Abstract
We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.
Highlights
In the author’s PhD thesis [Kar13], the study of quasiperiodic cocycles in Td × G, with G a semisimple compact Lie group, over a Diophantine rotation and satisfying a closeness to constants assumption was revisited
The basic reference for the subject is [Kri99], where the corresponding local density theorem is proved in the C∞ category by means of a K.A.M. scheme
The problem of loss of periodicity was settled outside the iterative step of the scheme and made necessary the combination of the local almost quasi-reducibility theorem with the reducibility theorem in a positive measure set of parameters in order to obtain a proof of local almost reducibility
Summary
In the author’s PhD thesis [Kar13], the study of quasiperiodic cocycles in Td × G, with G a semisimple compact Lie group, over a Diophantine rotation and satisfying a closeness to constants assumption was revisited. We were able to deal with this complication by proving a more efficient local conjugation lemma, in which the phenomenon of longer periods is no longer present, and which can serve as an iterative step of a K.A.M. scheme This improved scheme can be used in the proof of a local differentiable rigidity theorem which improves the one obtained in [HY09]. The assumption that deg(α, A(·)) = 0, which we will not define here, assures that renormalization (see [Kri01],[AFK01],[AK06],[Kar13]) converges to constants This fact, combined with the assumption α ∈ RDC ( of full measure in T), implies that there exists D (·) ∈ C∞(T, G), such that ConjD(·)(α, A(·)) satisfies the assumptions of theorem 1.1. We will not come back to the proof of this theorem, since its details exceed the scope of a short note
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