Differentiability at the tip of Arnold tongues for Diophantine rotations: numerical studies and renormalization group explanations

We study numerically the regularity of Arnold tongues corresponding to Diophantine rotation numbers of circle maps at the edge of validity of KAM theorem. This serves as a good test for the numerical stability of two different algorithms. We conclude that Arnold tongues are only finitely differentiable and we also provide a renormalization group explanation of the borderline regularity. Furthermore, we study numerically the breakdown of Sobolev regularity of the conjugacy close to the critical point and we provide explanations of asymptotic formulas found in terms of the scaling properties of the renormalization group. We also uncover empirically some other regularity properties which seem to require explanations.