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Science China Mathematics

, Volume 60, Issue 1, pp 59–82 | Cite as

Multi-recurrence and van der Waerden systems

  • Dominik Kwietniak
  • Jian LiEmail author
  • Piotr Oprocha
  • XiangDong Ye
Articles

Abstract

We explore recurrence properties arising from dynamical approach to the van der Waerden theorem and similar combinatorial problems. We describe relations between these properties and study their consequences for dynamics. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity of topologically weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

Keywords

multi-recurrent points van der Waerden systems multiple recurrence theorem multiple IP-recurrence property multi-non-wandering points 

MSC(2010)

37B20 37B05 37A25 05D10 

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Notes

Acknowledgements

This work was supported by the National Science Centre (Grant No. DEC-2012/07/E/ST1/00185), National Natural Science Foundation of China (Grant Nos. 11401362, 11471125, 11326135, 11371339 and 11431012), Shantou University Scientific Research Foundation for Talents (Grant No. NTF12021) and the Project of LQ1602 IT4Innovations Excellence in Science. The authors thank Jie Li for the careful reading and helpful comments. A substantial part of this paper was written when the authors attended the activity “Dynamics and Numbers”, June–July 2014, held at the Max Planck Institute f¨ur Mathematik (MPIM) in Bonn, Germany. Some part of the work was continued when the authors attended a conference held at the Wuhan Institute of Physics and Mathematics, China, and the satellite conference of 2014 ICM at Chungnam National University, South Korea. The authors are grateful to the organizers for their hospitality.

References

  1. 1.
    Akin E, Auslander J, Berg K. When is a transitive map chaotic? In: Convergence in Ergodic Theory and Probability. Berlin: de Gruyter, 1996, 25–40Google Scholar
  2. 2.
    Akin E, Auslander J, Glasner E. The topological dynamics of Ellis actions. Mem Amer Math Soc, 2008, 195: 913MathSciNetzbMATHGoogle Scholar
  3. 3.
    Auslander J. Minimal Flows and Their Extensions. In: North-Holland Mathematics Studies, vol. 153. Amsterdam: North-Holland, 1988Google Scholar
  4. 4.
    Auslander J, Yorke J. Interval maps, factors of maps, and chaos. Tohoku Math J, 1980, 32: 177–188MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banks J, Nguyen T, Oprocha P, et al. Dynamics of spacing shifts. Discrete Contin Dyn Syst, 2013, 33: 4207–4232MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchard F, Host B, Ruette S. Asymptotic pairs in positive-entropy systems. Ergodic Theory Dynam Systems, 2002, 22: 671–686MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Blokh A, Fieldsteel A. Sets that force recurrence. Proc Amer Math Soc, 2002, 130: 3571–3578MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    De D, Hindman N, Strauss D. A new and stronger central sets theorem. Fund Math, 2008, 199: 155–175MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Downarowicz T. Survey of odometers and Toeplitz flows. In: Algebraic and Topological Dynamics. Contemp Math, vol. 385. Providence: Amer Math Soc, 2005, 7–37CrossRefGoogle Scholar
  10. 10.
    Forys M, Huang W, Li J, et al. Invariant scrambled sets, uniform rigidity and weak mixing. Israel J Math, 2016, 211: 447–472MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Furstenberg H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Anal Math, 1977, 31: 204–256MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Furstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton University Press, 1981CrossRefzbMATHGoogle Scholar
  13. 13.
    Furstenberg H, Katznelson Y. An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J Anal Math, 1985, 45: 117–168MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Furstenberg H, Weiss B. Topological dynamics and combinatorial number theory. J Anal Math, 1978, 34: 61–85MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Glasner E. Topological ergodic decompositions and applications to products of powers of a minimal transformation. J Anal Math, 1994, 64: 241–262MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Glasner E. Ergodic Theory via Joinings. Providence: Amer Math Soc, 2003Google Scholar
  17. 17.
    Glasner E, Weiss B. Sensitive dependence on initial conditions. Nonlinearity, 1993, 6: 1067–1075MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Glasner S, Maon D. Rigidity in topological dynamics. Ergodic Theory Dynam Systems, 1989, 9: 309–320MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hindman N. Small sets satisfying the central sets theorem. In: Combinatorial Number Theory B. Berlin: de Gruyter, 2009, 57–63Google Scholar
  20. 20.
    Host B, Kra B, Maass A. Variations on topological recurrence. Monatsh Math, 2016, 179: 57–89MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Huang W, Park K, Ye X. Dynamical systems disjoint from all minimal systems with zero entropy. Bull Soc Math France, 2007, 135: 259–282MathSciNetGoogle Scholar
  22. 22.
    Huang W, Shao S, Ye X. Pointwise convergence of multiple ergodic averages and strictly ergodic models. ArXiv:1406.5930, 2014Google Scholar
  23. 23.
    Huang W, Ye X, Zhang G. Lowering topological entropy over subsets revisted. Trans Amer Math Soc, 2014, 366: 4423–4442MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Johnson J. A dynamical characterization of C sets. ArXiv:1112.0715, 2011Google Scholar
  25. 25.
    Lehrer E. Topological mixing and uniquely ergodic systems. Israel J Math, 1987, 57: 239–255MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Li J. Transitive points via Furstenberg family. Topology Appl, 2011, 158: 2221–2231MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Li J. Dynamical characterization of C-sets and its applications. Fund Math, 2012, 216: 259–286MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Li J, Zhang R. Levels of generalized expansiveness. J Dynam Differential Equations, 2015, doi:10.1007/s10884-015-9502-6Google Scholar
  29. 29.
    Lind D, Marcus B. An Introduction to Symbolic Dynamics and Coding. Cambridge: Cambridge University Press, 1995CrossRefzbMATHGoogle Scholar
  30. 30.
    Moothathu T. Diagonal points having dense orbit. Colloq Math, 2010, 120: 127–138MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Neumann D. Central sequences in dynamical systems. Amer J Math, 1978, 100: 1–18MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Rado R. Studien zur kombinatorik. Math Z, 1933, 36: 242–280MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Stanley B. Bounded density shifts. Ergodic Theory Dynam Systems, 2013, 33: 1891–1928MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Szemerédi E. On sets of integers containing no k elements in arithmetic progression. Acta Arith, 1975, 27: 199–245MathSciNetzbMATHGoogle Scholar
  35. 35.
    Van der Waerden L. Beweis eine baudetschen vermutung nieus arch. Wisk, 1927, 15: 212–216zbMATHGoogle Scholar
  36. 36.
    Weiss B. Multiple recurrence and doubly minimal systems. In: Topological Dynamics and Applications. Contemp Math, vol. 215. Providence: Amer Math Soc, 1998, 189–196CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Dominik Kwietniak
    • 1
  • Jian Li
    • 2
    Email author
  • Piotr Oprocha
    • 3
    • 4
  • XiangDong Ye
    • 5
  1. 1.Faculty of Mathematics and Computer ScienceJagiellonian University in KrakówKrakówPoland
  2. 2.Department of MathematicsShantou UniversityShantouChina
  3. 3.Faculty of Applied MathematicsAGH University of Science and TechnologyKrakówPoland
  4. 4.National Supercomputing Centre IT4Innovations, Division of the University of Ostrava, Institute for Research and Applications of Fuzzy ModelingUniversity of OstravaOstravaCzech Republic
  5. 5.Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina

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