Renormalization and universality of the Hofstadter spectrum

Abstract

We consider a renormalization transformation $R$ for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of $R$ determine universal constants analogous to the critical exponents in the theory of phase transitions. Restricting to skew-product maps over a circle-rotations by the golden mean, we find several periodic orbits for $R$, and we conjecture that there are infinitely many. Interestingly, all scaling factors that have been determined to high accuracy appear to be algebraically related to the circle-rotation number. We present evidence that these values describe (among other things) local scaling properties of the Hofstadter spectrum.

References in corpus (2)

• A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets
• Golden mean renormalization for the almost Mathieu operator and related skew products

Cited by in corpus (2)

• Golden mean renormalization for the almost Mathieu operator and related skew products
• Asymptotic scaling and universality for skew products with factors in SL(2,R)