A relation between a.c. 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_{α, λ, θ}u)(n)=u(n+1)+u(n-1)+λ cos(2παn+θ)u(n), and prove the existence of a.c. spectrum for sufficiently small λ, all irrational α's, and a.e. θ. Moreover, for 0≤λ<2 and (Lebesgue) a.e. pair α, θ, we prove the explicit equality of measures: |σ_ac|=|σ|=4 -2λ.