  We consider skew-product maps over circle rotations $x\mapsto x+\alpha$ (mod
1) with factors that take values in SL(2,R) In numerical experiments with
$\alpha$ the inverse golden mean, Fibonacci iterates of almost Mathieu maps
with rotation number $1/4$ and positive Lyapunov exponent exhibit asymptotic
scaling behavior. We prove the existence of such asymptotic scaling for
"periodic" rotation numbers and for large Lyapunov exponent. The phenomenon is
universal, in the sense that it holds for open sets of maps, with the scaling
limit being independent of the maps. The set of maps with a given periodic
rotation number is a real analytic codimension $1$ manifold in a suitable space
of maps.
