Golden mean renormalization for the almost Mathieu operator and related skew products (partly in collaboration with Sasa Kocic).

Hans Koch

We renormalize SL(2,R) skew-product maps over circle rotations. Such maps arise e.g. in the spectral analysis of the Hofstadter Hamiltonian and the almost Mathieu operator. For rotations by the inverse golden mean, we prove the existence of two renormalization-periodic orbits. We conjecture that there are infinitely many such orbits, and that the associated universal constants describe local scaling properties of the Hofstadter spectrum and of the corresponding generalized eigenvectors.

http://scgp.stonybrook.edu/video_portal/video.php?id=4509

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory