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.

25 pages, 1 figure. arXiv admin note: text overlap with arXiv:1407.4763

References in corpus (1)

• Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups

Cited by in corpus (3)

• Self-duality triggered dynamical transition
• Fibered rotation vector and hypoellipticity for quasiperiodic cocycles in compact Lie groups
• Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,R)$ cocycles