On cocycles with values in the group SU ( 2 )
نویسنده
چکیده
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 : T → SU (2) over an irrational rotation by α has nonzero degree, then the skew product T × SU (2) ∋ (x, g) → (x + α, g ϕ(x)) ∈ T × 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.
منابع مشابه
Institute for Mathematical Physics on Cocycles with Values in the Group Su (2) 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 ϕ : T → SU (2) over an irrational rotation by α has nonzero degree, then the skew product T × SU (2) (x, g) → (x + α, g ϕ(x)) ∈ T × SU (2) is not ergodic and the group of essential values of ϕ is equal to the maximal Abelia...
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