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Effective bounds for the measure of rotations

Hamiltonian systems, from topology to applications through analysis I October 08, 2018 - October 12, 2018

October 11, 2018 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): Alex Haro (University of Barcelona)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability, linearization, etc).  In this talk, 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 which are valid for a global set of rotations.

This is joint work with Jordi Lluis Figueras and Alejandro Luque.

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