Anderson localization for the almost Mathieu equation, III. Semi-uniform localization, continuity of gaps, and measure of the spectrum

We show that the almost Mathieu operator, (Hωλθψ)(η) = ψ(η+1) + ψ(η-1)+ λ cos(πωη+θ)ψ(η), has semi-uniform (and thus dynamical) localization for λ > 15 and a.e. ω, θ. We also obtain a new estimate on gap continuity (in ω) for this operator with λ > 29 (or λ < 4/29), and use it to prove that the measure of its spectrum is equal to |4 -2|λ|| for A in this range and all irrational ωs.