Renormalization, unstable manifolds, and the fractal structure of mode locking

The apparent universality of the fractal dimension of the set of quasiperiodic windings at the onset of chaos in a wide class of circle maps is described by construction of a universal one-parameter family of maps which lies along the unstable manifold of the renormalization group. The manifold generates a universal ``devil's staircase'' whose dimension agrees with direct numerical calculations. Applications to experiments are discussed.