Circle Extensions of Z d -Rotations on the d -Dimensional Torus

Abstract

Let T be an ergodic and free Zdrotation on the d-dimensional torus Td given by T ( m 1 , … , m d ) ( z 1 , … , z d ) = ( e 2 π i ( α 11 m 1 + … + α 1 d m d ) z 1 , … , e 2 π i ( α d 1 m 1 + … + α d d m d ) z d ) , where (m1, …, md) ∈ Zd, (z1, …, zd) ∈ Td and [αjk]j,k=1 …, d ∈ Md(R). For a continuous circle cocycle Φ:Zd × Td → T(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 TΦ:Zd×Td×T→Td×T, (TΦ)m(z,ω)=(TmZ,Φm(z)ω) 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.