Skip to main content
SpringerLink
Log in

Convergence of polynomial ergodic averages

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove theL 2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. Auslander, L. Green and G. Hahn,Flows on Homogeneous Spaces, Annals of Mathematics Studies53, Princeton University Press, 1963.

  2. V. Bergelson,Weakly mixing PET, Ergodic Theory and Dynamical Systems7 (1987), 337–349.

    MATH  MathSciNet  Google Scholar 

  3. V. Bergelson and A. Leibman,Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, Journal of the American Mathematical Society9 (1996), 725–753.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Frantzikinakis and B. Kra,Polynomial averages converge to the product of integrals, Israel Journal of Mathematics148 (2005), 267–276.

    MATH  MathSciNet  Google Scholar 

  5. H. Furstenberg,Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Mathématique31 (1977), 204–256.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Furstenberg and B. Weiss,A mean ergodic theorem for 1/NΣ n n=1 f(T n x)g(T n 2 x), inConvergence in Ergodic Theory and Probability (V. Bergelson, P. March and J. M. Rosenblatt, eds.), Walter de Gruyter & Co, Berlin, New York, 1996, pp. 193–227.

    Google Scholar 

  7. B. Host and B. Kra,Non conventional ergodic averages and nilmanifolds, Annals of Mathematics, to appear.

  8. A. Leibman,Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergodic Theory and Dynamical Systems, to appear.

  9. E. Lesigne,Sur une mil-variété, les parties minimales associées à une translation sont uniquement ergodiques, Ergodic Theory and Dynamical Systems11 (1991), 379–391.

    MATH  MathSciNet  Google Scholar 

  10. W. Parry,Ergodic properties of affine transformations and flows on nilmanifolds, American Journal of Mathematics91 (1969), 757–771.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Parry,Dynamical systems on nilmanifolds, The Bulletin of the London Mathematical Society2 (1970), 37–40.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Équipe d’analyse et de mathématiques appliquées, Université de Marne la Vallée, 77454, Marne la Vallée Cedex, France

    Bernard Host

  2. Department of Mathematics, McAllister B uilding, The Pennsylvania State University, 16802, University Park, PA, USA

    Bryna Kra

Authors
  1. Bernard Host
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bryna Kra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bernard Host.

Additional information

Dedicated to Hillel Furstenberg upon his retirement

The second author was partially supported by NSF grant DMS-0244994.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Host, B., Kra, B. Convergence of polynomial ergodic averages. Isr. J. Math. 149, 1–19 (2005). https://doi.org/10.1007/BF02772534

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02772534

Keywords

Search

Navigation