Renormalization of Diophantine skew flows, with applications to the reducibility problem

Abstract

We introduce a renormalization group framework for the study of quasiperiodic skew flows on Lie groups of real or complex ռxռ matrices, for arbitrary Diophantine frequency vectors in ℝᵈ and dimensions d,n. In cases where the Lie algebra component of the vector field is small, it is shown that there exists an analytic manifold of reducible skew systems, for each Diophantine frequency vector. More general near-linear flows are mapped to this case by increasing the dimension of the torus. This strategy is applied for the group of unimodular 2x2 matrices, where the stable manifold is identified with the set of skew systems having a fixed fibered rotation number. Our results apply to vector fields of class Cγ, with γ depending on the number of independent frequencies, and on the Diophantine exponent.