On the multiplier of a repelling fixed point
type: Item
instance of: scholarly article
title: On the multiplier of a repelling fixed point
zbMATH Open document ID: 0837.30020
DOI: 10.1007/BF01231527
author: Lisa R. Goldberg
published in: Inventiones Mathematicae
publication date: 1996-05-24T00:00:00Z
full work available at URL: https://eudml.org/doc/144230
review text: The author estimates the multiplier \(\lambda\) of a repelling fixed point with rational rotation number for a polynomial in one complex variable. It is shown that \(\lambda\) lies in a small disk tangent to the imaginary axis. The radius of the disk is determined by the degree of the polynomial, the rotation number of the fixed point and the escape rates and external arguments of the critical values. Theorem 2.8. Let \(f: \mathbb{C}\to \mathbb{C}\), be a degree \(d\) polynomial with \(d\geq 2\). Let \(z_0\) be a repelling fixed point with multiplier \(\lambda\). Suppose that the type \(T\) of \(z_0\) has rotation number \(p/q\) and cardinality \(kq\). Let \(\beta(f, T)\) be the space constant of \(f\) and \(T\). There is a complex number \(L\) for which \(\exp(L+ 2\pi i\cdot p/q)= \lambda\) and \(|L|^2\cdot \beta(f, T)\). This result generalizes an equality proved by J. C. Yoccoz for polynomials whose filled Julia sets are connected [\textit{J. C. Yoccoz}, Sur la taille des membres de l'ensemble de Mandelbrot (preprint)]. Let \({\mathcal E}_d\) be the escape locus consisting of those degree \(d\) polynomials whose critical values all lie in the set of point \(\Omega\) whose orbits are unbounded, the escape rate of \(f\), \(\rho(f)\), to be the maximum of the values \(e(v)\), as \(v\in {\mathcal V}\) ranges over the critical values \(f\) in \(\Omega\). It is given an estimate for the multiplier of a fixed point that is uniform over bounded regions in \({\mathcal E}_d\). Theorem 3.1. Let \(f\in {\mathcal E}_d\) and \(\rho(f)= r\). Let \(z_0\) be a repelling fixed point of \(f\) with multiplier \(\lambda\). Suppose that the type \(T\) of \(z_0\) has rotation number \(p/q\) and cardinality \(kq\). Let \(B(r, d, q)= \arctan 1/r(d^q- 1)\). Then there is a complex number \(L\) such that \(\exp(L+ 2\pi i\cdot p/q)= \lambda\) and \(|L|^2/\text{Re } L\leq 2\pi\cdot \log d/kq\cdot B(r, d, q)\). Finally, the Bers inequality for a quasi-Fuchsian manifold \(M\) is discussed [\te…
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 37F99
zbMATH DE Number: 703877
zbMATH Keywords: holomorphic dynamics
zbMATH Keywords: iterations
zbMATH Keywords: differentials on Riemann surfaces
zbMATH Keywords: repelling fixed point
MaRDI profile type: Publication
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OpenAlex ID: W2155625157
description: scientific article; zbMATH DE number 703877