Multidimensional continued fractions, dynamical renormalization and KAM theory

K Khanin, J Lopes Dias, J Marklof

Research output: Contribution to journalArticle (Academic Journal)peer-review

30 Citations (Scopus)


The disadvantage pf '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 SL(d,Z)\SL(d,R) (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. We explicitly construct renormalization schemes for (a) the linearization of vector fields on tori of arbitrary dimension and (b) the construction of invariant tori for Hamiltonian systems.
Translated title of the contributionMultidimensional continued fractions, dynamical renormalization and KAM theory
Original languageEnglish
Pages (from-to)197 - 231
Number of pages35
JournalCommunications in Mathematical Physics
Volume270 (1)
Publication statusPublished - Feb 2007

Bibliographical note

Publisher: Springer
Other identifier: IDS number 117PN


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