Let T be an ergodic and free Z(d)-rotation on the d-dimensional torus T-d given by T-(m1,T-...,T- md)(z(1),..., z(d)) = (e(2 pi i(alpha 11m1 +..+ alpha 1dmd)) z(1),..., e(2 pi i(alpha d1m1 +...+ alpha ddmd))z(d)), where (m(1),..., m(d)) epsilon Z(d), (z(1),..., z(d)) epsilon T-d and [alpha(jk)](j,k-1,...,d) epsilon M-d(R). For a continuous circle cocycle phi:Z(d) x T-d --> T (phi(m1n)(z) = phi(m)(T(n)z) phi(n)(z) for any m, n epsilon Z(d)), the winding matrix W(phi) of a cocycle phi, which is a generalization of the topological degree, is defined. Spectral properties of extensions given by T-phi:Z(d) x T-d x T --> T-d x T, (T-phi)(m) (z, omega) = (T-m z, phi,(z) omega) are studied. It is shown that if phi is smooth (for example phi is of class C-1) and det W(phi) not equal 0, then T-phi is mixing on the orthocomplement of the eigenfunctions of T. For d = 2 it is shown that if phi is smooth (for example phi is of class C-4), det W(phi) not equal 0 and T is a Z(2)-rotalion of finite type, then T-phi has countable Lebesgue spectrum on the orthocomplement of the eigenfunctions of T. If rank W(phi) = 1, then T-phi has singular spectrum. [References: 13]
展开▼
