HARMONIC SCALING FOR SMOOTH FAMILIES OF DIFFEOMORPHISMS OF THE CIRCLE

We consider one-parameter families f(t, z) of orientation-preserving diffeomorphisms of the circle which satisfy some natural assumptions. A prototype mapping of this type is the sine map y = x + OMEGA + (A/2-pi) sin(2-pi-x) (mod 1). Our goal is to give a mathematical proof of the universality of harmonic scaling in the case of families of diffeomorphisms. As a corollary we obtain that the rotation number depends Holder continuously on the parameter value with an exponent alpha-greater-than-or-equal-to 1/2. We also discuss the asymptotic behaviour of the considered scaling. As a conclusion we obtain that the typical Holder exponent is equal to 1/2.