Derivatives of rotation number of one parameter families of circle diffeomorphisms

Abstract

We consider the rotation number ρ(t) of a diffeomorphism f_t = R_t $\circ$ f, where R_t is the rotation by t and f is an orientation preserving C^∞ diffeomorphism of the circle S¹. We shall show that if ρ(t) is irrational $$\limsup_{t'\to t}(ρ(t′) − ρ(t)) / (t′ − t) ≥ 1.$$