Numerical study of rotation numbers and its variationals for parametric families of circle diffeomorphisms

Abstract

We present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy. Our methodology is an extension of an existing approach to compute rotation numbers, that it is based on suitable averages of the iterates of the map and Richardson extrapolation. In order to justify the method, we require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine. The method is used to compute the Taylor expansions of Arnold tongues with high precision.