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 note 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. Moreover, we prove that for $|b|$ small enough 
all spectral gaps predicted by the Gap Labelling theorem are open.

KEYWORDS: Almost Mathieu Operator, Cantor Spectrum, Ten Martini Problem.