Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T05:09:20.992Z Has data issue: false hasContentIssue false
  • English
  • Français

Mixing and asymptotic distribution modulo 1

Published online by Cambridge University Press:  19 September 2008

Russell Lyons
Russell Lyons
Affiliation:
Department of Mathematics, Stanford University, Stanford CA 94305, USA
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.

If μ is a probability measure which is invariant and ergodic with respect to the transformation x↦qx on the circle ℝ/ℤ, then according to the ergodic theorem, {qnx} has the asymptotic distribution μ for μ-a.e. x. On the other hand, Weyl showed that when μ is Lebesgue measure, λ, and {mj} is an arbitrary sequence of integers increasing strictly to ∞, the asymptotic distribution of {mjx} is λ for λ-a.e. x. Here, we investigate the asymptotic distributions of {mjx} μ-a.e. for fairly arbitrary {mj} under some strong mixing conditions on μ. The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to μ. The ideas extend to the situation of a sequence of transformations x↦qnx where invariance is not present. This gives us information about many Riesz products and Bernoulli convolutions. Finally, we apply the theory to resolve some questions about H-sets.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 4 , December 1988, pp. 597 - 619
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Bari, Nina K.. A Treatise on Trigonometric Series. Translated from the Russian by Mullins, M. F.. Volumes I, II. New York: Macmillan, 1964.Google Scholar
[2]Blum, J. R. & Epstein, Bernard. On the Fourier-Stieltjes coefficients of Cantor-type distributions. Israel J Math. 17 (1974), 35–45.CrossRefGoogle Scholar
[3]Cornfeld, I. P., Fomin, S. V. & Sinai, Ya. G.. Ergodic Theory. Translated from the Russian by Sossinskii, A. B.. New York: Springer, 1982.Google Scholar
[4]Graham, Colin C. & McGehee, O. Carruth. Essays in Commutative Harmonic Analysis. New York: Springer, 1979.Google Scholar
[5]Katznelson, Yitzhak. An Introduction to Harmonic Analysis. Second corrected edition. New York: Dover, 1976.Google Scholar
[6]Lyons, Russell. Measure-theoretic quantifiers and Haar measure. Proc. Amer. Math. Soc. 86 (1982), 67–70;Google Scholar
Erratum, Proc. Amer. Math. Soc. 91 (1984), 329–330.Google Scholar
[7]Lyons, Russell. A Characterization of Measures whose Fourier-Stieltjes Transforms Vanish at Infinity. Dissertation, The University of Michigan, 1983.CrossRefGoogle Scholar
[8]Lyons, Russell, Fourier-Stieltjes coefficients and asymptotic distribution modulo 1. Ann. of Math. 122 (1985), 155–170.Google Scholar
[9]Lyons, Russell. The size of some classes of thin sets. Studia Math., 86 (1987), 59–78.CrossRefGoogle Scholar
[10]Lyons, Russell. The local structure of some measure-algebra homomorphisms. Preprint (1988).Google Scholar
[11]Meyer, Yves. Les produits de Riesz sont des Bernoulli-shifts, (d'après Yitzhak Katznelson et Benjamin Weiss), unpublished.Google Scholar
[12]Oxtoby, John C.. Ergodic sets. Bull. Amer. Math. Soc. 58 (1952), 116–136.Google Scholar
[13]Petit, B.. Schémas de Bernoulli et g-mesures. C. R. Acad. Sc. Paris 280, série A (1975), 17–20.Google Scholar
[14]Pjateckiǐ-Šapiro, I. I.. On the problem of uniqueness of expansion of a function in a trigonometric series, (in Russian). Moscov. Gos. Univ. Uč. Zap. 155, Mat. 5 (1952), 54–72.Google Scholar
Supplement to the work ‘On the problem…,’ (in Russian), Moscov. Gos. Univ. Uč. Zap. 165, Mat. 7 (1954), 79–97.Google Scholar
[15]Rokhlin, V. A.. Exact endomorphisms of a Lebesgue space. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499–530 (Russian);Google Scholar
English transl., Amer. Math. Soc. Transl. (2) 39 (1964), 1–36.Google Scholar
[16]Smorodinsky, Meir. Ergodic Theory Entropy. Berlin-Heidelberg-New York: Springer, 1971.Google Scholar
[17]Weyl, Hermann. Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916), 313–352.CrossRefGoogle Scholar
Also in ‘Gesammelte Abhandlungen’, vol. 1, pp. 563–599, Berlin: Springer, 1968.Google Scholar
[18]Zygmund, Antoni. Trigonometric Series. Second edition, reprinted. Volumes I, II. Cambridge: Cambridge University Press, 1979.Google Scholar