Piecewise Absolutely Continuous Cocycles Over Irrational Rotations |
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Authors: | Iwanik, A. Lemanczyk, M. Mauduit, C. |
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Affiliation: | Institute of Mathematics, Technical University of Wrocaw Wybrzee Wyspiaskiego 27, 50-370 Wrocaw, Poland. E-mail: iwanik{at}im.pwr.wroc.pl Department of Mathematics and Computer Science, Nicholas Copernicus University ul. Chopina 12/18, 87-100 Toru, 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 |
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Abstract: | For an irrational rotation of the circle group T=R/Z and apiecewise absolutely continuous function f:TR, the unitary operatorVh(x)=e2if(x)h(x+) on L2(T) is studied. It is shown that iff has a single discontinuity with non-integer jump then V is-weakly mixing for some with 0<||<1. In particular Vhas continuous singular spectrum. The property of -weak mixing(with possible change of the value of , 0<||<1) holdsfor all irrational rotations and, given , is stable under perturbationsof f by functions with sufficiently small O(1/n)-norm. On theother hand, there exists a piecewise linear function f withtwo non-integer jumps such that the spectrum of V is continuoussingular for one value of and Lebesgue for another. |
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