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

TL;DR

This paper investigates the asymptotic scaling and universal behavior of skew-product maps with factors in SL(2,R), demonstrating critical phenomena and universality near the almost Mathieu family through theoretical proofs and numerical evidence.

Contribution

It proves the occurrence and universality of critical behavior in skew-product maps with SL(2,R) factors near the almost Mathieu family, using renormalization techniques.

Findings

Critical behavior is proven to occur and is universal near the almost Mathieu family.

Numerical experiments show asymptotic scaling reminiscent of phase transitions.

A second periodic orbit of the renormalization transformation attracts supercritical maps.

Abstract

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.