A RELATION BETWEEN AC SPECTRUM OF ERGODIC JACOBI MATRICES AND THE SPECTRA OF PERIODIC APPROXIMANTS

We study ergodic Jacobi matrices on l2(Z), and prove a general theorem relating their a.c. spectrum to the spectra of periodic Jacobi matrices, that are obtained by cutting finite pieces from the ergodic potential and then repeating them. We apply this theorem to the almost Mathieu operator: (H(alpha,lambda,theta)u)(n) = u(n + 1) + u(n - 1) + lambda cos(2pialphan + theta)u(n), and prove the existence of a.c. spectrum for sufficiently small lambda, all irrational alpha's, and a.e. theta. Moreover, for 0 less-than-or-equal-to lambda < 2 and (Lebesgue) a.e. pair alpha, theta, we prove the explicit equality of measures: \sigma(ac)\ = Absolute value of sigma = 4 - 2lambda.