Inventiones mathematicae

, Volume 113, Issue 1, pp 617–631

Quantitative recurrence results

  • Michael D. Boshernitzan


LetX be a probability measure spaceX=(X, Φ, μ) endowed with a compatible metricd so that (X,d) has a countable base. It is well-known that ifTXX is measure-preserving, then μ-almost all pointsxX are recurrent, i.e.,\(\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} d(x, T^n (x)) = 0\). We show that, under the additional assumption that the Hausdorff α-measureHα(X) ofX is σ-finite for some α>0, this result can be strengthened:\(\lim \begin{array}{*{20}c} {\inf } \\ {n \geqq 1} \\ \end{array} \left\{ {n^{1/\alpha } . d(x, T^n (x))} \right\}< \infty \), for μ-almost all pointsxX. A number of applications are considered.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ba] Baker, A.: Transcendental Number Theory. Cambridge: Cambridge University Press 1975Google Scholar
  2. [Bo] Boshernitzan, M.: Rank two interval exchange transformations. Ergodic Theory Dyn. Syst.8, 379–394 (1988)Google Scholar
  3. [Ca] Cassels, J.W.S.: An Introduction to Diophantine Approximations. Cambridge: Cambridge University Press 1957Google Scholar
  4. [C-F-S] Cornfeld, I.P., Fomin, S.V., Sinai, Ya. G.: Ergodic Theory. Berlin Heidelberg New York: Springer 1982Google Scholar
  5. [Co] Colebrook, C.M.: The Hausdorff dimension of certain sets of nonnormal numbers. Mich. Math. J.17, 103–116 (1970)Google Scholar
  6. [Fa] Falconer, K.: Fractal Geometry, Mathematical Foundations and Applications. New York: Wiley 1990Google Scholar
  7. [Fu] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, NJ: Princeton University Press 1981Google Scholar
  8. [M] Mañé, R.: Ergodic Theory and Differentiable Dynamics. Berlin Heidelberg New York: Springer 1987Google Scholar
  9. [O-W] Ornstein, D., Weiss, B.: Entropy and data compression schemes. (Preprint 1990)Google Scholar
  10. [Sp] Spitzer, F.: Principles of Random Walk. Van Nostrand, 1964Google Scholar
  11. [Sm] Schmidt, W.M.: Diophantine approximations. Lect. Note in Math. 785, 1980Google Scholar
  12. [V1] Veech W.A.: Interval exchange transformations. Anal. Math.33, 222–272 (1978)Google Scholar
  13. [V2] Veech, W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math.115, 201–242 (1982)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Michael D. Boshernitzan
    • 1
  1. 1.Department of MathematicsRice UniversityHoustonUSA

Personalised recommendations