DOCUMENTA MATHEMATICA, Extra Volume ICM II (1998), 779-787

L. H. Eliasson

Title: Reducibility and Point Spectrum for Linear Quasi-Periodic Skew-Products

We consider linear quasi-periodic skew-product systems on ${\Bbb T}^d\times G$ 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 has been 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. Fr\"ohlich-Spencer-Wittver and Sinai, independently, were able to prove that certain discrete Schr\"odinger equations sufficiently far from constant coefficients have pure point spectrum, which implies a dynamics completely different from systems with constant coefficients. In recent years these methods have been improved and in particular $SL(2,\Bbb{R})$ --- related to the the Schr\"odinger equation --- and $SO(3,\Bbb{R})$ have been well studied.

1991 Mathematics Subject Classification: 34, 58, 81

Keywords and Phrases: skew-product, quasi-periodicity, reducibility, point spectrum

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