# Maksimenko S. I.

### Topological stability of the averagings of functions

Maksimenko S. I., Marunkevych O. V.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 5. - pp. 625-633

We present sufficient conditions for the topological stability of the averagings of piecewise smooth functions $f : R \rightarrow R$ with finitely many extrema with respect to discrete measures with finite supports.

### Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus

Feshchenko B. G., Maksimenko S. I.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1205–1212

Let *f* : *T* ^{2} → 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 \( \mathcal{D} \) (*T* ^{2}) of diffeomorphisms of *T* ^{2}, let \( \mathcal{D} \) _{id}(*T* ^{2}), be the identity path component of the group \( \mathcal{D} \) (*T* ^{2}), and let *S*′(*f*) = *S*(*f*) ∩ \( \mathcal{D} \) _{id}(*T* ^{2}). We present sufficient conditions under which $$ {\uppi}_1\mathcal{O}(f)={\uppi}_1{\mathcal{D}}_{\mathrm{id}}\left({T}^2\right)\times {\uppi}_0S^{\prime }(f)\equiv {\mathrm{\mathbb{Z}}}^2\times {\uppi}_0S^{\prime }(f). $$ The obtained result is true for a larger class of functions whose critical points are equivalent to homogeneous polynomials without multiple factors.

### Homotopic types of right stabilizers and orbits of smooth functions on surfaces

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1204

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

### Deformations of circle-valued Morse functions on surfaces

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1360–1366

Let $M$ be a smooth connected orientable compact surface and let $\mathcal{F}_{\text{cov}}(M,S^1)$ 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 $\mathcal{F}_{\text{cov}}(M,S^1)$ is endowed with the $C^{∞}$-topology. We present the classification of connected components of the space $\mathcal{F}_{\text{cov}}(M,S^1)$. This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on $∂M$.

### Period functions for $\mathcal{C}^0$- and $\mathcal{C}^1$-flows

Ukr. Mat. Zh. - 2010. - 62, № 7. - pp. 954–967

Let $F:\; M×R→M$ be a continuous flow on a manifold $M$, let $V ⊂ M$ 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 topologically conjugate to some $\mathcal{C}^1$-flow. It is shown in this paper that, in this case, the period functions of $F$ satisfy some additional conditions that, generally speaking, are not satisfied for general continuous flows.

### Kernel of a map of a shift along the orbits of continuous flows

Ukr. Mat. Zh. - 2010. - 62, № 5. - pp. 651–659

Let $F: M × R → M$ be a continuous flow on a topological manifold $M$. For every subset $V ⊂ M$, we denote by $P(V)$ the set of all continuous functions $ξ: V → R$ such that $F(x,ξ(x)) = x$ for all $x ∈ V$. These functions vanish at nonperiodic points of the flow, while their values at periodic points are integer multiples of the corresponding periods (in general, not minimal). In this paper, the structure of $P(V)$ is described for an arbitrary connected open subset $V ⊂ M$.

### Classification of*m*-functions on surfaces

Ukr. Mat. Zh. - 1999. - 51, № 8. - pp. 1129–1135

We establish a necessary and sufficient condition of conjugacy of*m*=functions on surfaces.

### On topological spaces with π_{2} = 0

Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1144–1146

We consider one construction over topological spaces and study its influence on the group π_{2}.