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 contribution||Multidimensional continued fractions, dynamical renormalization and KAM theory|
|Pages (from-to)||197 - 231|
|Number of pages||35|
|Journal||Communications in Mathematical Physics|
|Publication status||Published - Feb 2007|
Bibliographical notePublisher: Springer
Other identifier: IDS number 117PN