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Acta Mathematica Sinica, English Series

, Volume 34, Issue 1, pp 79–90 | Cite as

Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

  • Yonatan GutmanEmail author
  • Wen Huang
  • Song Shao
  • Xiang Dong Ye
Article
  • 145 Downloads

Abstract

The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages
$$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{f_1}} \left( {{T^n}x} \right)...{f_d}\left( {{T^{dn}}x} \right),N \to \infty $$
almost surely converge.

Keywords

Multiple ergodic average PID Rokhlin conjecture 

MR(2010) Subject Classification

37A05 37B05 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Yonatan Gutman
    • 1
    Email author
  • Wen Huang
    • 2
    • 3
  • Song Shao
    • 2
    • 3
  • Xiang Dong Ye
    • 2
    • 3
  1. 1.Institute of MathematicsPolish Academy of ScienceWarszawaPoland
  2. 2.Wu Wen-Tsun Key Laboratory of Mathematics, USTCChinese Academy of SciencesHefeiP. R. China
  3. 3.Department of MathematicsUniversity of Science and Technology of ChinaHefeiP. R. China

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