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C. e. 13) holds as |n| → ∞. Remark. This strengthens the result originally proven in [31] and improved in [7] in two ways. 13). e. distribution. Proof. e. e. k ∈ (0, π), (1 − 2α)λ2 log T (n) = . 13). The theorem follows from general principles on rank one perturbations [12, 4, 5, 28]. The case α = 21 has an extra subtlety we will need to deal with, using an argument modeled on Kotani-Ushiroya [21]. 7. Let uθ = (cos θ, sin θ) in R2 . For any unimodular matrix A with A > 1, let θ(A) be the unique θ ∈ (− π2 , π2 ] with Auθ = A −1 .

E. c. spectrum on (−2, 2). Remarks. 1. c. spectrum recovering results of Delyon, et al. [7]. 2. 1) becomes n E(Vω (n)2 ) < ∞; we state the general bound because unbounded V ’s are so easy to accommodate. 3. The case E(Vω (n)2 )1/2 = n−1/2 log(n)−1 is of some interest. c. spectrum. Kotani-Ushiroya [21] cannot handle such borderline cases. Proof. Fix θ0 . Then Rω (n) and θω (n) become random variables which are measurable functions of {Vω (j)}j≤n−1 and so independent of {Vω (j)}j≥n . 12c), R(n + 1)4 = R(n)4 1 + Vω (n) sin(2θ¯ω (n)) + O(Vω2 + Vω4 ) .

Chulaevsky, V. : Positive Lyapunov exponents for a class of deterministic potentials. Commun. Math. Phys. 168, 455–466 (1995) 3. , Last, Y. : Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization. J. d’Analyse Math. 69, 153–200 (1996) 4. , Makarov, N. : Operators with singular continuous spectrum, II. Rank one operators. Commun. Math. Phys. 165, 59–67 (1994) 5. , Simon, B. : Stability of spectral types for Sturm-Liouville operators. Math.

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