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.
Highlights
In the present article we investigate the dynamical properties of the complement of this class in the particular case of the cocycles whose frequency α satisfies a Diophantine condition and where the cocycle is C∞-close to a constant one
The question of differentiable rigidity of measurable reducibility was investigated in [Kar17], where we proved that measurable reducibility to a full measure set of constants DCα ⊂ G implies smooth reducibility
The Proof of Theorem 1.1 is based on the use of the K.A.M. normal form, in order to prove that the existence of an eigenfunction of the Koopman operator associated to a given cocycle implies the condition for measurable reducibility, provided that the eigenfunction depends non-trivially on the variable in the fibers
Summary
Conjugations of the size of those produced by the K.A.M. scheme without any resonances or of those who reduce a given cocycle to its K.A.M. normal form will be referred to as close-to-the-identity conjugations and will be denoted by V Such conjugations satisfy a condition of the type Y 0 < C and Y s0−ξγ < 1, for some constants ε ≤ C < 1 and 1 ≤ ξ < s0/γ depending on γ , τ and d. The Proof of Theorem 1.1 is based on the use of the K.A.M. normal form, in order to prove that the existence of an eigenfunction of the Koopman operator associated to a given cocycle implies the condition for measurable reducibility, provided that the eigenfunction depends non-trivially on the variable in the fibers. The theorem is slightly stronger than a corollary of the almost reducibility theorem, and we stress it since the analysis of the K.A.M. normal form shows that, any class of cocycles can serve as the linear model, admittedly using as a basis the class of constant cocycles
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
