Universal scaling in circle maps

Abstract

We discuss scaling, renormalization, and universality for critical and subcritical circle maps. Using the structure of the rationals provided by the Farey tree, we construct a pair of renormalization group operators for circle maps with all values of the winding number. In this general formulation, the renormalization equations define a universal invariant set which determines all universal quantities for all cubic maps with all irrational winding numbers. As expected, the manifold of pure rotations defines the subcritical attracting set. We show that the invariant set governing the critical circle maps cannot be embedded in less than 3 dimensions. However, we are able to obtain a differentiable parametrization of the unstable manifold of the critical invariant set, which we believe is universal up to Lipschitz homeomorphisms; the derivative of this function describes the scaling structure of all small gaps in the devil's staircase for all critical maps. Our work can be viewed as providing numerical evidence for Lanford's conjectures on the strange set underlying renormalizations of critical circle maps.