Scaling properties of ? k- 1 actions on the circle

Abstract

We derive universal scaling properties for ℤk−1 actions on the circle whose generators have rotation numbers algebraic of degreek. As fork=2 these properties can be explained for arbitraryk in terms of a renormalization group transformation. It has at least one trivial fixed point corresponding to an action whose generators are pure rotations. The spectrum of the linearized transformation in this fixed point is analyzed completely. The fixed point is hyperbolic with a (k−1)-dimensional unstable manifold. In the casek=2 the known results are therefore recovered.