  We consider a renormalization transformation $R$ for skew-product maps of the
type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic
orbits of $R$ determine universal constants analogous to the critical exponents
in the theory of phase transitions. Restricting to skew-product maps over a
circle-rotations by the golden mean, we find several periodic orbits for $R$,
and we conjecture that there are infinitely many. Interestingly, all scaling
factors that have been determined to high accuracy appear to be algebraically
related to the circle-rotation number. We present evidence that these values
describe (among other things) local scaling properties of the Hofstadter
spectrum.
