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.