space of leaves, and $p: X \to Y$ be the natural projection. Endow $Y$ with the factor topology with respect to $p$. Then the group $\mathcal{H}(X, \Delta)$ of foliated (i.e. mapping leaves onto leaves) homeomorphisms of $X$ naturally acts on the space of leaves $Y$, which gives...

Let M be a connected orientable compact surface, \(f:M\rightarrow {\mathbb {R}}\) be a Morse function, and Open image in new window be the group of diffeomorphisms of M isotopic to the identity. Denote by Open image in new window the subgroup of Open image in new window consisting of diffeomorphisms “preserving” f, i.e., the stabilizer of f with re...

Let f:M→R be a Morse–Bott function on a closed manifold M, so the set Σf of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f)={h∈D(M)∣f∘h=h} the group of diffeomorphisms of M preserving f and let D(Σf) be the group of diffeomorphisms of Σf. We prove that the “restriction to Σf” map ρ...

Let $f:S^2\to \mathbb{R}$ be a Morse function on the $2$-sphere and $K$ be a connected component of some level set of $f$ containing at least one saddle critical point. Then $K$ is a $1$-dimensional CW-complex cellularly embedded into $S^2$, so the complement $S^2\setminus K$ is a union of open $2$-disks $D_1,\ldots, D_k$. Let $\mathcal{S}_{K}(f)$...

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$.
Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of i...

We introduce the first homotopic Baire class of maps as a homotopical counterpart of a usual first Baire class of maps between topological spaces and show that those classes with values in ANR spaces coincide.

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.
Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabili...

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and...

Let $B$ be a M\"obius band and $f:B \to \mathbb{R}$ be a Morse map taking a constant value on $\partial B$, and $\mathcal{S}(f,\partial B)$ be the group of diffeomorphisms $h$ of $B$ fixed on $\partial B$ and preserving $f$ in the sense that $f\circ h = f$. Under certain assumptions on $f$ we compute the group $\pi_0\mathcal{S}(f,\partial B)$ of is...

Let $M$ be a connected orientable compact surface, $f:M\to\mathbb{R}$ be a Morse function, and $\mathcal{D}_{\mathrm{id}}(M)$ be the group of difeomorphisms of $M$ isotopic to the identity. Denote by $\mathcal{S}'(f)=\{f\circ h = f\mid h\in\mathcal{D}_{\mathrm{id}}(M)\}$ the subgroup of $\mathcal{D}_{\mathrm{id}}(M)$ consisting of difeomorphisms "p...

Let $f:M\to\mathbb{R}$ be a Morse-Bott function on a closed manifold $M$, so the set $\Sigma_f$ of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by $\mathcal{S}(f) = \{h \in \mathcal{D}(M) \mid f\circ h=f \}$ the group of diffeomorphisms of $M$ preserving $f$ and let $\mathcal{D}(\Sigma_...

We introduce the first homotopic Baire class of maps as a homotopical counterpart of a usual first Baire class of maps between topological spaces and show that those classes with values in ANR spaces coincide

Let $Z$ be a non-compact two-dimensional manifold and $\Delta$ be a one-dimensional foliation of $Z$ such that $\partial Z$ consists of leaves of $\Delta$ and each leaf of $\Delta$ is a non-compact closed subset of $Z$. We obtain a characterization of a subclass of such foliated surfaces $(Z,\Delta)$ glued from open strips $\mathbb{R}\times(0,1)$ w...

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals, whence we get a foliation $\Delta$ on all of $Z...

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$ and $f:M\to P$ be a $C^{\infty}$ Morse map. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identit...

We present sufficient conditions for the topological stability of the averages of piecewise smooth functions f : \( \mathbb{R}\to \mathbb{R} \) with finitely many extrema with respect to discrete measures with finite supports.

Let $X$ be an $(n+1)$-dimensional manifold, $\Delta$ be a one-dimensional foliation on $X$, and $p: X \to X / \Delta$ be a quotient map. We will say that a leaf $\omega$ of $\Delta$ is special whenever the space of leaves $X / \Delta$ is not Hausdorff at $\omega$. We present necessary and sufficient conditions for the map $p: X \to X / \Delta$ to b...

Let $X$ be a connected non-compact $2$-dimensional manifold possibly with boundary and $\Delta$ be a foliation on $X$ such that each leaf $\omega\in\Delta$ is homeomorphic to $\mathbb{R}$ and has a trivially foliated neighborhood. Such foliations on the plane were studied by W. Kaplan who also gave their topological classification. He proved that t...

Let $\mu$ be a measure on $[-1,1]$. Then for every continuous function
$f:\mathbb{R}\to\mathbb{R}$ and $\alpha>0$ one can define its averaging
$f_{\alpha}:\mathbb{R}\to\mathbb{R}$ by the formula: \[ f_{\alpha}(x) =
\int_{-1}^{1} f(x+t\alpha)d\mu. \] In arXiv:1509.06064 the authors studied the
problem when $f_{\alpha}$ is topologically equivalent to...

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 f...

We present sufficient conditions for topological stability of real continuous functions in one variable having finitely many local extrema with respect to averagings by discrete measures with finite supports.

We present sufficient conditions for topological stability of continuous functions $f:\mathbb{R}\to\mathbb{R}$ having finitely many local extrema with respect to averagings by discrete measures with finite supports.

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 o...

Let f : T2 → ℝ be a Morse function on a 2-torus, let S(f) and \( \mathcal{O} \)(f) be, respectively, its stabilizer and orbit with respect to the right action of the group \( \mathcal{D} \)(T2) of diffeomorphisms of T2, let \( \mathcal{D} \)id(T2), be the identity path component of the group \( \mathcal{D} \)(T2), and let S′(f) = S(f) ∩ \( \mathcal...

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 o...

Let M be a connected orientable surface, P be either a real line ℝ or a circle S<sup>1</sup>, and f:M → P be a Morse map. Denote by D<sub>id</sub> the group of diffeomorphisms of M isotopic to the identity. This group acts from the right on the space of smooth maps C<sup>∞</sup>(M,P) and one can define the stabilizer S = {h ∈ D<sub>id</sub> | f ∘ h...

Let f:T<sup>2</sup> → ℝ be a Morse function on a 2-torus such that its Kronrod-Reeb graph has exactly one cycle, i.e. it is homotopy equivalent to S<sup>1</sup>.
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<sup>2</sup>.
This result holds for a larger c...

Let M be a smooth compact connected surface, P be either the real line ℝ or the circle S<sup>1</sup>. Let also f:M → P be a Morse map, and S(f) and O(f) be respectively the stabilizer and the orbit of f with respect to the right action of the group D(M) of diffeomorphisms of M. In a series of papers the author described homotopy types of S(F) and c...

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...

Let f:T<sup>2</sup> → ℝ be a Morse function on a 2-torus, S(f) and O(f) be its stabilizer and orbit with respect to the right action of the group D(T<sup>2</sup>) of diffeomorphisms of T<sup>2</sup>, D<sub>id</sub>(T<sup>2</sup>) be the identity path component of D(T<sup>2</sup>), and S'(f) = S(f) ∩ D<sub>id</sub>(T<sup>2</sup>).
We give sufficient...

Let $M$ be a smooth compact connected surface, $P$ be either the real line or
the circle, $f:M\to P$ be a smooth map, and $O(f)$ be the orbit of $f$ with
respect to the right action of the group $\mathrm{Diff}(M)$ of diffeomorphisms
of $M$. Assume that at each of its critical point the map $f$ is equivalent to
a homogeneous polynomial in two variab...

Let M be a connected smooth compact surface and let P be either the number line ℝ or a circle S<sup>1</sup>. For a closed subset X ⊂ M denote by D(M,X) the group of diffeomorphisms of M fixed on X. We consider a special class F(M,P) of smooth mappings f:M → P with isolated singularities containing all Morse mappings. For each mapping f ∈ F(M,P) we...

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...

In the paper [Probab. Theory Relat. Fields, 100 (1994) 417-428] Xue-Mei Li
studied stability of stochastic differential equations and the interplay
between the moment stability of a SDE and the topology of the underlying
manifold. In particular, she gave sufficient condition on SDE on a manifold M
under which the fundamental group \pi_1(M)=0. We pr...

The thesis is devoted to the following three topics:
1) smooth self maps of a manifold with a flow obtained by shifts of points along non-constant time;
2) deformations of smooth functions on surfaces;
3) homotopy types of stabilizers and orbits of smooth functions on compact surfaces under the natural right action of diffeomorphisms groups.

Let M be a smooth connected orientable compact surface and let \( {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) \) be a space of all Morse functions f : M → S
1 without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S
1 is either a constant map or a covering map. The space \( {\mathcal{F}_{{\rm co...

Let F:M × ℝ → M be a continuous flow on a topological manifold M, V ⊂ M be an open subset, and let ξ:V → ℝ be a continuous function. We say that ξ is a period function if F(x, ξ(x)) = x for all x ∈ V. Recently, for any open connected subset V ⊂ M; the author has described the structure of the set P(V) of all period functions on V. Assume that F is...

Let F: M x R -> M be a continuous flow on a topological manifold M: For every subset V subset of M; we denote by P (V) the set of all continuous functions xi : V -> R such that F(x, xi(x)) = x for all x is an element of V: These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the co...

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 tha...

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 $\alpha:V \to R$ such that $F_{\alpha(z)}(z)=z$ 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 minima...

Let D<sup>2</sup> ⊂ ℝ<sup>2</sup> be a closed unit 2-disk centered at the origin 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 D<sup>+</sup>(F) be the group of all diffeomorphisms of D<sup>2<...

Let D<sup>2</sup> ⊂ ℝ<sup>2</sup> be a closed unit 2-disk centered at the origin 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 q:D<sup>2</sup> \ {O} → (0, +∞) be the function associating wit...

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-->M of the following form: h(x) = F_{f(x)}(x), where f:M-->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...

Let F be a smooth vector field defined near the origin O ∈ ℝ<sup>n</sup>.
Suppose O is a singular point of F, so F(O) = 0, and let (F<sub>t</sub>) be the local flow of F.
Denote E(F) by the set of germs of orbit preserving diffeomorphisms h of ℝ<sup>n</sup> at O,
and let E<sup>r</sup><sub>id</sub>, (r ≥ 0), be the identity path component of E(F) wi...

Let (F<sub>t</sub>) be a smooth flow on a smooth manifold M and h:M → 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 smooth function f<sub>z</sub> at z such that near z h(x)=F(z, f<sub>z</sub>(x)). Can the functions (f<sub>z</sub>) be glued together to give a smooth...

Let $f:\mathbb{R}^2 \to \mathbb{R}$ be a real homogeneous polynomial and
$S(f)$ be the group of diffeomorphisms $h:\mathbb{R}^2 \to \mathbb{R}^2$
preserving $f$, i.e. $f \circ h = f$. Denote by $S(f,r)$, $(0\leq r \leq
\infty)$, the identity path component of $S(f)$ with respect to the weak
Whitney $C^{r}_{W}$-topology. We prove that $S(f,\infty) =...

Let M be a smooth connected compact surface, P be either the real line R<sup>1</sup> or the circle S<sup>1</sup>, 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...

Let $M$ be a smooth manifold and $F$ be a vector field on $M$. My article
["Smooth shifts along trajectories of flows", Topol. Appl. 130 (2003) 183-204,
arXiv:math/0106199] concerning the homotopy types of the group of
diffeomorphisms preserving orbits of $F$ contains two errors. They imply that
the principal statement of that paper holds under add...

Let $f$ be a real- or circle-valued Morse function on a compact surface M
having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with
respect to the right action of the group of diffeomorphisms of $M$. We show
that the connected components of $O$ have the homotopy type of a
finite-dimensional CW-complex. Actually, these connected comp...

Let $g:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial of degree
$p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the
local flow generated by $G$. Denote by $E(G,O)$ the space of germs of
$C^{\infty}$ diffeomorphisms $(\mathbb{R}^2,O)\to(\mathbb{R}^2,O)$ that
preserve orbits of $G$. Let also $E_{\mathrm{id}}(G,O)$ b...

Let F be a smooth vector field defined in a neighborhood of the origin in R^n, F(O)=0, and let F_t be its local flow. Denote by E the set of germs of diffeomorphisms h:R^n --> R^n preserving orbits of F and let E_{id}^r be the identity component of E with respect to C^r-topology. Then every E_{id}^{r} contains a subset Sh consisting of mappings of...

Let $M$ be a smooth finite-dimensional manifold, $G$ be a Lie group, and
$\Phi:G \times M \to M$ be a smooth action. Consider the following mapping
$\phi: C^{\infty}(M,G) \to C^{\infty}(M,M)$, defined by $\phi(\alpha)(x) =
\alpha(x)\cdot x$, for $\alpha\in C^{\infty}(M,G)$ and $x\in M$. In this paper
we describe the structure of inverse images of e...

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line ℝ or the circle S, and D(M) the group of diffeomorphisms of M acting on C^∞(M, P) by the rule h⋅ f = f ∘ h^{−1} for h ∊ D(M) and f ∊ C^{∞} (M,P).
Let f: M → P be an arbitrary Morse mapping, Σ(f) the set of critical points of f, D(M,Σ(f)) the s...

Let $M$ be a smooth ($C^{\infty}$) manifold, $F_1,...,F_n$ be vector fields on $M$ generating the corresponding flows $\Phi_1,...,\Phi_n$, and $\alpha_1,...,\alpha_{n}:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$ by $$f(x)= \Phi_n (... (\Phi_2 (\Phi_1 (x,\alpha_1(x)), \alpha_2(x)), ..., \alpha_n(x)).$$ In this note we give...

Let $M$ be a smooth compact manifold and $P$ be either $R^1$ or $S^1$. There is a natural action of the groups $Diff(M)$ and $Diff(M) \times Diff(P)$ on the space of smooth mappings $C^{\infty}(M,P)$. For $f\in C^{\infty}(M,P)$ let $S_f$, $S_{MP}$, $O_f$, and $O_{MP}$ be the stabilizers and orbits of $f$ under these actions. Recently, the author pr...

Let f:ℝ<sup>m</sup> → ℝ be a smooth function such that f(0)=0.
We give a condition on f when for arbitrary preserving orientation diffeomorphism 𝜑:ℝ → ℝ such that 𝜑(0)=0 the function 𝜑 ∘ f is right equivalent to f, i.e. there exists a diffeomorphism h:ℝ<sup>m</sup> → ℝ<sup>m</sup> such that 𝜑 ∘ f = f ∘ h at 0\in ℝ<sup>m</sup>. The requirement is th...

Let M be a smooth compact surface, orientable or not, with boundary or without it, P either the real line R 1 or the circle S 1, and D(M) the group of diffeomorphisms of M acting on C ∞ (M, P) by the rule h · f ↦ → f ◦ h −1 for h ∈ D(M) and f ∈ C ∞ (M, P). Let f: M → P be a Morse mapping, S(f) the stabilizer, and O(f) the orbit of f under this acti...

Let $M$ be a smooth compact surface, orientable or not, with boundary or without it, $P$ either the real line $R^1$ or the circle $S^1$, and $Diff(M)$ the group of diffeomorphisms of $M$ acting on $C^{\infty}(M,P)$ by the rule $h\cdot f\mapsto f \circ h^{-1}$, where $h\in Diff(M)$ and $f \in C^{\infty}(M,P)$. Let $f:M \to P$ be a Morse function and...

Let Φ be a flow on a smooth, compact, finite-dimensional manifold M. Consider the subset of C∞(M,M) consisting of diffeomorphisms of M preserving the foliation of the flow Φ. Let also be the identity path component of with compact-open topology. We prove that under mild conditions on fixed points of Φ the space is either contractible or homotopical...

Let $\Phi$ be a flow on a smooth, compact, finite-dimensional manifold $M$. Consider the subsets $E(\Phi)$ and $D(\Phi)$ of $C^{\infty}(M,M)$ consisting of smoothh mappings and diffeomorphisms (respectively) of $M$ preserving the foliation of the flow $\Phi$. Let also $E_{0}(\Phi)$ and $D_{0}(\Phi)$ be the identity path components of $E(\Phi)$ and...

The paper is devoted to study of the following question: when a k-dimensional subset of R n (0 < k < n) containes a set homeomorphic to k-dimensional disk.

The thesis is devoted to study deformations of Morse functions on surfaces. There two principal results:
1) classification of the connected components of the space of Morse mappings from a compact orientable surface into the circle; 2) topological classification of smooth functions with non-degenerate critical points on compact orientable surface s...

Let $M$ be a compact surface and $P$ be a one dimensional manifold without
boundary, that is the line $\mathbb{R}^1$ or a circle $S^1$. The classification
of path-components of the space of Morse maps from $M$ into $P$ was recently
obtained by S. V. Matveev and V. V. Sharko for the case $P=\mathbb{R}$. For
$P=S^1$ the classification was obtained by...

We classify the path-components of the space of circle-valued Morse functions on compact surfaces: two Morse functions $f, g: M\to S^1$ belong to same path-component of this space if and only if they are homotopic and have equal numbers of critical points at each index.