| Abstract: |
We consider skew-product maps over circle rotations
x→ 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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