Analytic smoothing of geometric maps with applications to KAM theory
type: Item
instance of: scholarly article
title: Analytic smoothing of geometric maps with applications to KAM theory
zbMATH Open document ID: 1160.37024
author: A. González-Enríquez
author: Rafael de la Llave
published in: Journal of Differential Equations
publication date: 2008-09-04T00:00:00Z
review text: It is shown that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact, and that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in \(C^r\) -norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. This result is applied to give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. The main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. And the symplectic, volume-preserving and contact forms are assumed to be analytic. It is interesting that the use of generating functions is avoided here. As it is well known, generating functions may fail to be globally defined for some maps. One advantage of not using generating functions is that the result given here can be applied directly to non-twist maps.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J10
zbMATH DE Number: 5321409
zbMATH Keywords: approximation
zbMATH Keywords: smoothing
zbMATH Keywords: symplectic maps
zbMATH Keywords: volume-preserving maps
zbMATH Keywords: contact maps
zbMATH Keywords: KAM tori
zbMATH Keywords: uniqueness
zbMATH Keywords: bootstrap of regularity
reviewed by: Xiaoping Yuan
MaRDI profile type: Publication
full work available at URL: j.jde.2008.05.009
OpenAlex ID: W2015592938
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DOI: 10.1016/J.JDE.2008.05.009
description: scientific article; zbMATH DE number 5321409