A classical result by Pál Turán, estimates the global behavior
of an exponential polynomial on an interval by its supremum on any
arbitrary subinterval. In this talk we discuss Nazarov's extension of this
``global to local reduction'' to arbitrary Borel sets of positive Lebesgue
measure.

Using the idea of "acceleration", which is first introduced by Avila, I will construct a
cerntain class of analytic quasiperiodic Szego cocycles with uniformly positive LE.

In this talk, we will introduce some advances on non-analytic quasi-periodic cocycles, including Schrodinger and non-Schrodinger situations. Moreover, we will discuss some conjectures in this area.

The talk is split into two parts. In the first half we present a strategy
to prove absence of point spectrum, on the example of the self-dual regime
of extended Harper's model for all but countably many phases and almost
all frequencies. The starting point is a dynamical formulation of Aubry
duality via rotation reducibility, previously used by Avila and
Jitomirskaya for the almost Mathieu operator.

The second half of the seminar is devoted to some on-going work on the
Lyapunov exponent (LE) of a quasi-periodic Schr\"odinger cocycle whose
potential is a trigonometric polynomial. Based on the strategy of ``almost
constant cocycles,'' we obtain upper bounds for the phase-complexified LE.
This allows to give an estimate on the regime of sub-critical behavior,
therefore complementing the classical results of Herman's on positivity of
the LE. Within the framework of Avila's global theory, sub-critical
behavior implies purely absolutely continuous spectrum for all phases.

We discuss d+1 dimensional percolation models with d dimensional
quasiperiodic disorder. A multiscale scheme is introduced which is suited
to the spatial structure of quasiperiodic disorder. In this case we will
show almost sure stretched exponential decay of correlations as compared
to faster than polynomial decay of correlations obtained for similar
models with random disorder. We mention in this case a disorder-rated
transition of phase structure.