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

AUTHOR:
Joaquim Puig (puig@maia.ub.es)
Dept. de Matematica Aplicada, Universitat de Barcelona
Gran Via de les Corts Catalanes, 585, 08007 Barcelona (Spain)

ABSTRACT:
In this paper we use results on reducibility, localization and duality
for the Almost Ma\-thieu 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.