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

Let T be an ergodic and free Z d rotation on the d ‐dimensional torus T d given byT ( m 1 , … , m d ) ( z 1 , … , z d ) = ( e 2 π i ( α 11m 1 + … + α 1 dm d )z 1 , … , e 2 π i ( α d 1m 1 + … + α d dm d )z d ) ,where ( m 1 , …, m d ) ∈ Z d , ( z 1 , …, z d ) ∈ T d and [α jk ] j,k =1 …, d ∈ M d (R). For a continuous circle cocycle Φ:Z d × T d → T(Φ m+n (z) = Φ m (T n z)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 T Φ :Z d ×T d ×T→T d ×T, ( T Φ ) m ( z ,ω)=( T m Z ,Φ m ( z )ω) are studied. It is shown that if Φ is smooth (for example Φ is of class C 1 ) 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 C 4 ), det W (Φ) ≠ 0 and T is a Z 2 ‐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.