The moduli space of rational maps and surjectivity of multiplier representation
type: Item
instance of: scholarly article
title: The moduli space of rational maps and surjectivity of multiplier representation
zbMATH Open document ID: 1151.30002
DOI: 10.1007/BF03321649
author: Masayo Fujimura
published in: Computational Methods and Function Theory
publication date: 2008-05-30T00:00:00Z
review text: Let \(\text{Rat}_n\) consist of functions \(f=p/q\) where \(p\) and \(q\) are coprime polynomials of the maximal degree \(n\). The moduli space of \(\text{Rat}_n\) denoted my \(\mathcal M_n\) is the set of all Möbius conjugacy classes of maps in \(\text{Rat}_n\). Let \(\text{Poly}_n\) be the space of all polynomials of degree \(n\). The moduli space of \(\text{Poly}_n\) denoted by \(M_n\) is the set of all affine conjugacy classes of maps in \(\text{Poly}_n\). The map \(\Psi_{\text{Rat}_n}: \mathcal M_n\to\mathbb C^n\) is a natural map to an \(n\)-tuple of multipliers at fixed points. The map \(\Psi_{\text{Poly}_n}\) is defined similarly. The author proves Theorem 1: For every \(n\geq2\), the map \(\Psi_{\text{Rat}_n}: \mathcal M_n\to\mathbb C^n\) is surjective. Theorem 2: The exceptional set \(\mathcal E(n)=\mathbb C^{n-1}\setminus\Psi_{\text{Poly}_n}(M_n)\) is non-empty for every \(n\geq4\). In the case of Theorem 2, the author gives a sufficient condition and a necessary one for points to be not in the image of \(\Psi_{\text{Poly}_n}\), and presents an explicit parametrization for all such points if \(n=4\) or 5. Also, it is shown that the preimage of a generic point by \(\Psi_{\text{Poly}_n}\) consists of \((n-2)!\) points.
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 5281673
zbMATH Keywords: fixed points
zbMATH Keywords: multiplier
reviewed by: Dmitri V. Prokhorov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5281673