Topological correspondence of multiple ergodic averages of nilpotent group actions

  • Wen Huang
  • Song Shao
  • Xiangdong YeEmail author


Let (X,Γ) be a topological system, where Γ is a nilpotent group generated by T1,...,Td such that for each T ∈ Γ, TeΓ, (X,T) is weakly mixing and minimal. For d,k ∈ ℕ, let pi,j(n),1 ≤ ik,1 ≤ jd be polynomials with rational coefficients taking integer values on the integers and pi,j(0) = 0. We show that if the expressions \(g_i(n)=T_1^{{p}_{i,1}(n)}\cdots{T_d^{{p}_{i,d}(n)}}\) depend nontrivially on n for i = 1,2,...,k, and for all ij ∈ {1,2,...,k} the expressions gi(n)gj(n)-1 depend nontrivially on n, then there is a residual set X0 of X such that for all xX0
$$\{(g_1(n)x, g_2(n)x, ...., g_k(n)x)\in{X^k}:n\in\mathbb{Z}\}$$
is dense in Xk.


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  1. [1]
    V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems 7 (1987), 337–349.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V. Bergelson and S. Leibman, Polynomial extensions of van der Waerden’s and Szemerédi’s theorems, J. Amer.Math. Soc. 9 (1996), 725–753.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. Bourgain, Pointwise ergodic theorems for arithmetic sets, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math. 404 (1990), 140–161.MathSciNetzbMATHGoogle Scholar
  5. [5]
    S. Donoso and W. Sun, Pointwise convergence of some multiple ergodic averages, Adv. Math. 330 (2018), 946–996.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.CrossRefzbMATHGoogle Scholar
  8. [8]
    H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, NJ, 1981.CrossRefzbMATHGoogle Scholar
  9. [9]
    H. Furstenberg, Nonconventional ergodic averages, in The legacy of John von Neumann (Hempstead, NY, 1988), American Mathematical Soceity, Providence, RI, 1990, pp. 43–56.CrossRefGoogle Scholar
  10. [10]
    H. Furstenberg, Ergodic Structures and Non-Conventional Ergodic Theorems, in Proceedings of the International Congress ofMathematicians. Vol. I, Hindustan Book Agency, New Delhi, 2010, pp. 286–298Google Scholar
  11. [11]
    H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Anal. Math., 34 (1978), 61–85.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241–262.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, in Handbook of dynamical systems. Vol. 1B, Elsevier, Amsterdam, 2006, pp. 597–648.CrossRefGoogle Scholar
  14. [14]
    B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161 (2005), 398–488.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity 15 (2002), 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    W. Huang, S. Shao and X. Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models arXiv:1406.5930.Google Scholar
  17. [17]
    D. Kwietniak and P. Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Ergodic Theory Dynam. Systems, 32 (2012), 1661–1672.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    A. Leibman, Multiple recurrence theorem for nilpotent group actions, Geom. Funct. Anal. 4 (1994), 648–659.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Leibman, Multiple recurrence theorem for measure preserving actions of a nilpotent group, Geom. Funct. Anal. 8 (1998), 853–931.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    T. K. S. Moothathu, Diagonal points having dense orbit, Colloq. Math. 120 (2010), 127–138.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657–688.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    M. Walsh, Norm convergence of nilpotent ergodic averages, Ann. of Math. 175 (2012) 1667–1688.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    T. Ziegler, Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), 53–97.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    P. Zorin-Kranich, A nilpotent IP polynomial multiple recurrence theorem, J. Anal. Math., 123 (2014), 183–225.MathSciNetCrossRefzbMATHGoogle Scholar

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© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity Of Science and Technology of China, Chinese Academy of SciencesHefei, AnhuiP. R. China

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