Continuity of the measure of the spectrum for discrete quasiperiodic operators

We study discrete Schrödinger operators $(H_{\alpha,\theta}\psi)(n)= \psi(n-1)+\psi(n+1)+f(\alpha n+\theta)\psi(n)$ on $l^2(Z)$, where $f(x)$ is a real analytic periodic function of period 1. %For any irrational $\alpha$ and real $\theta$, we show that %if the corresponding Lyapunov exponent is a.e. positive then %$|S(\alpha,\theta)|=\lim_{n\to\infty}|\cup_{\theta\in\cal R} %S(p_n/q_n,\theta)|$, where $S(\beta,\theta)$ is the spectrum of %$H_{\beta,\theta}$, $|S(\beta,\theta)|$, its Lebesgue measure, and %${p_n/q_n}$ is the sequence of canonical rational approximants to $\alpha$. We prove a general theorem relating the measure of the spectrum of $H_{\alpha,\theta}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{\alpha,\theta}$ are positive. For the almost Mathieu operator ($f(x)=2\lambda\cos 2\pi x$) it follows that the measure of the spectrum is equal to $4|1-|\lambda||$ for all real $\theta$, $\lambda\ne\pm 1$, and all irrational $\alpha$.

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Published 1 January 2002