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Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi‐periodic cocycles in Td×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 Td, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield–Wallach conjecture in PDEs. [ABSTRACT FROM AUTHOR]

We continue our study of the local theory for quasiperiodic cocycles in Td × G, where G = SU(2), over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the space orthogonal to L2(Td) → L2(Td × G). Subsequently, we describe the equivalence classes of cocycles under smooth conjugacy, as a function of the parameters defining their K.A.M. normal form. Finally, we derive a complete classification of the dynamics of one-frequency (d = 1) cocycles over a recurrent Diophantine rotation. [ABSTRACT FROM AUTHOR]