Analytic smoothing of geometric maps with applications to KAM theory

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:245

Article

Gonzalez-Enriquez, A.¹  de la Llave, R.² 

[1] Univ Camerino, Dipartimento Matemat & Informat, I-62032 Camerino, MC, Italy

[2] Univ Texas Austin, Dept Math, Austin, TX 78712 USA

关键词: approximadon;    smoothing;    symplectic maps;    volume-preserving maps;    contact maps;    KAM tori;    uniqueness;    bootstrap of regularity;   

DOI  :  10.1016/j.jde.2008.05.009

来源: Elsevier

【 摘 要 】

We show 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. We prove 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. As an application to this result, we 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. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-clegeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic. (C) 2008 Elsevier Inc. All rights reserved.

【 授权许可】

Free