Effective bounds for the measure of rotations

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.

Normal form for lower dimensional elliptic tori in Hamiltonian systems

“…Again, by imposing further the first Melnikov condition (see Definition 2.1), we obtain analytic solutions. Finally, according to Equation (15), the new parametrization is…”

“…Another advantage of the parametrization method is that it can be quite easily converted to a computer-assisted proof, once that an analytical proof of the convergence of the algorithm to the solution is provided. This has already been done for what concerns KAM Lagrangian tori in [14] and [15], therefore the extension to the lower dimensional case appears quite natural.…”