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).