On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product is not ergodic and the group of essential values of ϕ is equal to the maximal Abelian subgroup of SU(2). Moreover, if ϕ is of class C ² (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity. Possible values of degree are discussed, too. (Received 8 February 2000; in revised form 26 September 2000)