Rapidly convergent iteration method for simultaneous normal forms of commuting maps
type: Item
instance of: scholarly article
title: Rapidly convergent iteration method for simultaneous normal forms of commuting maps
zbMATH Open document ID: 0931.65055
DOI: 10.1007/PL00004751
author: Todor Gramchev
author: Masafumi Yoshino
published in: Mathematische Zeitschrift
publication date: 2000-02-23T00:00:00Z
review text: We propose a new rapidly convergent KAM type iteration scheme under weak simultaneous arithmetic conditions for overdetermined systems related to simulataneous conjugation of commuting maps. The first application is for the simultaneous linearization of commuting holomorphic maps \(\phi(x),\dots, \Phi_d(x)\) in \(\mathbb{C}^n\) fixing the origin, not necessarily diffeomorphisms if \(n\geq 2\), \(d\geq 2\). The second one is for simultaneous local reduction of orientation preserving Gevrey \(G^\sigma\) \((\sigma\geq 1)\) circle mappings. Here one of the crucial ingredients of the proof is to show some new nonlinear superposition estimates in scales of Banach spaces of Gevrey functions.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65E05
Mathematics Subject Classification ID: 32A19
zbMATH DE Number: 1336873
zbMATH Keywords: convergence
zbMATH Keywords: normal forms
zbMATH Keywords: KAM type iteration scheme
zbMATH Keywords: overdetermined systems
zbMATH Keywords: simultaneous linearization
zbMATH Keywords: simultaneous local reduction
zbMATH Keywords: Banach spaces
zbMATH Keywords: Gevrey functions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1336873