RELATIVE DENSITY OF IRRATIONAL ROTATION NUMBERS IN FAMILIES OF CIRCLEDIFFEOMORPHISMS

Consider a one-parameter family of circle diffeomorphisms which unfold s a saddle-node periodic orbit at the edge of an 'Arnold tongue'. Rece ntly it has been shown that homoclinic orbits of the saddle-node perio dic points induce a 'transition map' which completely describes the sm ooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-nod e periodic points. We show that after the bifurcation the relative den sity (measure) of parameter values corresponding to irrational rotatio n numbers is completely determined by the transition map and give a fo rmula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases havi ng infinite co-dimension.