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Israel Journal of Mathematics

, Volume 69, Issue 1, pp 65–74 | Cite as

Skew products over the irrational rotation

  • D. A. Pask
Article

Abstract

Here we give conditions on a class of functions defining skew product extensions of irrational rotations on T which ensure ergodicity. These results produce extensions of the work done by P. Hellekalek and G. Larcher [HL1] and [HL2] to the larger class of functions which are piecewise absolutely continuous, have zero integral and have a derivative which is Riemann integrable with a non-zero integral.

Keywords

Borel Function Irrational Number Fixed Proportion Continue Fraction Expansion Irrational Rotation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [C]
    P. Cohn,Measure Theory, Birkhauser, Boston, 1980.zbMATHGoogle Scholar
  2. [HL1]
    P. Hellekalek and G. Larcher,On the ergodicity of a class of skew products, Isr. J. Math.54 (1986), 301–306.zbMATHMathSciNetGoogle Scholar
  3. [HL2]
    P. Hellekalek and G. Larcher,On Weyl sums and skew products over irrational rotations, preprint, Salzburg, 1987.Google Scholar
  4. [K]
    Y. Katznelson,Sigma-finite invariant measures for smooth mappings of the circle, J. Analyse33 (1977), 1–18.CrossRefGoogle Scholar
  5. [O]
    I. Oren,Ergodicity of cylinder flows arising from irregularities of distribution, Isr. J. Math.44 (1983), 127–138.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [R]
    W. Rudin,Fourier Analysis on Groups, Interscience, Wiley, 1967.Google Scholar
  7. [S]
    K. Schmidt,Cocycles of ergodic transformation groups, Lecture Notes in Mathematics, Vol. 1, MacMillan Co. of India, New Delhi, 1977.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1990

Authors and Affiliations

  • D. A. Pask
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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