# Golden mean renormalization for the almost Mathieu operator and related skew products

## TL;DR

This paper studies a renormalization transformation for certain mathematical operators, revealing a periodic orbit that helps understand spectral properties and scaling behaviors in related quantum systems.

## Contribution

It introduces a new renormalization approach with a specific periodic orbit for the almost Mathieu operator and related skew products, linking to spectral scaling phenomena.

## Key findings

- Existence of a nontrivial period 3 orbit in the renormalization transformation.
- Numerical evidence connecting the period 3 orbit to Hofstadter butterfly scaling.
- Insights into eigenfunction scaling near spectral edges.

## Abstract

Considering SL(2,R) skew-product maps over circle rotations, we prove that a renormalization transformation associated with the golden mean alpha has a nontrivial periodic orbit of length 3. We also present some numerical results, including evidence this period 3 describes scaling properties of the Hofstadter butterfly near the top of the spectrum at alpha, and scaling properties of the generalized eigenfunction for this energy.