Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory

被引:0
|
作者
Kostya Khanin
João Lopes Dias
Jens Marklof
机构
[1] University of Toronto,Department of Mathematics
[2] Universidade Técnica de Lisboa,Departamento de Matemática, ISEG
[3] University of Bristol,School of Mathematics
来源
Communications in Mathematical Physics | 2007年 / 270卷
关键词
Homogeneous Space; Continue Fraction; Fundamental Domain; Fourier Mode; Renormalization Scheme;
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摘要
The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL(d, \mathbb{Z}) \backslash SL(d, \mathbb{R})$$\end{document} (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension.
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页码:197 / 231
页数:34
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