CIRCLE EXTENSIONS OF ℤ^d-ROTATIONS ON THE d-DIMENSIONAL TORUS

Abstract

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

formula here

where (m₁, …, m_d) ∈ ℤ^d, (z₁, …, z_d) ∈ [ ]^d and [α_jk]_{j,k=1 …, d} ∈ M_d(ℝ). For a continuous circle cocycle ϕ[ratio ]ℤ^d × [ ]^d → [ ](ϕ_m+n(z) = ϕ_m(T_nz)ϕ_n(z) for any m, n ∈ ℤ^d), 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 T_ϕ 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 T_ϕ has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(ϕ) = 1, then T_ϕ has singular spectrum.