Abstract:We give new examples of discrete Schrδdinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an x is that there is a z G X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all x G X if X derives from a primitive substitution. For potentials defined by circle maps, x n = ly($o + nu), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J.

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We study Birkhoff sums S k (α) = k j=1 X j (α) with X j (α) = g(jα) = log |2 − 2 cos(2πjα)| at the golden mean rotation number α = ( √ 5 − 1)/2 with periodic approximants pn/qn. The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena. We relate the boundedness of log averaged Birkhoff sums S k / log(k) and the convergence of Sq n (α) with the existence of an experimentally established limit function f (x) = limn→∞ S [xqn] (p n+1 /q n+1 ) − S [xqn] (pn/qn) on [0, 1] which satisfies a functional equation f (α) + α 2 f = β with a monotone function β. The limit limn→∞ Sq n (α) can be expressed in terms of f .

We consider Dirichlet series ζg,α(s) = P ∞ n=1 g(nα)e −λns for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series P ∞ n=1 g(nα)z n . We prove that a Dirichlet series ζ(s) = P ∞ n=1 g(nα)/n s has an abscissa of convergence σ0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence is smaller or equal than 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ(s) has an analytic continuation to the entire complex plane.

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