Effective bounds for the measure of rotations

TL;DR

This paper develops effective, computer-assisted methods to estimate the measure of parameter sets for which analytic circle diffeomorphisms are conjugate to rotations, providing explicit bounds and applying them to the Arnold family.

Contribution

It introduces an a-posteriori KAM scheme with verifiable conditions to obtain non-asymptotic measure bounds for conjugacy to rotations in analytic circle diffeomorphisms.

Findings

Derived explicit lower bounds for measure of conjugate parameters.

Applied methodology to the Arnold family case.

Produced asymptotic estimates valid for a broad set 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 (\`a la Arnold) which are valid for a global set of rotations.