On cocycles with values in the group SU(2)

Abstract

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 $\phi:S^1\to SU(2)$ over an irrational rotation by $\alpha$ has nonzero degree, then the skew product $S^1\times SU(2)\ni(x,g)\mapsto (x+\alpha,g\phi(x))\in S^1\times SU(2)$ is not ergodic and the group of essential values of $\phi$ is equal to the maximal Abelian subgroup of SU(2). Moreover, if $\phi$ 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.