COVERING PROPERTY OF HOFSTADTERS BUTTERFLY

Based on a thorough numerical analysis of the spectrum of Harper's ope rator, 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 a of the operator its spectrum can be covered by the bands of the spectrum for every rational approximant of a after stretching t hem by factors with a common upper bound. We show that this conjecture has the following important consequences: For all irrational values o f cr the spectrum is (i) a zero measure Cantor set and has (ii) a Haus dorff dimension less than or equal to 1/2. We propose that our numeric al approach may be a guide in finding;a rigorous proof of these result s. [S0163-1829(98)04036-3].