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.