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