Cantor Spectrum for the Almost Mathieu Operator. Corollaries of localization, reducibility and duality

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, (Hb,φx)n = xn+1 + xn−1 + b cos (2πnω + φ) xn on l2(Z) and its associated eigenvalue equation to deduce that for b 6= 0,±2 and ω Diophantine the spectrum of the operator is a Cantor subset of the real line. This solves the so-called “Ten Martini Problem” for these values of b and ω. Moreover, we prove that for |b| 6 = 0 small enough or large enough all spectral gaps predicted by the Gap Labelling theorem are open.