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More on Tame Dynamical Systems

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Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2213))

Abstract

In this work, on the one hand, we survey and amplify old results concerning tame dynamical systems and, on the other, prove some new results and exhibit new examples of such systems. In particular, we study tame symbolic systems and establish a neat characterization of tame subshifts. We also provide sufficient conditions which ensure that certain coding functions are tame. Finally we discuss examples where certain universal dynamical systems associated with some Polish groups are tame.

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Notes

  1. 1.

    An earlier version of this work is posted on the Arxiv (arXiv:1405.2588).

  2. 2.

    Actually this possibility can not occur, as is shown in the first step of the proof.

  3. 3.

    E.g., this latter condition is always satisfied when Y is distal. Another example of such (non-distal) system is the Sturmian system.

  4. 4.

    Modulo an extension of Weiss’ theorem, which does not yet exist, a similar idea would work for any locally compact group. The more general statement would be: A locally compact group which is intrinsically tame is compact.

References

  1. E. Akin, Enveloping linear maps, in Topological Dynamics and Applications. Contemporary Mathematics, vol. 215 (a volume in honor of R. Ellis) (American Mathematical Society, Providence, 1998), pp. 121–131

    Google Scholar 

  2. E. Akin, E. Glasner, WAP systems and labeled subshifts. Mem. AMS (2014, to appear). arXiv:1410.4753

    Google Scholar 

  3. J.B. Aujogue, Ellis enveloping semigroup for almost canonical model sets of an Euclidean space. Algebr. Geom. Topol. 15(4), 2195–2237 (2015)

    Article  MathSciNet  Google Scholar 

  4. I. Ben Yaacov, T. Tsankov, Weakly almost periodic functions, model-theoretic stability, and minimality of topological groups. Trans. Am. Math. Soc. 368, 8267–8294 (2016)

    Article  MathSciNet  Google Scholar 

  5. J.F. Berglund, H.D. Junghenn, P. Milnes, Analysis on Semigroups (Wiley, New York, 1989)

    Google Scholar 

  6. V. Berthe, L. Vuillon, Palindromes and two-dimensional Sturmian sequences. J. Autom. Lang. Combin. 6, 121–138 (2001)

    Google Scholar 

  7. V. Berthe, S. Ferenczi, L.Q. Zamboni, Interactions between dynamics, arithmetics and combinatorics: the good, the bad, and the ugly. Contemp. Math. 385, 333–364 (2005)

    Google Scholar 

  8. J. Bourgain, D.H. Fremlin, M. Talagrand, Pointwise compact sets in Baire-measurable functions. Am. J. Math. 100(4), 845–886 (1977)

    Article  MathSciNet  Google Scholar 

  9. B. Cascales, I. Namioka, J. Orihuela, The Lindelof property in Banach spaces. Stud. Math. 154, 165–192 (2003)

    Article  MathSciNet  Google Scholar 

  10. D. Cenzer, A. Dashti, F. Toska, S. Wyman, Computability of countable subshifts in one dimension. Theory Comput. Syst. 51(3), 352–371 (2012)

    Article  MathSciNet  Google Scholar 

  11. A. Chernikov, P. Simon, Definably amenable NIP groups (2015). arXiv:1502.04365v1

    Google Scholar 

  12. P. de la Harpe, Moyennabilité de quelques groupes topologiques de dimension infinie. C.R. Acad. Sci. Paris Sér. A 277, 1037–1040 (1973)

    Google Scholar 

  13. R. Ellis, Universal minimal sets. Proc. Am. Math. Soc. 11, 540–543 (1960)

    Article  MathSciNet  Google Scholar 

  14. R. Ellis, The Veech structure theorem. Trans. Am. Math. Soc. 186, 203–218 (1973)

    Article  MathSciNet  Google Scholar 

  15. R. Ellis, The enveloping semigroup of projective flows. Ergod. Theory Dyn. Syst. 13, 635–660 (1993)

    Google Scholar 

  16. M. Fabian, Gateaux Differentiability of Convex Functions and Topology. Weak Asplund Spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts (Wiley, New York, 1997)

    Google Scholar 

  17. T. Fernique, Multi-dimensional Sturmian sequences and generalized substitutions. Int. J. Found. Comput. Sci. 17 575–600 (2006)

    Article  MathSciNet  Google Scholar 

  18. H. Furstenberg, A poisson formula for semi-simple Lie groups. Ann. Math. 77, 335–386 (1963)

    Article  MathSciNet  Google Scholar 

  19. E. Ghys, Groups acting on the circle. Enseign. Math. (2) 47, 329–407 (2001)

    Google Scholar 

  20. S. Glasner, Compressibility properties in topological dynamics. Am. J. Math. 97, 148–171 (1975)

    Article  MathSciNet  Google Scholar 

  21. E. Glasner, Proximal Flows. Lecture Notes, vol. 517 (Springer, Berlin, 1976)

    Google Scholar 

  22. E. Glasner, Ergodic Theory via Joinings. Mathematical Surveys and Monographs, vol. 101 (American Mathematical Society, Providence, 2003)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. E. Glasner, The structure of tame minimal dynamical systems. Ergod. Theory Dyn. Syst. 27, 1819–1837 (2007)

    Google Scholar 

  25. E. Glasner, Translation-finite sets (2011). ArXiv: 1111.0510

    Google Scholar 

  26. E. Glasner, The structure of tame mini systems for general groups. Invent. Math. 211, 213–244 (2018). https://doi.org/10.1007/s00222-017-0747-z

  27. E. Glasner, M. Megrelishvili, Linear representations of hereditarily non-sensitive dynamical systems. Colloq. Math. 104(2), 223–283 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. E. Glasner, M. Megrelishvili, Representations of dynamical systems on Banach spaces not containing l1. Trans. Am. Math. Soc. 364, 6395–6424 (2012). ArXiv e-print: 0803.2320

    Google Scholar 

  30. E. Glasner, M. Megrelishvili, On fixed point theorems and nonsensitivity. Isr. J. Math. 190, 289–305 (2012). ArXiv e-print: 1007.5303

    Article  MathSciNet  Google Scholar 

  31. E. Glasner, M. Megrelishvili, Banach representations and affine compactifications of dynamical systems, in Fields Institute Proceedings Dedicated to the 2010 Thematic Program on Asymptotic Geometric Analysis, ed. by M. Ludwig, V.D. Milman, V. Pestov, N. Tomczak-Jaegermann (Springer, New York, 2013). ArXiv version: 1204.0432

    Chapter  Google Scholar 

  32. E. Glasner, M. Megrelishvili, Representations of dynamical systems on Banach spaces, in Recent Progress in General Topology III, ed. by K.P. Hart, J. van Mill, P. Simon (Springer/Atlantis Press, New York/Pairs, 2014), pp. 399–470

    Google Scholar 

  33. E. Glasner, M. Megrelishvili, Eventual nonsensitivity and tame dynamical systems (2014). arXiv:1405.2588

    Google Scholar 

  34. E. Glasner, M. Megrelishvili, Circularly ordered dynamical systems. Monatsh. Math. (2016, to appear). arXiv:1608.05091

    Google Scholar 

  35. E. Glasner, B. Weiss, Quasifactors of zero-entropy systems. J. Am. Math. Soc. 8, 665–686 (1995)

    MathSciNet  MATH  Google Scholar 

  36. E. Glasner, B. Weiss, Minimal actions of the group \(S({{\mathbb Z}})\) of permutations of the integers. Geom. Funct. Anal. 12, 964–988 (2002)

    Google Scholar 

  37. E. Glasner, B. Weiss, The universal minimal system for the group of homeomorphisms of the Cantor set. Fundam. Math. 176, 277–289 (2003)

    Article  MathSciNet  Google Scholar 

  38. E. Glasner, B. Weiss, On Hilbert dynamical systems. Ergod. Theory Dyn. Syst. 32(2), 629–642 (2012)

    Article  MathSciNet  Google Scholar 

  39. E. Glasner, X. Ye, Local entropy theory. Ergod. Theory Dyn. Syst. 29, 321–356 (2009)

    Article  MathSciNet  Google Scholar 

  40. E. Glasner, M. Megrelishvili, V.V. Uspenskij, On metrizable enveloping semigroups. Isr. J. Math. 164, 317–332 (2008)

    Article  MathSciNet  Google Scholar 

  41. W. Huang, Tame systems and scrambled pairs under an abelian group action. Ergod. Theory Dyn. Syst. 26, 1549–1567 (2006)

    Article  MathSciNet  Google Scholar 

  42. T. Ibarlucia, The dynamical hierarchy for Roelcke precompact Polish groups. Isr. J. Math. 215, 965–1009 (2016)

    Article  MathSciNet  Google Scholar 

  43. J.E. Jayne, C.A. Rogers, Borel selectors for upper semicontinuous set-valued maps. Acta Math. 155, 41–79 (1985)

    Article  MathSciNet  Google Scholar 

  44. J.E. Jayne, J. Orihuela, A.J. Pallares, G. Vera, σ-Fragmentability of multivalued maps and selection theorems. J. Funct. Anal. 117(2), 243–273 (1993)

    Article  MathSciNet  Google Scholar 

  45. T. Jolivet, Combinatorics of Pisot Substitutions, TUCS Dissertations No 164 (2013)

    Google Scholar 

  46. A.S. Kechris, Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156 (Springer, Berlin, 1991)

    Google Scholar 

  47. A.S. Kechris, V.G. Pestov, S. Todorc̆ević, Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15(1), 106–189 (2005)

    Article  Google Scholar 

  48. P.S. Kenderov, W.B. Moors, Fragmentability of groups and metric-valued function spaces. Top. Appl. 159, 183–193 (2012)

    Article  MathSciNet  Google Scholar 

  49. D. Kerr, H. Li, Dynamical entropy in Banach spaces. Invent. Math. 162, 649–686 (2005)

    Article  MathSciNet  Google Scholar 

  50. D. Kerr, H. Li, Independence in topological andC∗-dynamics. Math. Ann. 338, 869–926 (2007)

    Article  MathSciNet  Google Scholar 

  51. A. Köhler, Enveloping semigroups for flows. Proc. R. Ir. Acad. 95A, 179–191 (1995)

    Google Scholar 

  52. A. Komisarski, H. Michalewski, P. Milewski, Bourgain-Fremlin-Talagrand dichotomy and dynamical systems. Preprint (2004)

    Google Scholar 

  53. M. Megrelishvili, Free topological G-groups. N. Z. J. Math. 25, 59–72 (1996)

    MathSciNet  MATH  Google Scholar 

  54. M. Megrelishvili, Fragmentability and continuity of semigroup actions. Semigroup Forum 57, 101–126 (1998)

    Article  MathSciNet  Google Scholar 

  55. M. Megrelishvili, Fragmentability and representations of flows. Topol. Proc. 27(2), 497–544 (2003). See also: www.math.biu.ac.il/~megereli

  56. M. Megrelishvili, Topological transformation groups: selected topics, in Open Problems in Topology II, ed. by E. Pearl (Elsevier Science, Amsterdam, 2007), pp. 423–438

    Chapter  Google Scholar 

  57. M. Megrelishvili, A note on tameness of families having bounded variation. Topol. Appl. 217, 20–30 (2017)

    Article  MathSciNet  Google Scholar 

  58. M. Megrelishvili, L. Polev, Order and minimality of some topological groups. Topol. Appl. 201, 131–144 (2016)

    Article  MathSciNet  Google Scholar 

  59. I. Namioka, Radon-Nikodým compact spaces and fragmentability. Mathematika 34, 258–281 (1987)

    Article  MathSciNet  Google Scholar 

  60. I. Namioka, R.R. Phelps, Banach spaces which are Asplund spaces. Duke Math. J. 42, 735–750 (1975)

    Article  MathSciNet  Google Scholar 

  61. L. Nguyen van Thé, More on the Kechris-Pestov-Todorcevic correspondence: precompact expansions. Fund. Math. 222(1), 19–47 (2013)

    Article  MathSciNet  Google Scholar 

  62. L. Nguyen van Thé, Fixed points in compactifications and combinatorial counterparts (2017). arXiv:1701.04257

    Google Scholar 

  63. V.G. Pestov, On free actions, minimal flows, and a problem by Ellis. Trans. Am. Math. Soc. 350, 4149–4165 (1998)

    Article  MathSciNet  Google Scholar 

  64. V.G. Pestov, Dynamics of Infinite-Dimensional Groups. The Ramsey-Dvoretzky-Milman Phenomenon. University Lecture Series, vol. 40 (American Mathematical Society, Providence, 2006)

    Google Scholar 

  65. R. Pikula, Enveloping semigroups of affine skew products and sturmian-like systems, Dissertation, The Ohio State University, 2009

    Google Scholar 

  66. W. Roelcke, S. Dierolf, Uniform Structures on Topological Groups and Their Quotients (McGraw-Hill, New York, 1981)

    MATH  Google Scholar 

  67. A.V. Romanov, Ergodic properties of discrete dynamical systems and enveloping semigroups. Ergod. Theory Dyn. Syst. 36, 198–214 (2016)

    Article  MathSciNet  Google Scholar 

  68. H.P. Rosenthal, A characterization of Banach spaces containing l1. Proc. Natl. Acad. Sci. U.S.A. 71, 2411–2413 (1974)

    Article  Google Scholar 

  69. W. Ruppert, On weakly almost periodic sets. Semigroup Forum 32, 267–281 (1985)

    Article  MathSciNet  Google Scholar 

  70. S.A. Shapovalov, A new solution of one Birkhoff problem. J. Dyn. Control Syst. 6(3), 331–339 (2000)

    Article  MathSciNet  Google Scholar 

  71. M. Talagrand, Pettis integral and measure theory. Mem. AMS 51 (1984)

    Article  MathSciNet  Google Scholar 

  72. S. Todorc̆ević, Topics in Topology. Lecture Notes in Mathematics, vol. 1652 (Springer, Berlin, 1997)

    Google Scholar 

  73. V.V. Uspenskij, A universal topological group with countable base. Funct. Anal. Appl. 20, 160–161 (1986)

    Article  Google Scholar 

  74. V.V. Uspenskij, Compactifications of topological groups, in Proceedings of the Ninth Prague Topological Symposium (Prague, August 19–25, 2001), ed. by P. Simon. Published April 2002 by Topology Atlas (electronic publication), pp. 331–346. ArXiv:math.GN/0204144

    Google Scholar 

  75. W.A. Veech, Point-distal flows. Am. J. Math. 92, 205–242 (1970)

    Article  MathSciNet  Google Scholar 

  76. B. Weiss, Minimal models for free actions, in Dynamical Systems and Group Actions. Contemporary Mathematics, vol. 567 (American Mathematical Society, Providence, 2012), pp. 249–264

    Google Scholar 

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Acknowledgements

This research was supported by a grant of Israel Science Foundation (ISF 668/13). The first named author thanks the Hausdorff Institute at Bonn for the opportunity to participate in the Program “Universality and Homogeneity” where part of this work was written, November 2013.

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Authors and Affiliations

  1. Department of Mathematics, Tel-Aviv University, Ramat Aviv, Israel

    Eli Glasner

  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

    Michael Megrelishvili

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  1. Eli Glasner
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Correspondence to Eli Glasner .

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  1. CNRS UMR 7373, Institut de Mathématiques de Marseille, Marseille, France

    Sébastien Ferenczi

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Joanna Kułaga-Przymus

  3. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Mariusz Lemańczyk

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Glasner, E., Megrelishvili, M. (2018). More on Tame Dynamical Systems. In: Ferenczi, S., Kułaga-Przymus, J., Lemańczyk, M. (eds) Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 2213. Springer, Cham. https://doi.org/10.1007/978-3-319-74908-2_18

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