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.

We use the Lippmann Schwinger equations to derive a relation between the transfer and
the scattering matrix for a quasi one-dimensonal scattering problem with a periodic background
operator.
If the background operator has hyperbolic channels, then the scattering matrix is of smaller
dimension than the transfer matrix and related to a 'reduced' transfer matrix.

The continuity of Lyapunov exponent plays an important role for many problems in quasi-periodic cocycles. One example is Ten Martini problem. It is well known that the Lyapunov exponent is continuous in analytic topology and discontinuous in C^0-topology. In this talk, we will provide quasi-periodic cocycles at which the Lyapunov exponent is not continuous in C^l-topology with 0 ≤ l ≤ +∞. This is joint work with Jiangong You.

Discrete quasiperiodic Schrodinger operators have been researched extensively over the past thirty years to produce a rather complete spectral analysis when the potential is defined by analytic functions. However, the nature of the spectral measures for less than $C^\infty$ regularity of the potential is largely unknown. We demonstrate that, with only minimal assumptions on the regularity of the potential, in the regime of positive Lyapunov exponents, the spectral measures are always of
Hausdorff dimension zero.