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On cocycles with values in the group SU(2)

Krzysztof Fraczek
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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 $ϕ:S^1\to SU(2)$ over an irrational rotation by $α$ has nonzero degree, then the skew product $S^1\times SU(2)\ni(x,g)\mapsto (x+α,gϕ(x))\in S^1\times SU(2)$ 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.

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Krzysztof Fraczek. 2000-02-15. On cocycles with values in the group SU(2). https://arxiv.org/abs/math/0002120

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