  We consider skew-product maps over circle rotations $x\mapsto x+\alpha$ (mod
1) with factors that take values in SL(2,R). This includes maps of almost
Mathieu type. In numerical experiments, with $\alpha$ the inverse golden mean,
Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic
scaling behavior that is reminiscent of critical phase transitions. In a
restricted setup that is characterized by a symmetry, we prove that critical
behavior indeed occurs and is universal in an open neighborhood of the almost
Mathieu family. This behavior is governed by a periodic orbit of a
renormalization transformation. An extension of this transformation is shown to
have a second periodic orbit as well, and we present some evidence that this
orbit attracts supercritical almost Mathieu maps.
