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Pavel Gumenyuk

Publications and source records attributed to Pavel Gumenyuk.

At least 19 recordsLinked to original sources
arXiv2026

Simultaneous linearization and centralizers of parabolic self-maps II: positive hyperbolic step

The study of holomorphic self-mappings of the unit disc commuting under the composition goes back to A.L. Shields (1964), W.A. Pranger (1970), D.F. Behan (1973), and C.C. Cowen (1984). In many situations, the centralizer of a holomorphic self-map ${φ:\mathbb D \to \mathbb D}$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ\in\mathsf{Hol}(\mathbb D):ψ\circφ=φ\circψ\}$ turns out to be commutative. However, this does not hold for the case of a parabolic self-map $φ$ of positive hyperbolic step, which is analyzed in detail in this paper. We investigate the relationships among commutativity, simultaneous linearization, and holomorphic models. In particular, we obtain existence and uniqueness results for the simultaneous linearization of commuting pairs $φ$, $ψ\in \mathcal Z_\forall(φ)$. Furthermore, extending this notion to arbitrary families of holomorphic self-mappings, we show that a given (finite or infinite) family in the centralizer ${Δ\subset\mathcal Z_\forall(φ)}$ can be simultaneously linearized together with$~φ$ if and only if any two elements of$~Δ$ commute with each other. This gives a far reaching extension of Cowen's result concerning commuting pairs in$~\mathsf{Hol}(\mathbb D)$.

math.CV↗
arXiv2025

Loewner Theory for Bernstein functions II: applications to inhomogeneous continuous-state branching processes

This paper continues the research project launched in [Constr. Approx. (2025) https://doi.org/10.1007/s00365-023-09675-9] and aimed at studying time-inhomogeneous one-dimensional branching processes (mainly on a continuous but also on a discrete state space) with the help of recent achievements in Loewner Theory dealing with evolution families of holomorphic self-maps in simply connected domains of the complex plane. Under a suitable stochastic continuity condition, we show that the families of the Laplace exponents of branching processes on$~[0,\infty]$ can be characterized as topological (i.e. depending continuously on the time parameters) reverse evolution families whose elements are Bernstein functions. For the case of a stronger regularity w.r.t. time, we establish a Loewner-Kufarev type ODE for the Laplace exponents and characterize branching processes with finite mean in terms of the vector field driving this ODE. Similar results are obtained for families of probability generating functions of branching processes on the discrete state space $\{0,1,2,\ldots\}\cup\{\infty\}$. In addition, we find a necessary and sufficient condition for "spatial" embeddability of such branching processes into branching processes on$~[0,\infty]$. Finally, we give some probabilistic interpretations of the Denjoy-Wolff point at$~0$ and at$~\infty$.

math.PR↗
arXiv2025

Simultaneous linearization and centralizers of parabolic self-maps I: zero hyperbolic step

Let $φ:\mathbb D \to \mathbb D$ be a parabolic self-map of the unit disc $\mathbb D$ having zero hyperbolic step. We study holomorphic self-maps of $\mathbb D$ commuting with $φ$. In particular, we answer a question from Gentili and Vlacci (1994) by proving that $ψ\in\mathsf{Hol(\mathbb D,\mathbb D)}$ commutes with $φ$ if and only if the two self-maps have the same Denjoy-Wolff point and $ψ$ is a pseudo-iterate of $φ$ in the sense of Cowen. Moreover, we show that the centralizer of $φ$, i.e. the semigroup $\mathcal Z_\forall(φ):=\{ψ:ψ\circφ=φ\circψ\}$ is commutative. We also prove that if $φ$ is univalent, then all elements of $\mathcal Z_\forall(φ)$ are univalent as well, and if $φ$ is not univalent, then the identity map is an isolated point of $\mathcal Z_\forall(φ)$. The main tool is the machinery of simultaneous linearization, which we develop using holomorphic models for iteration of non-elliptic self-maps originating in works of Cowen and Pommerenke.

math.CV↗
arXiv2024

Hyperbolic convexity of holomorphic level sets

We prove that the sublevel set $\big\{z\in\mathbb D\colon k_{\mathbb D}\big(z,z_0\big)-k_{\mathbb D}\big(f(z),w_0\big)<μ\big\}$, ${μ\in\mathbb R}$, is geodesically convex with respect to the Poincaré distance $k_{\mathbb D}$ in the unit disk $\mathbb D$ for every ${z_0,w_0\in\mathbb D}$ and every holomorphic ${f:\mathbb D\to\mathbb D}$ if and only if ${μ\leqslant0}$. An analogous result is established also for the set $\{z\in\mathbb D \colon 1-|f(z)|^2<λ(1-|z|^2)\}$, ${λ>0}$. This extends a result of Solynin (2007) and solves a problem posed by Arango, Mej\'ıa and Pommerenke (2019). We also propose several open questions aiming at possible extensions to more general settings.

math.CV↗
arXiv2024

Hyperbolic distortion and conformality at the boundary

We characterize two classical types of conformality of a holomorphic self-map of the unit disk at a boundary point - existence of a finite angular derivative in the sense of Carathéodory and the weaker property of angle preservation - in terms of the non-tangential asymptotic behaviour of the hyperbolic distortion of the map. These characterizations are given purely with reference to the intrinsic metric geometry of the unit disk. In particular, we relate the classical Julia-Wolff-Carathéodory theorem with the case of equality in the Schwarz-Pick lemma at the boundary. We also provide an operator-theoretic characterization of the existence of a finite angular derivative based on Hilbert space methods. As an application we study the backward dynamics of discrete dynamical systems induced by holomorphic self-maps, and characterize the regularity of the associated pre-models in terms of a Blaschke-type condition involving the hyperbolic distortion along regular backward orbits.

math.CV↗
arXiv2024

Criteria for extension of commutativity to fractional iterates of holomorphic self-maps in the unit disc

Let $φ$ be a univalent non-elliptic self-map of the unit disc $\mathbb D$ and let $(ψ_{t})$ be a continuous one-parameter semigroup of holomorphic functions in $\mathbb D$ such that $ψ_{1}\neq\mathrm{id}_{\mathbb D}$ commutes with $φ$. This assumption does not imply that all elements of the semigroup $(ψ_t)$ commute with $φ$. In this paper, we provide a number of sufficient conditions that guarantee that ${ψ_t\circφ=φ\circψ_t}$ for all ${t>0}$: this holds, for example, if $φ$ and $ψ_1$ have a common boundary (regular or irregular) fixed point different from their common Denjoy-Wolff point $τ$, or when $ψ_1$ has a boundary regular fixed point ${σ\neqτ}$ at which $φ$ is isogonal, or when $(φ-\mathrm{id}_{\mathbb D})/(ψ_1-\mathrm{id}_{\mathbb D})$ has an unrestricted limit at $τ$. In addition, we analyze how $φ$ behaves in the petals of the semigroup $(ψ_t)$.

math.CV↗
arXiv2024

Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map $φ$ of the unit disc $\mathbb D$ in order to be contained in a continuous one-parameter semigroup. In this paper, we tackle this embedding problem by establishing different dichotomy results about the centralizer of $φ$ (i.e. the set of all univalent self-maps commuting with $φ$) which depend strongly on the dynamical character of $φ$. Our approach is, in part, based on a new technique to obtain simultaneous linearizations of two non-elliptic univalent self-maps of the unit disc, which might be interesting on their own. We also introduce and study several closed additive subsemigroups of the complex plane that collect the main features of the centralizer of $φ$ and which play a prominent position in those dichotomy results.

math.CV↗
arXiv2024

The angular derivative problem for petals of one-parameter semigroups in the unit disk

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy-Wolff point.

math.CV↗
arXiv2023

A note on composition operators on model spaces

Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $ϕ\not\in\mathsf{Aut}(\mathbb D)$, find a non-constant inner function $Ψ$ satisfying the functional equation $Ψ\circϕ=τΨ$, where $τ$ is a unimodular constant. We prove that this problem has a solution if and only if $ϕ$ is of positive hyperbolic step. More precisely, if this condition holds, we show that there is an infinite Blaschke product $B$ satisfying the equation for $τ=1$. If in addition, $ϕ$ is parabolic, we prove that the problem has a solution $Ψ$ for $any$ unimodular $τ$. Finally, we show that if $ϕ$ is of zero hyperbolic step, then no non-constant Bloch function $f$ and no unimodular constant $τ$ satisfy $f\circϕ=τf$.

math.CV↗
arXiv2023

Loewner Theory for Bernstein functions I: evolution families and differential equations

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups in the inhomogeneous setting is the notion of a (reverse) evolution family. In this paper we study evolution families formed by Bernstein functions, which play the role of Laplace exponents for inhomogeneous continuous-state branching processes. In particular, we characterize all Herglotz vector fields that generate such evolution families and give a complex-analytic proof of a qualitative description equivalent to Silverstein's representation formula for the infinitesimal generators of one-parameter semigroups of Bernstein functions. We also establish several sufficient conditions for families of holomorphic self-maps, satisfying the algebraic part in the definition of an evolution family, to be absolutely continuous and hence to be described as solutions to the generalized Loewner - Kufarev differential equation. Most of these results are then applied in the sequel paper [https://doi.org/10.48550/arXiv.2211.12442] to study continuous-state branching processes.

math.CV↗
arXiv2021

On the squeezing function for finitely connected planar domains

In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions.

math.CV↗
arXiv2020

Angular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk

We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.

math.CV↗
arXiv2020

Non-diffeomorphic Reeb foliations and modified Godbillon-Vey class

The paper deals with a modified Godbillon-Vey class defined by Losik for codimension-one foliations. This characteristic class takes values in the cohomology of the second order frame bundle over the leaf space of the foliation. The definition of the Reeb foliation depends upon two real functions satisfying certain conditions. All these foliations are pairwise homeomorphic and have trivial Godbillon-Vey class. We show that the modified Godbillon-Vey is non-trivial for some Reeb foliations and it is trivial for some other Reeb foliations. In particular, the modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and it provides more information than the classical Godbillon-Vey class. We also show that this class is non-trivial for some foliations on the two-dimensional surfaces.

math.DG↗
arXiv2020

On existence of Becker extension

A well-known theorem by J. Becker states that if a normalized univalent function $f$ in the unit disk $\mathbb{D}$ can be embedded as the initial element into a Loewner chain $(f_t)_{t\geqslant 0}$ such that the Herglotz function $p$ in the Loewner -- Kufarev PDE $$\partial f_t(z)/\partial f=zf'_t(z)p(z,t),\qquad z\in\mathbb{D},\quad\mathrm{a.e.}~t\ge0,$$ satisfies $\big|(p(z,t)-1)/(p(z,t)+1)\big|\le k<1$, then $f$ admits a $k$-q.c. (="$k$-quasiconformal") extension $F:\mathbb{C}\to\mathbb{C}$. The converse is not true. However, a simple argument shows that if $f$ has a $q$-q.c. extension with $q\in(0,1/6)$, then Becker's condition holds with $k:=6q$. In this paper we address the following problem: find the largest $k_*\in(0,1]$ with the property that for any $q\in(0,k_*)$ there exists $k_0(q)\in(0,1)$ such that every normalized univalent function $f:\mathbb D\to\mathbb C$ with a $q$-q.c. extension to $\mathbb C$ satisfies Becker's condition with $k:=k_0(q)$. We prove that $k_*\ge1/3$.

math.CV↗
arXiv2020

Infinitesimal generators of semigroups with prescribed boundary fixed points

We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb D$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning.vWe also give a new proof of the classical Cowen-Pommerenke inequalities for univalent self-maps of $\mathbb D$.

math.CV↗
arXiv2019

Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficient

We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by Kühnau and Niske [Math. Nachr. 78 (1977) 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972) 23-43]. Another problem we consider is to find the maximal $k_*\in(0,1]$ such that every univalent function $f$ in $\mathbb D$ having a $k$-q.c. extension to $\mathbb C$ with $k\leqslant k_*$ admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that $k_*>1/6$. Moreover, we show that in some cases, Becker's extension turns out to be the optimal one. Namely, given any $k\in(0,1)$, to each finite Blaschke product there corresponds a univalent function $f$ in $\mathbb D$ that admits a Becker $k$-q.c. extension but no $k'$-q.c. extensions to $\mathbb C$ with $k'<k$.

math.CV↗
arXiv2017

Quasiconformal extensions, Loewner chains, and the lambda-Lemma

In 1972, J. Becker [J. Reine Angew. Math. 255] discovered a sufficient condition for quasiconformal extendibility of Loewner chains. Many known conditions for quasiconformal extendibility of holomorphic functions in the unit disk can be deduced from his result. We give a new proof of (a generalization of) Becker's result based on Slodkowski's Extended lambda-Lemma. Moreover, we characterize all quasiconformal extensions produced by Becker's (classical) construction and use that to obtain examples in which Becker's extension is extremal (i.e. optimal in the sense of maximal dilatation) or, on the contrary, fails to be extremal.

math.CV↗