Let T be an ergodic and free ℤdrotation on the d-dimensional torus [ ]d given by

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where (m1, …, md) ∈ ℤd, (z1, …, zd) ∈ [ ]d and [αjk]j,k=1 …, d ∈ Md(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤd × [ ]d → [ ](ϕm+n(z) = ϕm(Tnz)ϕn(z) for any m, n ∈ ℤd), the winding matrix W(ϕ) of a cocycle ϕ, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by

formula here

are studied. It is shown that if ϕ is smooth (for example ϕ is of class C1) and det W(ϕ) ≠ 0, then Tϕ is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if ϕ is smooth (for example ϕ is of class C4), det W(ϕ) ≠ 0 and T is a ℤ2-rotation of finite type, then Tϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then Tϕ has singular spectrum.