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http://hdl.handle.net/2445/192675
Title: | Effective bounds for the measure of rotations |
Author: | Haro, Àlex Luque, Alejandro, 1974- Figueras, Jordi-Lluís |
Keywords: | Sistemes dinàmics de baixa dimensió Teoria ergòdica Anàlisi numèrica Anàlisi d'intervals (Matemàtica) Low-dimensional dynamical systems Ergodic theory Numerical analysis Interval analysis (Mathematics) |
Issue Date: | 19-Dec-2019 |
Publisher: | IOP Publishing |
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 (à la Arnold) which are valid for a global set of rotations. |
Note: | Versió postprint del document publicat a: https://doi.org/10.1088/1361-6544/ab500d |
It is part of: | Nonlinearity, 2019, vol. 33, num. 2, p. 700-741 |
URI: | http://hdl.handle.net/2445/192675 |
Related resource: | https://doi.org/10.1088/1361-6544/ab500d |
ISSN: | 0951-7715 |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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