In some previous works, P. Duarte and I have developed an abstract scheme of
proving continuity properties of the Lyapunov exponents and of the Oseledets filtration
associated with general linear cocycles, by means of large deviation type estimates. The
purpose of this talk is to describe a recent result that fits this abstract scheme, concerning
analytic quasi-periodic cocycles on the higher dimensional torus. The main new feature
of this result is allowing the determinant of the matrix-valued function defining the linear
cocycle to vanish identically. As consequences of this result, we obtain sharp lower bounds
on the Lyapunov exponents of Schrödinger-type operators, as well as a sufficient condition
ensuring that they have multiplicity one.

[Joint work with Pedro Duarte from University of Lisbon.]