# Polulyakh E. O.

### Trees as set levels for pseudoharmonic functions in the plane. II

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 254-270

Let $T$ be a forest formed by finitely many locally finite trees. Let $V_0$ be the set of all vertices of $T$ of degree 1. We propose a sufficient condition for the image of an embedding $\Psi : T \setminus V_0 \rightarrow R^2$ to be a level set of a pseudoharmonic function.

### Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. II

Ukr. Mat. Zh. - 2015. - 67, № 10. - pp. 1398-1408

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema and if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that the function restricted to this neighborhood is topologically conjugate to Re $z^n$ in a certain neighborhood of zero. Given $f : M^2 → ℝ$, let $Γ_{K−R} (f)$ be a quotient space of $M^2$ with respect to its partition formed by the components of level sets of the function $f$. It is known that the space $Γ_{K−R} (f)$ is a topological graph if $M^2$ is compact. In the first part of the paper, we introduced the notion of graph with stalks that generalizes the notion of topological graph. For noncompact $M^2$ , we present three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks. In the second part, we prove that these conditions are also necessary in the case $M^2 = ℝ^2$. In the general case, one of our conditions is not necessary. We provide an appropriate example.

### Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. I

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 375-396

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema; if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that a function restricted to this neighborhood is topologically conjugate to Re $z^n$ in a certain neighborhood of zero. Given $f : M^2 → ℝ$, let $Γ_{K−R} (f)$ be a quotient space of $M^2$ with respect to its partition formed by the components of the level sets of $f$. It is known that, for compact $M^2$, the space $Γ_{K−R} (f)$ is a topological graph. We introduce the notion of graph with stalks, which generalizes the notion of topological graph. For noncompact $M^2$, we establish three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks.

### Trees as Level Sets for Pseudoharmonic Functions in the Plane

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 974–995

Let *T* be a finite or infinite tree and let *V* _{0} be the set of all vertices of *T* of valency 1. We propose a sufficient condition for the image of the imbedding ψ: *T* \*V* _{0} → \( {{\mathbb{R}}^2} \) to be a level set of a pseudoharmonic function.

### On iteration stability of the Birkhoff center with respect to power 2

Polulyakh E. O., Vlasenko I. Yu.

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 705–707

It is proved that the Birkhoff center of a homeomorphism on an arbitrary metric space coincides with the Birkhoff center of its power 2.

### On a fiber bundle over a disk with the cantor set as a fiber

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1567–1571

We construct a Pontryagin fiber bundle ξ = (*N, p, S* ^{1}), the total space *N* of which cannot be imbedded into any two-dimensional oriented manifold but can be imbedded into an arbitrary nonoriented two-dimensional manifold.