Skip to main content
SpringerLink
Log in

The dynamical hierarchy for Roelcke precompact Polish groups

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study several distinguished function algebras on a Polish group G, under the assumption that G is Roelcke precompact. We do this by means of the model-theoretic translation initiated by Ben Yaacov and Tsankov: we investigate the dynamics of No-categorical metric structures under the action of their automorphism group. We show that, in this context, every strongly uniformly continuous function (in particular, every Asplund function) is weakly almost periodic. We also point out the correspondence between tame functions and NIP formulas, deducing that the isometry group of the Urysohn sphere is Tame ∩ UC-trivial.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov, Model theory for metric structures, in Model Theory with Applications to Algebra and Analysis. Vol. 2, London Mathematical Society Lecture Note Series, Vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427.

    Book  MATH  Google Scholar 

  2. I. Ben Yaacov, Model theoretic stability and definability of types, after A. Grothendieck, Bull. Symb. Log. 20 (2014), 491–496.

    MathSciNet  Google Scholar 

  3. I. Ben Yaacov, On Roelcke precompact Polish groups which cannot act transitively on a complete metric space, 2015, arXiv:1510.00238 [math.LO].

    Google Scholar 

  4. S. K. Berberian, Lectures in Functional Analysis and Operator Theory, Springer–Verlag, New York–Heidelberg, 1974, Graduate Texts in Mathematics, No. 15.

    Book  MATH  Google Scholar 

  5. J. Bourgain, D. H. Fremlin and M. Talagrand, Pointwise compact sets of Bairemeasurable functions, Amer. J. Math. 100 (1978), 845–886.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, Vol. 232, Cambridge University Press, Cambridge, 1996.

    Book  MATH  Google Scholar 

  7. I. Ben Yaacov and A. Kai'chouh, Reconstruction of separably categorical metric structures, J. Symb. Log. 81 (2016), 216–224.

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Bourbaki, Éleménts de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971.

    MATH  Google Scholar 

  9. I. Ben Yaacov and T. Tsankov, Weakly almost periodic functions, model–theoretic stability, and minimality of topological groups, 2014, arXiv:1312.7757 [math.LO] (to appear in Trans. Amer. Math. Soc.).

    MATH  Google Scholar 

  10. I. Ben Yaacov and A. Usvyatsov, On d–Sniteness in continuous structures, Fund. Math. 194 (2007), 67–88.

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Ben Yaacov and A. Usvyatsov, Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010), 5213–5259.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. B. Conway, A Course in Functional Analysis, second ed., Graduate Texts in Mathematics, Vol. 96, Springer–Verlag, New York, 1990.

  13. A. Chernikov and P. Simon, Definably amenable NIP groups, 2015, arXiv: 1502.04365 [math.LO].

    Google Scholar 

  14. J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, Vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993.

    Book  MATH  Google Scholar 

  15. E. Glasner, On tame dynamical systems, Colloq. Math. 105 (2006), 283–295.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Glasner and M. Megrelishvili, Hereditarily non–sensitive dynamical systems and linear representations, Colloq. Math. 104 (2006), 223–283.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Glasner and M. Megrelishvili, New algebras of functions on topological groups arising from G–spaces, Fund. Math. 201 (2008), 1–51.

    Article  MathSciNet  MATH  Google Scholar 

  18. E. Glasner and M. Megrelishvili, Representations of dynamical systems on Banach spaces not containing l1, Trans. Amer. Math. Soc. 364 (2012), 6395–6424.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Glasner and M. Megrelishvili, Banach representations and affine compactiScations of dynamical systems, in Asymptotic Geometric Analysis, Fields Inst. Commun., Vol. 68, Springer, New York, 2013, pp. 75–144.

    MathSciNet  Google Scholar 

  20. E. Glasner and M. Megrelishvili, Eventual nonsensitivitv and tame dynamical systems, 2014, arXiv:1405.2588 [math.DS].

    Google Scholar 

  21. E. Glasner and M. Megrelishvili, Representations of dynamical systems on Banach spaces, in Recent progress in general topology. III, Atlantis Press, Paris, 2014, pp. 399–470.

    MATH  Google Scholar 

  22. T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc. (3) 29 (1974), 331–350.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Iovino, Stable models and reflexive Banach spaces, J. Symbolic Logic 64 (1999), 1595–1600.

    Article  MathSciNet  MATH  Google Scholar 

  24. D. Kerr and H. Li, Independence in topological and C*–dynamics, Math. Ann. 338 (2007), 869–926.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Kohler, Enveloping semigroups for flows, Proc. Roy. Irish Acad. Sect. A 95 (1995), 179–191.

    MathSciNet  MATH  Google Scholar 

  26. M. C. Laskowski, Vapnik Chervonenkis classes of definable sets, J. London Math. Soc. (2) 45 (1992), 377–384.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Megrelishvili, Every semitopological semigroup compactification of the group H+[0,1] is trivial, Semigroup Forum 63 (2001), 357–370.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Megrelishvili, Fragmentabilitv and representations of flows, in Proceedings of the 17th Summer Conference on Topology and its Applications, Vol. 27, 2003, pp. 497–544.

    MathSciNet  Google Scholar 

  29. M. Megrelishvili, Reflexively representable but not Hilbert representable compact flows and semitopological semigroups, Colloq. Math. 110 (2008), 383–407.

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Melleray, A note on Hjorth's oscillation theorem, J. Symbolic Logic 75 (2010), 1359–1365.

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Pestov, On free actions, minimal Bows, and a problem by Ellis, Trans. Amer. Math. Soc. 350 (1998), 4149–4165.

    Article  MathSciNet  MATH  Google Scholar 

  32. V. Pestov, The isometry group of the Urysohn space as a Levy group, Topology Appl. 154 (2007), 2173–2184.

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Pillay, Geometric Stability Theory, Oxford Logic Guides, Vol. 32, The Clarendon Press, Oxford University Press, New York, 1996, Oxford Science Publications.

    MATH  Google Scholar 

  34. B. Poizat, Cours dé theorie des modeles, Bruno Poizat, Lyon, 1985, Une introduction à la logique math6matique contemporaine. [An introduction to contemporary mathematical logic].

    MATH  Google Scholar 

  35. H.P. Rosenthal, A characterization of Banach spaces contaning l1, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411–2413.

    Google Scholar 

  36. S. Shelah, Stability, the fc.p., and superstability; model theoretic properties of formulas in first order theory, Ann. Math. Logic 3 (1971), 271–362.

    Google Scholar 

  37. S. Shelah, Classification theory and the number of nonisomorphic models, second ed., Studies in Logic and the Foundations of Mathematics, Vol. 92, North–Holland Publishing Co., Amsterdam, 1990.

    Google Scholar 

  38. P. Simon, A Guide to NIP Theories, Vol. 44, Cambridge University Press, 2015.

  39. P. Simon, Rosenthal compacta and NIP formulas, Fund. Math. 231 (2015), 81–92.

    Article  MathSciNet  MATH  Google Scholar 

  40. K. Tent and M. Ziegler, A Course in Model Theory, Lecture Notes in Logic, Vol. 40, Association for Symbolic Logic, La Jolla, CA; Cambridge University Press, Cambridge, 2012.

    Book  MATH  Google Scholar 

  41. V. Uspenskij, Compactifications of topological groups, in Proceedings of the Ninth Prague Topological Symposium (2001), Topol. Atlas, North Bay, ON, 2002, pp. 331–346.

    MATH  Google Scholar 

  42. A. Usvyatsov, Generic separable metric structures, Topology Appl. 155 (2008), 1607–1617.

    Article  MathSciNet  MATH  Google Scholar 

  43. V. N. Vapnik and A. Y. Chervonenkis, Theory of uniform convergence of frequencies of events to their probabilities and problems of search for an optimal solution from empirical data, Avtomat. i Telemeh. (1971), 42–53.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Lyon Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Tomás Ibarlucía

Authors
  1. Tomás Ibarlucía
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomás Ibarlucía.

Additional information

Research partially supported by GruPoLoCo (ANR-11-JS01-0008) and ValCoMo (ANR-13-BS01-0006)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ibarlucía, T. The dynamical hierarchy for Roelcke precompact Polish groups. Isr. J. Math. 215, 965–1009 (2016). https://doi.org/10.1007/s11856-016-1399-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-016-1399-1

Search

Navigation