# Univalent polynomials and Hubbard trees

@article{Lazebnik2019UnivalentPA,
title={Univalent polynomials and Hubbard trees},
author={Kirill Lazebnik and Nikolai G. Makarov and Sabyasachi Mukherjee},
journal={arXiv: Complex Variables},
year={2019}
}
• Published 16 August 2019
• Mathematics
• arXiv: Complex Variables
We study rational functions $f$ of degree $d$, univalent in the exterior unit disc, with $f(\mathbb{T})$ having the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. We connect this class of rational maps and their associated bi-angled trees to the class of anti-holomorphic… Expand
6 Citations

#### Figures and Tables from this paper

Classification of critically fixed anti-rational maps.
We show that there is a one-to-one correspondence between conjugacy classes of critically fixed anti-rational maps and equivalence classes of certain plane graphs. We furthermore prove thatExpand
David extension of circle homeomorphisms, mating, and removability.
• Mathematics
• 2020
We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go toExpand
• K. Lazebnik
• Mathematics
• Conformal Geometry and Dynamics of the American Mathematical Society
• 2021
We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of theExpand
Bers slices in families of univalent maps
• Mathematics
• Mathematische Zeitschrift
• 2021
We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions $\Sigma$. This embedding is such that the conformal mating of the reflectionExpand
Combination Theorems in Groups, Geometry and Dynamics
• Mathematics
• 2021
The aim of this article is to give a survey of combination theorems occurring in hyperbolic geometry, geometric group theory and complex dynamics, with a particular focus on Thurston’s contributionExpand
Circle packings, kissing reflection groups and critically fixed anti-rational maps
• Mathematics
• 2020
In this paper, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs,Expand

#### References

SHOWING 1-10 OF 38 REFERENCES
Schwarz reflections and the Tricorn
• Mathematics
• 2018
We continue our study of the family $\mathcal{S}$ of Schwarz reflection maps with respect to a cardioid and a circle which was started in [LLMM1]. We prove that there is a natural combinatorialExpand
Extreme points in a class of polynomials having univalent sequential limits
This paper concerns a class ^ (defined below) of polynomials of degree less than or equal to n having the properties: each polynomial which is Univalent in the unit disk and of degree n or less is inExpand
Hubbard forests
Abstract The theory of Hubbard trees provides an effective classification of nonlinear postcritically finite polynomial maps in ℂ. This note extends the classification to maps from a finite union ofExpand
Sharp bounds for the valence of certain harmonic polynomials
In Khavinson and Światek (2002) it was proved that harmonic polynomials z - p(z), where p is a holomorphic polynomial of degree n > 1, have at most 3n - 2 complex zeros. We show that this bound isExpand
The valence of harmonic polynomials
The paper gives an upper bound for the valence of harmonic polynomials. An example is given to show that this bound is sharp. Interest in harmonic mappings in the complex plane has increased due toExpand
Dynamics in one complex variable
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is largeExpand
Coefficient regions for univalent polynomials of small degree
Let be the class of normalised polynomials of the form: of degree n which are univalent in | z | be the class of “polynomials” of the form of degree n which are univalent in 0 z | ( k = 2, 3, 4) andExpand
Quasiconformal surgery in holomorphic dynamics
• Mathematics
• 2013
Preface Introduction 1. Quasiconformal geometry 2. Extensions and interpolations 3. Preliminaries on dynamical systems and actions of Kleinian groups 4. Introduction to surgery and first occurrencesExpand
Sharpness of connectivity bounds for quadrature domains
• Mathematics
• 2014
In this paper we prove the sharpness of connectivity bounds established in [15]. The proof depends on some facts in the theory of univalent polynomials. We also discuss applications to the equationExpand
MATH
Abstract: About a decade ago, biophysicists observed an approximately linear relationship between the combinatorial complexity of knotted DNA and the distance traveled in gel electrophoresisExpand