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


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.


Multiple ergodic average PID Rokhlin conjecture 

MR(2010) Subject Classification

37A05 37B05 


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  1. [1]
    El Abdalaoui, E. H.: On the pointwise convergence of multiple ergodic averages, arXiv:1406.2608v2 [math.DS]Google Scholar
  2. [2]
    Abbaspour, H., Moskowitz, M. A.: Basic Lie Theory, World Scientific, Singapore, 2007CrossRefzbMATHGoogle Scholar
  3. [3]
    Ageev, O., Silva, C.: Genericity of rigid and multiply recurrent infinite measure-preserving and nonsingular transformations. In Proceedings of the 16th Summer Conference on General Topology and its Applications (New York). Topology Proc., 26(2), 357–365 (2001)MathSciNetzbMATHGoogle Scholar
  4. [4]
    Assani, I.: Multiple recurrence and almost sure convergence for weakly mixing dynamical systems. Israel J. Math., 103, 111–124 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Assani, I.: Pointwise convergence of ergodic averages along cubes. J. Analyse Math., 110, 241–269 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Austin, T.: On the norm convergence of non-conventional ergodic averages. Ergod. Th. Dynam. Sys., 30, 321–338 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bashtanov, A. I.: Generic mixing transformations are rank 1. Math. Notes, 93, 209–216, (2013); Translation of Mat. Zametki, 93, 163–171 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Bergelson, V.: Combinatorial and Diophantine applications of ergodic theory, Appendix A by A. Leibman and Appendix B by Anthony Quas and Máté Wierdl. Handbook of dynamical systems. Vol. 1B, 745–869, Elsevier B. V., Amsterdam, 2006zbMATHGoogle Scholar
  9. [9]
    Birkhoff, G.: Proof of the ergodic theorem. Proc. Natn. Acad. Sci. U.S.A., 17, 656–660 (1931)CrossRefzbMATHGoogle Scholar
  10. [10]
    Bourgain, J.: Double recurrence and almost sure convergence. J. Reine Angew. Math., 404, 140–161 (1990)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Chu, Q., Frantzikinakis, N.: Pointwise convergence for cubic and polynomial ergodic averages of noncommuting transformations. Ergod. Th. and Dynam. Sys., 32, 877–897 (2012)CrossRefzbMATHGoogle Scholar
  12. [12]
    Cornfeld, I. P., Fomin, S. V., Sinai, Y. G.: Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 245. Springer-Verlag, New York, 1982zbMATHGoogle Scholar
  13. [13]
    Derrien, J., Lesigne, E.: Un théorème ergodique polynomial ponctuel pour les endomorphismes exacts et les K-systèmes. (French) [A pointwise polynomial ergodic theorem for exact endomorphisms and K-systems]. Ann. Inst. H. Poincar Probab. Statist., 32, 765–778 (1996)zbMATHGoogle Scholar
  14. [14]
    del Junco, A.: On minimal self-joinings in topological dynamics. Ergod. Th. Dynam. Sys., 7, 211–227 (1987)MathSciNetzbMATHGoogle Scholar
  15. [15]
    del Junco, A., Rudolph, D.: On ergodic actions whose self-joinings are graphs. Ergod. Th. Dynam. Sys., 7, 531–557 (1987)MathSciNetzbMATHGoogle Scholar
  16. [16]
    del Junco, A., Lemanczyk, M.: Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115, 725–C736 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Dekking, F. M., Keane, M.: Mixing properties of substitutions. Z. Wahrschein-lichkeitstheorie und Verw. Gebiete, 42, 23–33 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    de la Rue, T.: 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull. Braz. Math. Soc. (N.S.), 37, 503–521 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    de la Rue, T.: Joinings in ergodic theory. Mathematics of Complexity and Dynamical Systems. Vols. 1–3, Springer, New York, 2012, 796–809Google Scholar
  20. [20]
    Donoso, S., Sun, W.: Pointwise multiple averages for systems with two commuting transformations. arXiv:1509.09310 [math.DS], Ergod. Th. and Dynam. Sys., to appearGoogle Scholar
  21. [21]
    Donoso, S., Sun, W.: Pointwise convergence of some multiple ergodic averages, arXiv:1609.02529Google Scholar
  22. [22]
    Erdős, P., Turán, P.: On some sequences of integers. J. London Math. Soc., 11, 261–264 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Fayad, B., Kanigowski, A.: Multiple mixing for a class of conservative surface flows. Invent. Math., 203, 555–614 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1, 1–49 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Analyse Math., 31, 204–256 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Furstenberg, H.: Recurrence in ergodic theory and combinatorial number theory. M. B. Porter Lectures. Princeton University Press, Princeton, N. J., 1981CrossRefzbMATHGoogle Scholar
  27. [27]
    Furstenberg, H.: Nonconventional ergodic averages. The legacy of John von Neumann (Hempstead, NY, 1988), 43–56, Proc. Sympos. Pure Math., 50, Amer. Math. Soc., Providence, RI, 1990Google Scholar
  28. [28]
    Furstenberg, H.: Recurrent ergodic structures and Ramsey theory. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1057–1069, Math. Soc. Japan, Tokyo, 1991Google Scholar
  29. [29]
    Furstenberg, H.: Ergodic Structures and Non-Conventional Ergodic Theorems. Proceedings of the International Congress of Mathematicians. Volume I, 286–298, Hindustan Book Agency, New Delhi, 2010Google Scholar
  30. [30]
    Glasner, E.: Ergodic theory via joinings. Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003Google Scholar
  31. [31]
    Goodman, S., Marcus, B.: Topological mixing of higher degrees. Proc. Amer. Math. Soc., 72, 561–565 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Halmos, P. R.: Lectures on ergodic theory. Publications of the Mathematical Society of Japan, no. 3, The Mathematical Society of Japan, 1956Google Scholar
  33. [33]
    Host, B.: Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Israel J. Math., 76, 289–298 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Host, B.: Ergodic seminorms for commuting transformations and applications. Studia Math., 195, 31–49 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Host, B., Kra, B.: Nonconventional averages and nilmanifolds. Ann. of Math., 161, 398–488 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Huang, W., Shao, S., Ye, X.: Strictly ergodic models and pointwise ergodic averages for cubes. Comm. Math. Stat., 5, 93–122 (2017)CrossRefGoogle Scholar
  37. [37]
    Huang, W., Shao, S., Ye, X.: Pointwise convergence of multiple ergodic averages and strictly ergodic models. arXiv:1406.5930, J. Analyse Math., to appearGoogle Scholar
  38. [38]
    Kalikow, S.: Two-fold mixing implies threefold mixing for rank one transformations. Ergod. Th. and Dynam. Sys., 4, 237–59 (1984)CrossRefzbMATHGoogle Scholar
  39. [39]
    Katok, A.: Combinatorial constructions in ergodic theory and dynamics. University Lecture Series, 30 American Mathematical Society, Providence, RI, 2003Google Scholar
  40. [40]
    Katok, A., Stepin, A.: Approximations in ergodic theory. Russian Mathematical Surveys, 22, 77–102 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    Kamiński, B., Liardet, P.: Spectrum of multidimensional dynamical systems with positive entropy. Studia Math., 108, 77–85 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    Kifer, Y., Varadhan, S. R. S.: Nonconventional large deviations theorems. Probab. Theory Related Fields, 158, 197–224 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    Kra, B.: From combinatorics to ergodic theory and back again. International Congress of Mathematicians. Vol. III, 57–76, Eur. Math. Soc., Zürich, 2006MathSciNetzbMATHGoogle Scholar
  44. [44]
    Ledrappier, F.: Un champ markovien peut être dentropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B, 287, 561–563 (1978)MathSciNetzbMATHGoogle Scholar
  45. [45]
    Lemanczyk, M.: Spectral theory of dynamical systems. Mathematics of Complexity and Dynamical Systems. Vols. 1–3, Springer, New York, 2012, 1618–1638Google Scholar
  46. [46]
    Marcus, B.: The horocycle flow is mixing of all degrees. Invent. Math., 46, 201–209 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    Nadkarni, M. G.: Spectral theory of dynamical systems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 1998CrossRefGoogle Scholar
  48. [48]
    Ratner, M.: Factors of horocycle flows. Ergod. Th. Dynam. Sys., 2, 465–489, (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    Ratner, M.: Rigidity of horocycle flows. Annals of Math., 115, 597–614 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    Ratner, M.: Horocycle flows, joinings and rigidity of products. Annals of Math., 118, 277–313 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    Robinson, E. A.: The maximal abelian subextension determines weak mixing for group extensions. Proc. Amer. Math. Soc., 114, 443–450 (1992)MathSciNetzbMATHGoogle Scholar
  52. [52]
    Rokhlin, V. A.: On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat., 13, 329–340 (1949)MathSciNetGoogle Scholar
  53. [53]
    Rokhlin, V. A., Sinai, Ya. G.: Construction and properties of invariant measurable partitions [In Russian]. Dokl. Akad. Nauk SSSR, 141, 1038–1041 (1961)MathSciNetGoogle Scholar
  54. [54]
    Ryzhikov, V. V.: Joinings and multiple mixing of the actions of finite rank. (Russian) Funktsional. Anal. i Prilozhen., 27, 63–78 (1993); translation in Funct. Anal. Appl., 27, 128–140 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    Starkov, A. N.: Multiple mixing of homogeneous flows. Dokl. Akad. Nauk, 333, 442–445, (1993)zbMATHGoogle Scholar
  56. [56]
    Stepin, A. M.: Spectral properties of generic dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 50, 801–834 (1986)MathSciNetGoogle Scholar
  57. [57]
    Tao, T.: Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. and Dynam. Sys., 28, 657–688 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    Thouvenot, J. P.: Some properties and applications of joinings in ergodic theory, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), 205, 207–235 (1995)CrossRefGoogle Scholar
  59. [59]
    Tikhonov, S. V.: Complete metric on the set of mixing transformations. Uspekhi Mat. Nauk, 62, 209–210 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    Towsner H.: Convergence of diagonal ergodic averages. Ergod. Th. Dynam. Sys., 29, 1309–1326 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    Walsh, M.: Norm convergence of nilpotent ergodic averages. Ann. of Math., 175, 1667–1688 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics 79, Springer-Verlag, New York, 1982CrossRefGoogle Scholar
  63. [63]
    Ziegler, T.: Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc., 20, 53–97 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    Zimmer, R. J.: Extensions of ergodic group actions. Illinois J. Math., 20, 373–409 (1976)MathSciNetzbMATHGoogle Scholar
  65. [65]
    Zimmer, R. J.: Ergodic actions with generalized discrete spectrum. Illinois J. Math., 20, 555–588 (1976)MathSciNetzbMATHGoogle Scholar

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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