Golden mean renormalization for the almost Mathieu operator and related skew products
arXiv:1907.06804 · doi:10.1063/5.0005429
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.
The source code for our programs, as well as data files, can be found at http://web.ma.utexas.edu/users/koch/papers/skewrg/