ABSTRACT:

We prove that under some diophantine assumption on the fibered rotation number, an SL(2,R)-valued quasi-periodic cocycle over any irrational rotation on a circle (even Liouville) is conjugated to an SO(2,R)-valued cocycle. This result which is an extension of a theorem by Dinaburg and Sinai relies on a new KAM-type scheme which takes advantage of Liouvilean situation. This is a joint work with A. Avila and B. Fayad.