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Effective bounds for the measure of rotations

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A fundamental question in dynamical systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations.

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(c) IOP Publishing & London Mathematical Society , 2019

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Articles publicats en revistes (Matemàtiques i Informàtica)

# metadata.dc.title[].value
Effective bounds for the measure of rotations