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Variations on topological recurrence

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Abstract

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems.

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Notes

  1. After this paper was submitted, Wenbo Sun adapted the methods we use to generalize this theorem, showing that a set of \(s\)-recurrence for \(s\)-step nilsystems is also a set of \(t\)-recurrence for all \(t\ge s\).

References

  1. Auslander, J.: Minimal Flows and their Extensions. North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1988)

    MATH  Google Scholar 

  2. Auslander, L., Green, L., Hahn, F.: Flows on homogeneous spaces. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. Annals of Mathematics Studies, No. 53. Princeton University Press, Princeton (1963)

  3. Bergelson, V.: Ergodic Ramsey theory. Logic and combinatorics (Arcata, Calif., 1985). Contemp. Math., vol. 65, pp. 63–87. American Mathematical Society, Providence (1987)

  4. Bergelson, V.: Ergodic Ramsey theory-an update. Ergodic theory of \(\mathbb{Z}^d\) actions (Warwick, 1993–1994). London Math. Soc. Lecture Note Ser., vol. 228, pp. 1–61. Cambridge University Press, Cambridge (1996)

  5. Bergelson, V., Høland Knutson, I., McCutcheon, R.: IP-systems, generalized polynomials and recurrence. Ergod. Theory Dyn. Syst. 26(4), 999–1019 (2006)

    Article  MATH  Google Scholar 

  6. Bergelson, V., Leibman, A.: Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Am. Math. Soc. 9(3), 725–753 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bergelson, V., Ruzsa, I.: Squarefree numbers, IP sets and ergodic theory. Paul Erdős and his mathematics, I (Budapest, 1999). Bolyai Soc. Math. Stud., vol. 11, pp. 147–160. Já nos Bolyai Math. Soc., Budapest (2002)

  8. Blanchard, F., Glasner, E., Kolyada, S., Maass, A.: On Li-Yorke pairs. J. Reine Angew. Math. 547, 51–68 (2002)

    MATH  MathSciNet  Google Scholar 

  9. Boshernitzan, M., Glasner, E.: On two recurrence problems. Fund. Math. 206, 113–130 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brown, T., Graham, R., Landman, B.: On the set of common differences in van der Waerden’s theorem on arithmetic progressions. Can. Math. Bull. 42(1), 25–36 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Dong, P., Donoso, S., Maass, A., Shao, S., Ye, X.: Infinite-step nilsystems, independence and complexity. Ergod. Theory Dyn. Syst. 33(1), 118–143 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Frantzikinakis, N.: Multiple recurrence and convergence for Hardy sequences of polynomial growth. J. Anal. Math. 112, 79–135 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Frantzikinakis, N., Host, B., Kra, B.: Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. 611, 131–144 (2007)

    MATH  MathSciNet  Google Scholar 

  14. Frantzikinakis, N., Leisgne, E., Wierdl, M.: Sets of \(k\)-recurrence but not \((k+1)\)-recurrence. Ann. Inst. Fourier (Grenoble) 56(4), 839–849 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Frantzikinakis, N., McCutcheon, R.: Ergodic Theory: Recurrence. Encyclopedia of Complexity and System Science, vol. 5, pp. 3083–3095. Springer (2009)

  16. Frantzikinakis, N., Wierdl, M.: A Hardy field extension of Szemerédi’s theorem. Adv. Math. 222(1), 1–43 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  18. Furstenberg, H.: Poincaré recurrence and number theory. Bull. Am. Math. Soc. (N.S.) 5(3), 211–234 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  19. Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)

    Book  MATH  Google Scholar 

  20. Furstenberg, H., Weiss, B.: Topological dynamics and combinatorial number theory. J. Anal. Math. 34, 61–85 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Glasner, E.: Divisible properties and the Stone-Čech compactification. Can. J. Math. 32(4), 993–1007 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  22. Host, B., Kra, B.: Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161(1), 397–488 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Host, B., Kra, B.: Nil–Bohr sets of integers. Ergod. Theory Dyn. Syst. 31(1), 113–142 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Host, B., Kra, B., Maass, A.: Nilsequences and a topological structure theorem. Adv. Math. 224(1), 103–129 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. Host, B., Kra, B., Maass, A.: Complexity of nilsystems and systems lacking nilfactors. J. Anal. Math. 124, 261–295 (2014)

  26. Huang, W., Shao, S., Ye, X.: Nil Bohr-sets and almost automorphy of higher order. Mem. Am. Math. Soc. (to appear) (2014)

  27. Katznelson, Y.: Chromatic numbers of Cayley graphs on \({\mathbb{Z}}\) and recurrence. Combinatorica 21(2), 211–219 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kriz, I.: Large independent sets in shift-invariant graphs. Solution of Bergelson’s problem. Graphs Comb. 3, 145–158 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  29. Leibman, A.: Polynomial sequences in groups. J. Algebra 201(1), 189–206 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. McCutcheon, R.: Three results in recurrence. Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math. Soc. Lecture Note Ser., vol. 205, pp. 349–358. Cambridge University Press, Cambridge (1995)

  31. McCutcheon, R.: Elemental Methods in Ergodic Ramsey Theory. Lecture Notes in Mathematics, vol. 1722. Springer, Berlin (1999)

  32. Parry, W.: Ergodic properties of affine transformations and flows on nilmanifolds. Am. J. Math. 91, 757–771 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pavlov, R.: Some counterexamples in topological dynamics. Ergod. Theory Dyn. Syst. 28(4), 1291–1322 (2008)

    Article  MATH  Google Scholar 

  34. Sárkőzy, A.: On difference sets of sequences of integers. I. Acta Math. Acadm. Sci. Hungar. 31, 125–149 (1978)

    Article  Google Scholar 

  35. Shao, S., Ye, X.: Regionally proximal relation of order d is an equivalence one for minimal systems and a combinatorial consequence. Adv. Math. 231(3–4), 1786–1817 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Weiss, B.: Single orbit dynamics. In: CBMS Regional Conference Series in Mathematics, vol. 95. American Mathematical Society, Providence (2000)

  37. Wierdl, M.: Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64(3), 315–336 (1988)

    Article  MathSciNet  Google Scholar 

  38. van der Werden, B.L.: Beweis einer Baudetschen Vermutung. Nieuw Arch. Wish. 15, 212–216 (1927)

    Google Scholar 

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Authors and Affiliations

  1. Université de Paris-Est Marne-la-Vallée, Laboratoire d’analyse et de mathématiques appliquées CNRS UMR 8050, 5 Bd. Descartes, Champs sur Marne, 77454, Marne-la-Vallée Cedex 2, France

    Bernard Host

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Bryna Kra

  3. Department of Mathematical Engineering and Center for Mathematical Modeling, UMI 2071 UCHILE-CNRS, University of Chile, Beauchef 851, Santiago, Chile

    Alejandro Maass

Authors
  1. Bernard Host
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  2. Bryna Kra
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  3. Alejandro Maass
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Corresponding author

Correspondence to Bryna Kra.

Additional information

Communicated by H. Bruin.

The second author was partially supported by NSF Grant DMS-\(1200971\) and the third author was partially supported by the Bézout Chair of the Université Paris-Est Marne-la-Vallée.

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Host, B., Kra, B. & Maass, A. Variations on topological recurrence. Monatsh Math 179, 57–89 (2016). https://doi.org/10.1007/s00605-015-0765-0

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  • DOI: https://doi.org/10.1007/s00605-015-0765-0

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