  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 SL(2,Z)\SL(2,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 explicitely 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.
