Asymptotic scaling and universality for skew products with factors in SL(2, $\boldsymbol {\mathbb {R}}$ )

We consider skew-product maps over circle rotations x 7→ x+α (mod 1) with factors that take values in SL(2,R). This includes maps of almost Mathieu type. In numerical experiments, with α 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.

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