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Absolutely continuous cocycles over irrational rotations

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Abstract

For homeomorphisms

$$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$

(z, w ∈S 1,α is irrational,ϕ:S 1→S 1) of the torusS 1×S 1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

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Authors and Affiliations

  1. Institute of Mathematics, Technical University of Wroclaw, 50-370, Wroclaw, Poland

    A. Iwanik

  2. Institute of Mathematics, Nicholas Copernicus University, ul. Chopina 12/18, 87-100, Toruń, Poland

    M. Lemańczyk

  3. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    D. Rudolph

Authors
  1. A. Iwanik
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  2. M. Lemańczyk
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  3. D. Rudolph
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Additional information

Research supported by KBN grant PB 666/2/91.

Research supported by KBN grant PB 521/2/91.

Research supported by NSF grant DMS 01524351.

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Iwanik, A., Lemańczyk, M. & Rudolph, D. Absolutely continuous cocycles over irrational rotations. Israel J. Math. 83, 73–95 (1993). https://doi.org/10.1007/BF02764637

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  • DOI: https://doi.org/10.1007/BF02764637

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