A covering property of Hofstadter's butterfly

TL;DR

This paper conjectures a universal covering property of Hofstadter's butterfly spectrum for Harper's operator, linking rational approximations to the spectral structure of irrational cases, with implications for measure and fractal dimension.

Contribution

It introduces a conjecture about the spectral covering property of Harper's operator's spectrum across all incommensurability parameters, supported by numerical analysis.

Findings

Spectrum for irrational sigma is a zero measure Cantor set

Hausdorff dimension of the spectrum is at most 1/2

Numerical approach may guide rigorous proofs

Abstract

Based on a thorough numerical analysis of the spectrum of Harper's operator, which describes, e.g., an electron on a two-dimensional lattice subjected to a magnetic field perpendicular to the lattice plane, we make the following conjecture: For any value of the incommensurability parameter sigma of the operator its spectrum can be covered by the bands of the spectrum for every rational approximant of sigma after stretching them by factors with a common upper bound. We show that this conjecture has the following important consequences: For all irrational values of sigma the spectrum is (i) a zero measure Cantor set and has (ii) a Hausdorff dimension less or equal to 1/2. We propose that our numerical approach may be a guide in finding a rigorous proof of these results.