Consider a quasi-periodic Schr\"odinger operator
$H_{\alpha,\theta}$ with analytic potential and Diophantine frequency
$\alpha$. Given any rational approximating $\alpha$, let $S_+$ and $S_-$
denote the union, respectively, the intersection of the spectra taken over
$\theta$. We show that up to sets of zero Lebesgue measure, the absolutely
continuous spectrum can be obtained asymptotically from $S_-$ of the
periodic operators associated with the continued fraction expansion of
$\alpha$. Similarly, from the asymptotics of $S_+$, one recovers the
spectrum of $H_{\alpha,\theta}$ (modulo a set of zero Lebesgue measure).

I will review recent work linking quantum dynamical estimates
and rates of mixing in fluid flow. The main result is a sharp classification
of stationary or time periodic flows that are particularly efficient mixers.
I will also formulate some open questions.

The Ten Martini problem asked to show that the spectrum
of the Almost-Mathieu operator, that is, a Schroedinger operator with potential V (n) = 2 cos(2 pi alpha n), is a Cantor set. In particular, this means that the spectrum does not contain any intervals. The Ten Martini problem was solved in 2009 by Avila and Jitomirskaya. I will show that such a claim is false for the generalization to the potential V(n) =2 cos(2 pi alpha n^2), which is known as skew-shift Schroedinger operator. The proof relies on localization properties of this operator and that the phase space of the skew-shift is two dimensional, whereas it is one dimensional for the rotations underlying the Almost-Mathieu operator.