Derivatives of rotation number of one parameter families of circle diffeomorphisms

Abstract

We consider the rotation number $\rho (t)$ of a diffeomorphism $f_t=R_t\circ f$, where $R_t$ is the rotation by $t$ and $f$ is an orientation preserving $C^{\infty }$ diffeomorphism of the circle $S^1$. We shall show that if $\rho (t)$ is irrational $$\underset{t^{\prime }\to t}{\mathrm{limsup}}(\rho (t^{\prime })-\rho (t))/(t^{\prime }-t)\ge 1.$$

9 pages, 1 figure