Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T00:57:23.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic measures for the irrational rotation on the circle

Part of: Harmonic analysis in one variable Measure-theoretic ergodic theory

Published online by Cambridge University Press:  09 April 2009

William Moran
William Moran
Affiliation:
Department of Pure MathematicsThe University of AdelaideAdelaide, South Australia 5000, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Riesz products are employed to give a construction of quasi-invariant ergodic measures under the irrational rotation of T. By suitable choice of the parameters such measures may be required to have Fourier-Stieltjes coefficients vanishing at infinity. We show further that these are the unique quasi-invariant measures on T with their associated Radon-Nikodym derivative.

MSC classification

Secondary: 28D99: None of the above, but in this section 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 133 - 141
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Baggett, L., Carey, A. L., Ramsay, A. and Moran, W., in preparation.Google Scholar
[2]Brown, Gavin, ‘Riesz products and generalised characters’, Proc. London Math. Soc. (3) 27 (1973), 484–504.CrossRefGoogle Scholar
[3]Brown, G. and Moran, W., ‘A dichotomy for infinite convolutions of discrete measures’, Proc. Cambridge Philos. Soc. 73 (1973), 307–316.CrossRefGoogle Scholar
[4]Katznelson, Y. and Weiss, B., ‘The construction of quasi-invariant measures’, Israel J. Math. 12 (1972), 1–4.CrossRefGoogle Scholar
[5]Keane, M., ‘Sur les mesures quasi-ergodiques des translations irrationelles’, C. R. Acad. Sci. Paris 272 (1971), 54–55.Google Scholar
[6]Krieger, W., ‘On quasi-invariant measures in uniquely ergodic systems’, Invent. Math. 14 (1971), 184–196.CrossRefGoogle Scholar
[7]Salem, R., Algebraic numbers and Fourier analysis, (Heath, Boston, Mass., 1963).Google Scholar
[8]Zygmund, A., Trigonometric series, Vol. I, (Cambridge University Press 1959).Google Scholar