Numerical computation of rotation numbers of quasi-periodic planar curves

Recently, a new numerical method has been proposed to compute rotation numbers of analytic circle diffeomorphisms, as well as derivatives with respect to parameters, that takes advantage of the existence of an analytic conjugation to a rigid rotation. This method can be directly applied to the study of invariant curves of planar twist maps by simply projecting the iterates of the curve onto a circle. In this work we extend the methodology to deal with general planar maps. Our approach consists in computing suitable averages of the iterates of the map that allow us to obtain a new curve for which the direct projection onto a circle is well posed. Furthermore, since our construction does not use the invariance of the quasi-periodic curve under the map, it can be applied to more general contexts. We illustrate the method with several examples.