We describe some asymptotic properties of trigonometric skew-product maps over
irrational rotations of the circle. The limits are controlled using renormalization. The maps
considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero.
They are analogous to the almost Mathieu maps, but the factors commute. This allows us to
construct periodic orbits under renormalization, for every quadratic irrational, and to prove
that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits.
Analogous results are believed to be true for the self-dual almost Mathieu maps, but they seem
presently beyond reach.