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Mark Elin

Publications and source records attributed to Mark Elin.

32 records · Page 2Linked to original sources
arXiv2013

Parabolic type semigroups: asymptotics and order of contact

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the boundary Denjoy--Wolff point. The geometric content includes, in particular, the asymptotes to trajectories, the so-called limit curvature, the order of contact, and so on. We then establish asymptotic rigidity properties for a broad class of semigroups of parabolic type.

math.CV↗
arXiv2013

Growth estimates for pseudo-dissipative holomorphic maps in Banach spaces

In this paper we introduce a class of pseudo-dissipative holomorphic maps which contains, in particular, the class of infinitesimal generators of semigroups of holomorphic maps on the unit ball of a complex Banach space. We give a growth estimate for maps of this class. In particular, it follows that pseudo-dissipative maps on the unit ball of (infinite-dimensional) Banach spaces are bounded on each domain strictly contained inside the ball. We also present some applications.

math.CV↗
arXiv2012

Covering results and perturbed Roper--Suffridge operators

This work is devoted to the advanced study of Roper--Suffridge type extension operators. For a given non-normalized spirallike function (with respect to an interior or boundary point) on the open unit disk of the complex plane, we construct perturbed extension operators in a certain class of Banach spaces and prove that these operators preserve the spirallikeness property. In addition, we present an extension operator for semigroup generators. We use a new geometric approach based on the connection between spirallike mappings and one-parameter continuous semigroups. It turns out that the new one-dimensional covering results established below are crucial for our investigation.

math.CV↗
arXiv2011

Boundary behavior and rigidity of semigroups of holomorphic mappings

In this paper we give some quantative characteristics of boundary asymptotic behavior of semigroups of holomorphic self-mappings of the unit disk including the limit curvature of their trajectories at the boundary Denjoy--Wolff point. This enable us to establish an asymptotic rigidity property for semigroups of parabolic type.

math.CV↗
arXiv2010

Extension operators via semigroups

The Roper--Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is $A$-spirallike for a variety of linear operators $A$. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.

math.CV↗
arXiv2010

Separation of Boundary Singularities for Holomorphic Generators

We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates the existence and importance of a particular role of the binary operation $\circ$ given by $1 / f \circ g = 1/f + 1/g$ on generators.

math.CV↗
arXiv2009

Linearization models for parabolic dynamical systems via Abel's functional equation

We study linearization models for continuous one-parameter semigroups of parabolic type. In particular, we introduce new limit schemes to obtain solutions of Abel's functional equation and to study asymptotic behavior of such semigroups. The crucial point is that these solutions are univalent functions convex in one direction. In a parallel direction, we find analytic conditions which determine certain geometric properties of those functions, such as the location of their images in either a half-plane or a strip, and their containing either a half-plane or a strip. In the context of semigroup theory these geometric questions may be interpreted as follows: is a given one-parameter continuous semigroup either an outer or an inner conjugate of a group of automorphisms? In other words, the problem is finding a fractional linear model of the semigroup which is defined by a group of automorphisms of the open unit disk. Our results enable us to establish some new important analytic and geometric characteristics of the asymptotic behavior of one-parameter continuous semigroups of holomorphic mappings, as well as to study the problem of existence of a backward flow invariant domain and its geometry.

math.CV↗
arXiv2007

A rigidity theorem for holomorphic generators on the Hilbert ball

We present a rigidity property of holomorphic generators on the open unit ball $\mathbb{B}$ of a Hilbert space $H$. Namely, if $f\in\Hol (\mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{t\geq 0}$ on $\mathbb{B}$ such that for some boundary point $τ\in \partial\mathbb{B}$, the admissible limit $K$-$\lim\limits_{z\toτ}\frac{f(x)}{\|x-τ\|^{3}}=0$, then $f$ vanishes identically on $\mathbb{B}$.

math.CV↗
arXiv2006

Commuting semigroups of holomorphic mappings

Let $S_{1}=\left\{F_t\right\}_{t\geq 0}$ and $S_{2}=\left\{G_t\right\}_{t\geq 0}$ be two continuous semigroups of holomorphic self-mappings of the unit disk $Δ=\{z:|z|<1\}$ generated by $f$ and $g$, respectively. We present conditions on the behavior of $f$ (or $g$) in a neighborhood of a fixed point of $S_{1}$ (or $S_{2}$), under which the commutativity of two elements, say, $F_1$ and $G_1$ of the semigroups implies that the semigroups commute, i.e., $F_{t}\circ G_{s}=G_{s}\circ F_{t}$ for all $s,t\geq 0$. As an auxiliary result, we show that the existence of the (angular or unrestricted) $n$-th derivative of the generator $f$ of a semigroup $\left\{F_t\right\}_{t\geq 0}$ at a boundary null point of $f$ implies that the corresponding derivatives of $F_{t}$, $t\geq 0$, also exist, and we obtain formulae connecting them for $n=2,3$.

math.CV↗
arXiv2006

Controlled approximation and interpolation for some classes of holomorphic functions

This paper reports on constructive approximation methods for three classes of holomorphic functions on the unit disk which are closely connected each other: the class of starlike and spirallike functions, the class of semigroup generators, and the class of functions with positive real part. It is more-or-less known that starlike or spirallike functions can be defined as solutions of singular differential equations which, in general, are not stable under the motion of interior singular points to the boundary. At the same time, one can establish a perturbation formula which continuously transforms a starlike (or spirallike) function with respect to a boundary point to a starlike (or spirallike) function with respect to an interior point. This formula is based on an appropriate approximation method of holomorphic generators which determine the above-mentioned differential equations. In turn, the well-known Berkson--Porta parametric representation of holomorphic generators leads us to study an approximation-interpolation problem for the class of functions with positive real part. While this problem is of independent interest, the solution we present here is again based on the Berkson--Porta formula. Finally, we apply our results to solve a natural perturbation problem for one-parameter semigroups of holomorphic self-mappings, as well as the eigenvalue problem for the semigroup of composition operators.

math.CV↗
arXiv2005

A flower structure of backward flow invariant domains for semigroups

In this paper, we study conditions which ensure the existence of backward flow invariant domains for semigroups of holomorphic self-mappings of a simply connected domain $D$. More precisely, the problem is the following. Given a one-parameter semigroup $\mathcal S$ on $D$, find a simply connected subset $Ω\subset D$ such that each element of $\mathcal S$ is an automorphism of $Ω$, in other words, such that $\mathcal S$ forms a one-parameter group on $Ω$. On the way to solving this problem, we prove an angle distortion theorem for starlike and spirallike functions with respect to interior and boundary points.

math.CV↗