Multidimensional Continued Fractions,Dynamical Renormalization and KAM Theory |
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Authors: | Kostya Khanin João Lopes Dias Jens Marklof |
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Institution: | (1) Department of Mathematics, University of Toronto, 100 St George Street, Toronto, Ontario, M5S 3G3, Canada;(2) Departamento de Matemática, ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, Lisboa, 1200-781, Portugal;(3) School of Mathematics, University of Bristol, Bristol, BS8 1TW, U.K. |
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Abstract: | 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
(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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