Spherical hypersurfaces and Lattès rational maps
type: Item
instance of: scholarly article
title: Spherical hypersurfaces and Lattès rational maps
zbMATH Open document ID: 0913.58031
DOI: 10.1016/S0021-7824(98)80003-2
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1999-05-26T00:00:00Z
review text: A Lattès map is a rational map with an empty Fatou set. It is a chaotic rational map \(f\) on the Riemann sphere \(\widehat{{\mathbb C}}\) induced from an expanding complex multiplication \(s\) on some torus \(T\) by means of an elliptic function \(q: T\to\widehat{{\mathbb C}}\) that is a regular branched cover. These were discovered by S. Lattès in 1918. Let \(F\) be a nondegenerate homogeneous polynomial self-map of \({\mathbb C}^2\) that canonically induces \(f\) and let \(\Omega_F\) denote the basin of attraction for \(F\) at the origin. If \(f\) is a Lattès, map the authors show that the Brolin-Lyubich measure of \(f\) is smooth and strictly positive on some open set of \(\widehat{{\mathbb C}}\), that the intersection \(b\Omega_F\cap V\) (\(b\Omega_F\) is the boundary of the basin \(\Omega_F\)) is smooth and strictly pseudoconvex for some nonempty open set \(V\subset{\mathbb C}^2\), and \(b\Omega_F\) is spherical except on a finite union of circles that project onto a finite subset of \(\widehat{{\mathbb C}}\); in fact, these four conditions are all equivalent. The proof involves an investigation of the structure of the group \(G\) of automorphisms of \({\mathbb C}^2\) that preserve \(b\Omega_F\). A further study of this group \(G\) enables the authors to find examples of domains that are almost strictly pseudoconvex but do admit noninjective proper holomorphic maps.
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 1207283
zbMATH Keywords: rational maps
zbMATH Keywords: Fatou sets
zbMATH Keywords: basin of attraction
zbMATH Keywords: chaotic maps
author: François Berteloot
author: Jean-Jacques Loeb
MaRDI profile type: Publication
cites work: Q4172303
cites work: A proof of Thurston's topological characterization of rational functions
cites work: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions
cites work: Proper holomorphic images of strictly pseudoconvex domains
cites work: Q4310116
cites work: Q4869193
cites work: Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann
cites work: Q4301007
cites work: Q3336420
cites work: The dynamics of rational transforms: the topological picture
cites work: Parabolic orbifolds and the dimension of the maximal measure for rational maps
Wikidata QID: Q127191052
Recommended article: Q5201162
Recommended article: Rational surfaces and regular maps into the 2-dimensional sphere
Recommended article: Rational Sphere Maps
Recommended article: Q3771209
Recommended article: Q3109959
Recommended article: Q3736479
Recommended article: Explicit description of spherical rigid hypersurfaces in \(\mathbb C^2\)
Recommended article: Spherical hypersurfaces associated with the spherical Gauss map and Gauss map
Recommended article: Q3538576
Recommended article: Hypersurfaces in spheres with finite total curvature
description: scientific article; zbMATH DE number 1207283