S. Ya. Jitomirskaya, I. V. Krasovsky
Continuity of the measure of the spectrum for discrete 
quasiperiodic operators
(34K, LaTeX)

ABSTRACT.  We study discrete Schr\"odinger 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. 
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 $\alpha$.