Regularity of the Rotation Number for Circle Diffeomorphisms at Integrable Resonances

A one parameter family obtained by post-composing an orientation preserving circle diffeomorphism with rigid rotations is a simple and much studied model for tunable, periodically forced, non-linear oscillatory systems. The frequency, called in this context the rotation number, generically exhibits mode-locking at each rational value p/q (resonance), i.e., it remains unchanged under small perturbations. This results in the Devil's staircase-like non-smooth graph of the rotation number.
A seemingly obvious and embarrassingly wide-open conjecture asserts that absence of mode-locking, referred to as integrability, is only possible if the family is "linear" (in a suitable sense). A crucial part of the problem is showing that integrability implies infinite smoothness of the rotation number function. I will discuss a weak form of this function's higher-order regularity at resonances, in a simple result that extends the existing understanding of the first derivative. This is a part of a project with M.N. Graczyk (a master student at U. of Cambridge).