Continuous spectrum or measurable reducibility for quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$

Abstract

We continue our study of the local theory for quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, 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 $L^{2}(\mathbb{T} ^{d}) \hookrightarrow L^{2}(\mathbb{T} ^{d} \times G)$. 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. All theorems will be stated sharply in terms of the number of frequencies $d$, but in the proofs we will always assume $d=1$, for simplicity in expression and notation.