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.
