Joaquim Puig
Cantor Spectrum for the Almost Mathieu Operator. Corollaries of 
localization,reducibility and duality.
(348K, Postscript)

ABSTRACT.  In this paper we use results on reducibility, localization and duality 
for the Almost Mathieu operator, 
\[ 
\left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n-1} + b \cos\left(2 \pi n \omega + 
\phi\right)x_n 
\] 
on $l^2(\mathbb{Z})$ and its associated eigenvalue equation to deduce 
that for $b \ne 0,\pm 2$ and 
$\omega$ 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 $\omega$. Moreover, we prove that 
for $|b|\ne 0$ small enough or large enough 
all spectral gaps predicted by the Gap Labelling theorem are open. 
(This is a revised version of preprint mp_arc 03-145).