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Journal of the London Mathematical Society 2000 61(1):139-162; doi:10.1112/S0024610799008510
© 2000 by London Mathematical Society
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© The London Mathematical Society

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

Krzysztof Fraczek

Faculty of Mathematics and Computer Science, Nicholas Copernicus University ul. Chopina 12/18, 87-100 Torun, Poland, fraczek{at}mat.uni.torun.pl

Received 3 December 1997. Revision received 29 October 1998.

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

T(m1,...md)(z1,...zd)=(e2{pi}i({alpha}11m1+...+1dmd))z1,...,e2{pi}i({alpha}d1m1+...+{alpha}ddmd)zd),

where (m1, ..., md) isin Zd, (z1, ..., zd) isin Td and [{alpha}jk]j,k=1 ..., d isin Md(R). For a continuous circle cocycle {Phi}:Zd x Td -> T({Phi}m+n(z) = {Phi}m(Tnz){Phi}n(z) for any m, n isin Zd), 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}:ZdxTdxT->TdxT, (T{Phi})m(z,{omega})=(TmZ,{Phi}m(z){omega})

are studied. It is shown that if {Phi} is smooth (for example {Phi} is of class C1) and det W({Phi}) != 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 C4), det W({Phi}) != 0 and T is a Z2-rotation 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.


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