Effective bounds for the measure of rotations

Jordi-Lluís Figueras; Alex Haro; Alejandro Luque

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arXiv · 1806.05517

Effective bounds for the measure of rotations

Jordi-Lluís Figueras·Alex Haro·Alejandro Luque

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Abstract

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 (\`a la Arnold) which are valid for a global set of rotations.

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BibTeX RIS

Jordi-Lluís Figueras, Alex Haro, Alejandro Luque. 2018-06-14. Effective bounds for the measure of rotations. https://arxiv.org/abs/1806.05517

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