Last Y.
Zero Measure Spectrum for the Almost Mathieu Operator
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ABSTRACT. We study the almost Mathieu operator:
$(H_{\alpha,\lambda,\theta}u)(n)=
u(n+1)+u(n-1)+\lambda\cos (2\pi\alpha n+\theta)u(n)$,
on $l^2(Z)$, and show that for all $\lambda,\theta$, and
(Lebesgue) a.e.\ $\alpha$, the Lebesgue measure of its spectrum is
precisely $|4-2|\lambda||$. In particular, for $|\lambda|=2$ the
spectrum is a zero measure cantor set. Moreover, for a large set of
irrational $\alpha$'s (and $|\lambda|=2$) we show that the Hausdorff
dimension of the spectrum is smaller than or equal to $1/2$.