A renormalization group approach to quasiperiodic motion with Brjuno frequencies

We introduce a renormalization group scheme that applies to vector fields on Td×Rm with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, due to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued fractions expansion. Our results concerning invariant tori generalize those of reference [17] from Diophantine to Brjuno type frequency vectors. In particular, each Brjuno vector ω ∈R determines an analytic manifoldW of infinitely renormalizable vector fields, and each vector field onW is shown to have an elliptic invariant d-torus with frequencies ω1, ω2, . . . , ωd .