On cocycles with values in the group SU(2)

TL;DR

This paper introduces a degree concept for $C^1$-cocycles over irrational circle rotations with values in SU(2), revealing how nonzero degree affects ergodicity, spectral properties, and the structure of the cocycle.

Contribution

It defines the degree for $C^1$-cocycles in SU(2) and analyzes its implications on ergodicity, essential values, and spectral multiplicity, extending understanding of cocycle dynamics.

Findings

Nonzero degree implies non-ergodicity of the skew product.

The group of essential values equals the maximal Abelian subgroup of SU(2).

For $C^2$-cocycles, the spectrum has countable multiplicity.

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.