93-173 Last Y.

Zero Measure Spectrum for the Almost Mathieu Operator (31K, TeX (plain)) Jun 9, 93

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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 $1/2$.

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