Advertisement

Israel Journal of Mathematics

, Volume 229, Issue 1, pp 255–267 | Cite as

A double return times theorem

  • Pavel Zorin-KranichEmail author
Article

Abstract

We prove that for any bounded functions f1, f2 on a measure-preserving dynamical system (X, T) and any distinct integers a1, a2, for almost every x the sequence
$${f_1}\left( {{T^{{a_1}n}}x} \right){f_2}\left( {{T^{{a_2}n}}x} \right)$$
is a good weight for the pointwise ergodic theorem. 1. Introduction

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ADM]
    I. Assani, D. Duncan and R. Moore, Pointwise characteristic factors for Wiener–Wintner double recurrence theorem, Ergodic Theory and dynamical Systems 36 (2016), 1037–1066.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Ass]
    I. Assani, Multiple return times theorems for weakly mixing systems, Annales de l’Institut Henri Poincaré. Probabilités et Statistiques 36 (2000), 153–165.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BFKO]
    J. Bourgain, H. Furstenberg, Y. Katznelson and D. S. Ornstein, Appendix on return-time sequences, Institute des Hautes études Scientifiques. Publications mathématiques 69 (1989), 42–45.MathSciNetCrossRefGoogle Scholar
  4. [Bou]
    J. Bourgain, Double recurrence and almost sure convergence, Journal für die Reine und Angewandte Mathematik 404 (1990), 140–161.MathSciNetzbMATHGoogle Scholar
  5. [CFH]
    Q. Chu, N. Frantzikinakis and B. Host, Ergodic averages of commuting transformations with distinct degree polynomial iterates, Proceedings of the London Mathematical Society 102 (2011), 801–842.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Dem]
    C. Demeter, Pointwise convergence of the ergodic bilinear Hilbert transform, Illinois Journal of Mathematics 51 (2007), 1123–1158.MathSciNetzbMATHGoogle Scholar
  7. [DOP]
    Y. Do, R. Oberlin and E. A. Palsson, Variation-norm and fluctuation estimates for ergodic bilinear averages, Indiana University Mathematics Journal 66 (2017), 55–99.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [HKa]
    B. Host and B. Kra, Convergence of polynomial ergodic averages, Israel Journal of Mathematics 149 (2005), 1–19.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [HKb]
    B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Annals of Mathematics 161 (2005), 397–488.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [HKc]
    B. Host and B. Kra, Uniformity seminorms on ℓ8 and applications, Journal d’Analyse Mathématique 108 (2009), 219–276.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [LMM]
    E. Lesigne, C. Mauduit and B. Mossé, Le théorème ergodique le long d’une suite q-multiplicative, Compositio Mathematica 93 (1994), 49–79.MathSciNetzbMATHGoogle Scholar
  12. [ZK]
    P. Zorin-Kranich, Return times theorem for amenable groups, Israel Journal of Mathematics 204 (2014), 85–96.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BonnBonnGermany

Personalised recommendations