We consider averages $\kappa$ of spectral measures of rank one
perturbations with respect to a sigma-finite measure $\nu$. It is shownhow various degrees of continuity of $\nu$ with respect to Hausdorff measures are inherited by $\kappa$. This extends Kotani's trick where $\nu$ is simply the Lebesgue measure.

We recapture the orthofermion algebra by means of a q-deformed particle
algebra A_p(q) whose deformed number operator has a finite spectrum
which is given by the deformed integers 0,[1],...[p]

We consider one-dimensional Schrodinger operator with analytic potential
and a single frequency. Two ``local theories'' for such operators have
been developed extensively, and cover small and large potentials.
However, a global picture, which should in particular describe how one
moves from a regime to the other has remained elusive except in the case
of the almost Mathieu operator.

It turns out that, as in the case of the almost Mathieu, energies in the
spectrum can be always separated into three types (subcritical, critical
and supercritical), according to the Lyapunov exponent of the
(complexified) associated cocycle. Our focus is in the study of the
critical locus in the infinite dimensional parameter space.
Our analysis gives a detailed picture for the ``phase transitions''
between subcritical and supercritical regions in the spectrum of typical
operators. One of our tools is a surprising regularity property of the
Lyapunov exponent that emerges from a quantization phenomenon.