On cocyles with values in the group \(SU(2)\) (Q5931261)

This paper is devoted to cocycles with values in the group \(SU(2)\). To this end the author introduces the notion of degree for \(C^1\)-cocycles over irrational rotations on the circle with values in the group \(SU(2)\). He shows that if a \(C^1\)-cocycle \(\varphi:T\to SU(2)\) over an irrational rotation by \(\alpha\) has nonzero degree, then the skew product \[ T\times SU(2)\ni (x,g)\mapsto (x+\alpha, g\varphi(x))\in T\times SU(2) \] is not ergodic and the group of essential values of \(\varphi\) is equal to the maximal Abelian subgroup of \(SU(2)\). Moreover, if \(\varphi\in C^2\) (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity.