Reducibility and point spectrum for linear quasi-periodic skew-products

Abstract

The author considers linear quasi-periodic skew-product systems on \( \bbfT^d\times G \) where $G$ is some matrix group. When the quasi-periodic frequencies are Diophantine such systems can be studied by perturbation theory of KAM-type and it is known since the mid 60's that most systems sufficiently close to constant coefficients are reducible, i.e. their dynamics is basically the same as for systems with constant coefficients. In the late 80's a perturbation theory was developed for the other extreme. Fröhlich-Spencer-Wittver and Sinai, independently, were able to prove that certain discrete Schrödinger equations sufficiently far from constant coefficients have pure point spectra, which imply a dynamics completely different from systems with constant coefficients. In recent years these methods are improved and in particular \( {SL}(2,\bbfR) \) -- related to the Schrödinger equation -- and \( {SO}(3,\bbfR) \) are well studied.