ROTATION NUMBER AND ONE-PARAMETER FAMILIES OF CIRCLE DIFFEOMORPHISMS

We consider one-parameter families of circle diffeomorphisms, f(t)(x) = f(x) + t (t is-an-element-of S1), where f:S1 is a C(r)-diffeomorphism (r greater-than-or-equal-to 3). We show that, for Lebesgue almost every t is an element S1, the rotation number of f, is either a rational number or an irrational number of Roth type. In the former case, f(t) has periodic orbits and, in the latter case, f(t), is C(r-2)-conjugate to an irrational rigid rotation from well-known theorems of Herman and Yoccoz.