Continuity of the measure of the spectrum for discrete quasiperiodic operators

Abstract

We study discrete Schroedinger operators $(H_{α,θ}ψ)(n)= ψ(n-1)+ψ(n+1)+f(αn+θ)ψ(n)$ 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 $H_{α,θ}$ to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of $H_{α,θ}$ are positive. For the almost Mathieu operator ($f(x)=2λ\cos 2πx$) it follows that the measure of the spectrum is equal to $4|1-|λ||$ for all real $θ$, $λ\ne\pm 1$, and all irrational $α$.