Harmonic scaling for smooth families of diffeomorphisms of the circle

Abstract

The author considers 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). His goal is to give a mathematical proof of the universality of harmonic scaling in the case of families of diffeomorphisms. As a corollary he obtains that the rotation number depends Holder continuously on the parameter value with an exponent alpha >or=1/2. He also discusses the asymptotic behaviour of the considered scaling. As a conclusion he obtains that the typical Holder exponent is equal to 1/2.