Communications in Mathematical Physics

, Volume 358, Issue 2, pp 741–766 | Cite as

Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in \({\mathbb{T} ^{d} \times SU(2)}\)

  • Nikolaos Karaliolios
Open Access


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.


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Authors and Affiliations

  1. 1.Imperial College LondonLondonUK

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