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PIECEWISE ABSOLUTELY CONTINUOUS COCYCLES OVER IRRATIONAL ROTATIONS

Published online by Cambridge University Press:  01 February 1999

A. IWANIK ,
M. LEMAŃCZYK  and
C. MAUDUIT
A. IWANIK
Affiliation:
Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland. E-mail: [email protected]
M. LEMAŃCZYK
Affiliation:
Department of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. E-mail: [email protected]
C. MAUDUIT
Affiliation:
Institut de Mathématiques de Luminy, UPR 9016 CNRS, 163 av. de Luminy, 13288 Marseille Cedex 9, France. E-mail: [email protected]
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Abstract

For an irrational rotation α of the circle group T=R/Z and a piecewise absolutely continuous function f@[ratio ]T→R, the unitary operator Vh(x)=e2πif(x)h(x+α) on L2(T) is studied. It is shown that if f has a single discontinuity with non-integer jump then V is κ-weakly mixing for some κ with 0<[mid ]κ[mid ]<1. In particular V has continuous singular spectrum. The property of κ-weak mixing (with possible change of the value of κ, 0<[mid ]κ[mid ]<1) holds for all irrational rotations and, given α, 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 α and Lebesgue for another.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 59 , Issue 1 , February 1999 , pp. 171 - 187
Copyright
The London Mathematical Society 1999

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