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Hölder Continuity of the Rotation Number for Quasi-Periodic Co-Cycles in \({SL(2, \mathbb R)}\)

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Abstract

We prove two results on the rotation number of the skew-product system \({(\omega ,A):(\theta ,y)\in\mathbb T^d\times\mathbb R^2\mapsto (\theta +\omega ,A(\theta)y)\in\mathbb T^d\times\mathbb R^2,}\) where ω is Diophantine and \({A(\theta)\in SL(2, \mathbb R)}\) is homotopic to the identity. On the one hand, we prove that this function has the behavior of a \({\frac{1}{2}-}\) Hölder function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. We give also an estimate of the complement of the spectrum. These results are obtained by studying the reducibility of the quasi-periodic co-cycle (ω , A).

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  1. Ecole Nationale d’Ingénieurs de Monastir, Avenue Ibn Eljazzar, 5019, Monastir, Tunisia

    Sana Hadj Amor

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  1. Sana Hadj Amor
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Correspondence to Sana Hadj Amor.

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Communicated by B. Simon

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Hadj Amor, S. Hölder Continuity of the Rotation Number for Quasi-Periodic Co-Cycles in \({SL(2, \mathbb R)}\) . Commun. Math. Phys. 287, 565–588 (2009). https://doi.org/10.1007/s00220-008-0688-x

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