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Journal of the London Mathematical Society 1999 59(1):171-187; doi:10.1112/S0024610799006961
© 1999 by London Mathematical Society
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© The London Mathematical Society

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

A. Iwanik, M. Lemanczyk and C. Mauduit

Institute of Mathematics, Technical University of Wroclaw Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland. E-mail: iwanik{at}im.pwr.wroc.pl
Department of Mathematics and Computer Science, Nicholas Copernicus University ul. Chopina 12/18, 87-100 Torun, Poland. E-mail: mlem{at}mat.uni.torun.pl
Institut de Mathématiques de Luminy UPR 9016 CNRS, 163 av. de Luminy, 13288 Marseille Cedex 9, France. E-mail: mauduit{at}iml.univ-mrs.fr

Received 13 May 1996. Revision received 2 August 1996.

For an irrational rotation {alpha} of the circle group T=R/Z and a piecewise absolutely continuous function f:T->R, the unitary operator Vh(x)=e2{pi}if(x)h(x+{alpha}) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is {kappa}-weakly mixing for some {kappa} with 0<|{kappa}|<1. In particular V has continuous singular spectrum. The property of {kappa}-weak mixing (with possible change of the value of {kappa}, 0<|{kappa}|<1) holds for all irrational rotations and, given {alpha}, is stable under perturbations of f by functions with sufficiently small O(1/n)-norm. On the other hand, there exists a piecewise linear function f with two non-integer jumps such that the spectrum of V is continuous singular for one value of {alpha} and Lebesgue for another.


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