Abstract
AbstractWe consider skew-product maps over circle rotations
$x\mapsto x+\alpha \;(\mod 1)$
with factors that take values in
${\textrm {SL}}(2,{\mathbb {R}})$
. 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics