Smooth Conjugacies to Rotations

We have seen in Chap. 3 that every sufficiently smooth diffeomorphism of the circle without periodic points is topologically conjugate to a rigid rotation. In other words, the topological orbit structure of such a diffeomorphism is indistinguishable from that of a rigid rotation. The relative order of points of a given orbit on the circle is the same no matter which orbit we take; everything is determined by a single invariant, the rotation number.