Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups

Abstract

Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in ${\mathbb{T}}^d\times G$, $G$ a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in ${\mathbb{T}}^d$, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in PDEs.

14 pages