Silvius Klein: Continuity, positivity and simplicity of the Lyapunov exponents for linear quasi-periodic cocycles

Abstract: The purpose of this talk is to describe a recent result on the continuity of the Lyapunov exponents for 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. This in turn implies sharp lower bounds on the Lyapunov exponents of Schrödinger-type operators, as well as a sufficient condition ensuring that they have multiplicity one. Various kinds of estimates on pluri-subharmonic functions are essential in this work, and their use will be emphasized during the talk.

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