 03396 Joaquim Puig
 Cantor Spectrum for the Almost Mathieu Operator. Corollaries of
localization,reducibility and duality.
(348K, Postscript)
Sep 1, 03

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Abstract. In this paper we use results on reducibility, localization and duality
for the Almost Mathieu operator,
\[
\left(H_{b,\phi} x\right)_n= x_{n+1} +x_{n1} + b \cos\left(2 \pi n \omega +
\phi\right)x_n
\]
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 socalled ``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 03145).
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