We study relations between quantum dynamics and spectral properties, concentrating on spectral decompositions which arise from decomposing measures with respect to dimensional Hausdorff measures.

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We study the almost Mathieu operator: (H a^x^u ) (n) = u(n + 1) + u(n -1) + λ cos(2παn + θ)u(ri), on 1 2 (Z), and show that for all λ, 0, and (Lebesgue) a.e. α, the Lebesgue measure of its spectrum is precisely |4 -2|λ||. In particular, for |λ| = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational α's (and |λ| = 2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.

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We study ergodic Jacobi matrices on 1 2 (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 a xθ u) (n) = u(n + 1) + u(n -1) + λcos(2παn + θ)u(n), and prove the existence of a.c. spectrum for sufficiently small λ, all irrational α's, and a.e. 0. Moreover, for 0 < λ < 2 and (Lebesgue) a.e. pair α, 0, we prove the explicit equality of measures: |σ ac | = |σ| = 4 -2λ.