Skip to main content
SpringerLink
Log in

Mean equicontinuity, almost automorphy and regularity

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

Abstract

The aim of this article is to obtain a better understanding and classification of strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map of a minimal topological dynamical system has trivial (one point) fibers. In other words, we characterize when minimal mean equicontinuous systems are almost automorphic. Furthermore, we investigate another natural subclass of mean equicontinuous systems, so-called diam-mean equicontinuous systems, and show that a minimal system is diam-mean equicontinuous if and only if the maximal equicontinuous factor is regular (the points with trivial fibers have full Haar measure). Combined with previous results in the field, this provides a natural characterization for every step of a natural hierarchy for strictly ergodic topological models of ergodic systems with discrete spectrum. We also construct an example of a transitive almost diam-mean equicontinuous system with positive topological entropy, and we give a partial answer to a question of Furstenberg related to multiple recurrence.

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. E. Akin, J. Auslander and K. Berg, When is a transitive map chaotic?, in Convergence in Ergodic Theory and Probability (Colombus, OH, 1993), Ohio State University Mathematical Research Institute Publications, Vol. 5, de Gryter, Berlin, 1996, pp. 25–40.

    Chapter  Google Scholar 

  2. J. Auslander, Mean-L-stable systems, Illinois Journal of Mathematics 3 (1959), 566–579.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, Vol. 153, North-Holland, Amsterdam, 1988.

    MATH  Google Scholar 

  4. M. Baake, T. Jäger and D. Lenz, Toeplitz flows and model sets, Bulletin of the London Mathematical Society 48 (2016), 691–698.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Baake, D. Lenz and R. V. Moody, Characterization of model sets by dynamical systems, Ergodic Theory and Dynamical Systems 27 (2007), 341–382.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Downarowicz, Survey of odometers and Toeplitz flows, in Algebraic and Topological Dynamics, Contemporary Mathematics, Vol. 385, American Mathematical Society, Providence, RI, 2005, pp. 7–38.

    Chapter  MATH  Google Scholar 

  7. T. Downarowicz and E. Glasner, Isomorphic extension and applications, Topological Methods in Nonlinear Analysis 48 (2016), 321–338.

    Article  MathSciNet  MATH  Google Scholar 

  8. T. Downarowicz and S. Kasjan, Odometers and Toeplitz systems revisited in the context of Sarnak’s conjecture, Studia Mathematica 229 (2015), 45–72.

    MathSciNet  MATH  Google Scholar 

  9. R. Ellis, A semigroup associated with a transformation group, Transactions of the American Mathematical Society 94 (1960), 272–281.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Transactions of the American Mathematical Society 94 (1960), 258–271.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Fomin, On dynamical systems with a purely point spectrum, Proceedings of the USSR Academy of Sciences 77 (1951), 29–32.

    MathSciNet  Google Scholar 

  12. G. Fuhrmann, E. Glasner, T. Jäger and C. Oertel, Irregular model sets and tame dynamics, Transactions of the American Mathematical Society, to appear.

  13. G. Fuhrmann, M. Gröger and D. Lenz, The structure of mean equicontinuous group actions, Israel Journal of Mathematics, to appear.

  14. G. Fuhrmann and D. Kwietniak, On tameness of almost automorphic dynamical systems for general groups, Bulletin of the London Mathematical Society 52 (2020), 24–42.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Furstenberg, The structure of distal flows, American Journal of Mathematics 85 (1963), 477–515.

    Article  MathSciNet  MATH  Google Scholar 

  16. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Mathématique 31 (1977), 204–256.

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Furstenberg, Poincare recurrence and number theory, Bulletin of the American Mathematical Society 5 (1981), 211–234.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. García-Ramos, Weak forms of topological and measure theoretical equicontinuity: relationships with discrete spectrum and sequence entropy, Ergodic Theory and Dynamical Systtems 37 (2017), 1211–1237.

    Article  MathSciNet  MATH  Google Scholar 

  19. F. García-Ramos and D. Kwietniak, On topological models of zero entropy loosely Bernoulli systems, https://arxiv.org/abs/2005.02484.

  20. F. García-Ramos, J. Li and R. Zhang, When is a dynamical system mean sensitive?, Ergodic Theory and Dynamical Systems 39 (2019), 1608–1636.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. García-Ramos and B. Marcus, Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems, Discrete and Continuous Dynamical Systems. Series A 39 (2019), 729–746.

    Article  MathSciNet  MATH  Google Scholar 

  22. E. Glasner, The structure of tame minimal dynamical systems, Ergodic Theory and Dynamical Systems 27 (2007), 1819–1837.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Glasner, The structure of tame minimal dynamical systems for general groups, Inventiones Mathematicae 211 (2018), 213–244.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Glasner, W. Huang, S. Shao and X. Ye, Regionally proximal relation of order d along arithmetic progressions and nilsystems, Science China. Mathematics 63 (2020), 1757–1776.

    Article  MathSciNet  MATH  Google Scholar 

  25. T. N. T. Goodman, Topological sequence entropy, Proceedings of the London Mathematical Society 3 (1974), 331–350.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. Graf, Selected results on measurable selections, Rendiconti del Circolo Matematico di Palermo Suppl. 2 (1982), 87–122.

    MathSciNet  MATH  Google Scholar 

  27. P. Halmos and J. von Neumann, Operator methods in classical mechanics II, Annals of Mathematics 43 (1942), 332–350.

    Article  MathSciNet  MATH  Google Scholar 

  28. W. Huang, S. Kolyada and G. Zhang, Analogues of Auslander-Yorke theorems for multi-sensitivity, Ergodic Theory and Dynamical Systems 38 (2018), 651–665.

    Article  MathSciNet  MATH  Google Scholar 

  29. W. Huang, S. Li, S. Shao and X. Ye, Null systems and sequence entropy pairs, Ergodic Theory and Dynamical Systems 23 (2003), 1505–1523.

    Article  MathSciNet  MATH  Google Scholar 

  30. W. Huang, S. Shao and X. Ye, Pointwise convergence of multiple ergodic averages and strictly ergodic models, Journal d’Analyse Mathématique 139 (2019), 265–305.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. Huang, J. Li, J. P. Thouvenot, L. Xu and X. Ye, Bounded complexity, mean equicontinuity and discrete spectrum, Ergodic Theory and Dynamical Systems 41 (2021), 494–533.

    Article  MathSciNet  MATH  Google Scholar 

  32. K. Jacobs and M. Keane, 0-1-sequences of Toeplitz type, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 13 (1969), 123–131.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Kerr and H. Li, Dynamical entropy in Banach spaces, Inventiones Mathematica 162 (2005), 649–686.

    Article  MathSciNet  MATH  Google Scholar 

  34. D. Kerr and H. Li, Independence in topological and C*-dynamics, Mathematische Annalen 338 (2007), 869–926.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. G. Kushnirenko, On metric invariants of entropy type, Russian Mathematical Surveys 22 (1967), 53–61.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Kwietniak, J. Li, P. Oprocha and X. Ye, Multi-recurrence and van der Waerden systems, Science China. Mathematics 60 (2017), 59–82.

    Article  MathSciNet  MATH  Google Scholar 

  37. E. Lehrer, Topological mixing and uniquely ergodic systems, Israel Journal of Mathematics 57 (1972), 239–255.

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Li, S. Tu and X. Ye, Mean equicontinuity and mean sensitivity, Ergodic Theory and Dynamical Systems 35 (2015), 2587–2612.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Li, X. Ye and T. Yu, Mean equicontnuity, bounded complexity and applications, Discrete and Continuous Dynamical Systems. Series A 41 (2021), 359–393.

    Article  MathSciNet  Google Scholar 

  40. D. Ornstein and B. Weiss, Mean distality and tightness, Proceedings of the Steklov Institute of Mathematics 244 (2004), 312–319.

    MathSciNet  MATH  Google Scholar 

  41. M. E. Paul, Construction of almost automorphic symbolic minimal flows, General Topology and its Applications 6 (1976), 45–56.

    Article  MathSciNet  MATH  Google Scholar 

  42. P. A. B. Pleasants and C. Huck, Entropy and diffraction of the k-free points in n-dimensional lattices, Discrete & Computational Geometry 50 (2013), 39–68

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Qiu and J. Zhao, A note on mean equicontinuity, Journal of Dynamics and Differential Equations 32 (2020), 101–116.

    Article  MathSciNet  MATH  Google Scholar 

  44. B. Scarpellini, Stability properties of flows with pure point spectrum, Journal of the London Mathematical Society 2 (1982), 451–464.

    Article  MathSciNet  MATH  Google Scholar 

  45. W. A. Veech, The equicontinuous structure relation for minimal abelian transformation groups, American Journal of Mathematics 90 (1968), 723–732.

    Article  MathSciNet  MATH  Google Scholar 

  46. J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Annals of Mathematics 33 (1932), 587–642.

    Article  MathSciNet  MATH  Google Scholar 

  47. P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, Springer, New York-Berlin, 1982.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CONACyT - Institute of Physics, Universidad Autónoma de San Luis Potosí, Av. Manuel Nava #6, Zona Universitaria, C.P. 78290, San Luis Potosí, S.L.P., Mexico

    Felipe García-Ramos

  2. Institute of Mathematics, Friedrich Schiller University Jena, 07737, Jena, Germany

    Tobias Jäger

  3. Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, Anhui, 230026, China

    Xiangdong Ye

  4. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

    Xiangdong Ye

Authors
  1. Felipe García-Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobias Jäger
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiangdong Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felipe García-Ramos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García-Ramos, F., Jäger, T. & Ye, X. Mean equicontinuity, almost automorphy and regularity. Isr. J. Math. 243, 155–183 (2021). https://doi.org/10.1007/s11856-021-2157-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2157-6

Search

Navigation