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We consider skew-product maps over circle rotations $x\mapsto x+\alpha$ (mod1) with factors that take values in SL(2,R). This includes maps of almostMathieu type. In numerical experiments, with $\alpha$ the inverse golden mean,Fibonacci iterates of maps from the almost Mathieu family exhibit asymptoticscaling behavior that is reminiscent of critical phase transitions. In arestricted setup that is characterized by a symmetry, we prove that criticalbehavior indeed occurs and is universal in an open neighborhood of the almostMathieu family. This behavior is governed by a periodic orbit of arenormalization transformation. An extension of this transformation is shown tohave a second periodic orbit as well, and we present some evidence that thisorbit attracts supercritical almost Mathieu maps.

Asymptotic scaling and universality for skew products with factors in SL(2,R)