Skip to main content
SpringerLink
Log in

Rank-one actions, their (C, F)-models and constructions with bounded parameters

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Let G be a discrete countable infinite group. We show that each topological (C, F)-action T of G on a locally compact non-compact Cantor set is a free minimal amenable action admitting a unique up to scaling non-zero invariant Radon measure (answer to a question by Kellerhals, Monod and Rørdam). We find necessary and sufficient conditions under which two such actions are topologically conjugate in terms of the underlying (C, F)-parameters. If G is linearly ordered abelian, then the topological centralizer of T is trivial. If G is monotileable and amenable, denote by \({{\cal A}_G}\) the set of all probability preserving actions of G on the unit interval with Lebesgue measure and endow it with the natural topology. We show that the set of (C, F)-parameters of all (C, F)-actions of G furnished with a suitable topology is a model for \({{\cal A}_G}\) in the sense of Foreman, Rudolph and Weiss. If T is a rank-one transformation with bounded sequences of cuts and spacer maps, then we found simple necessary and sufficient conditions on the related (C, F)-parameters under which (i) T is rigid, (ii) T is totally ergodic. An alternative proof is found of Ryzhikov’s theorem that if T is totally ergodic and a non-rigid rank-one map with bounded parameters, then T has MSJ. We also give a more general version of the criterion (by Gao and Hill) for isomorphism and disjointness of two commensurate non-rigid totally ergodic rank-one maps with bounded parameters. It is shown that the rank-one transformations with bounded parameters and no spacers over the last subtowers is a proper subclass of the rank-one transformations with bounded parameters.

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. T. Adams, S. Ferenczi and K. Petersen, Constructive symbolic presentations of rank one measure-preserving systems, Colloq. Math. 150 (2017), 243–255.

    Article  MathSciNet  Google Scholar 

  2. C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C*-algebras, Trans. Amer. Math. Soc. 354 (2002), 4153–4178.

    Article  MathSciNet  Google Scholar 

  3. P. Berk and K. Frączek, On special flows over IETs that are not isomorphic to their inverses, Discrete Contin. Dyn. Syst. 35 (2015), 829–855.

    MathSciNet  MATH  Google Scholar 

  4. J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math. 120 (2013), 105–130.

    Article  MathSciNet  Google Scholar 

  5. A. I. Danilenko, Funny rank-one weak mixing for nonsingular abelian actions, Israel J. Math. 121 (2001), 29–54.

    Article  MathSciNet  Google Scholar 

  6. A. I. Danilenko, Strong orbit equivalence of locally compact Cantor minimal systems, Internat. J. Math. 12 (2001), 113–123.

    Article  MathSciNet  Google Scholar 

  7. A. I. Danilenko, Infinite rank one actions and nonsingular Chacon transformations, Illinois J. Math. 48 (2004), 769–786.

    Article  MathSciNet  Google Scholar 

  8. A. I. Danilenko, Explicit solution of Rokhlin’s problem on homogeneous spectrum and applications, Ergodic Theory Dynam. Systems 26 (2006), 1467–1490.

    Article  MathSciNet  Google Scholar 

  9. A. I. Danilenko, (C, F)-actions in ergodic theory, in Geometry and Dynamics of Groups and Spaces, Birkhäuser, Basel, 2008, pp. 325–351.

    Chapter  Google Scholar 

  10. A. I. Danilenko, Actions of finite rank: weak rational ergodicity and partial rigidity, Ergodic Theory Dynam. Systems 36 (2016), 2138–2171.

    Article  MathSciNet  Google Scholar 

  11. A. del Junco, A simple map with no prime factors, Israel J. Math. 104 (1998), 301–320.

    Article  MathSciNet  Google Scholar 

  12. A. del Junco, M. Rahe and L. Swanson, Chacon’s automorphism has minimal self joininings, J. Anal. Math. 27 (1980), 276–284.

    Article  Google Scholar 

  13. A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531–557.

    Article  MathSciNet  Google Scholar 

  14. E. H. El Abdalaoui, M. Lemańczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal. 266 (2014), 284–317.

    Article  MathSciNet  Google Scholar 

  15. S. Ferenczi, Systèmes de rang un gauche, Ann. Inst. Henri Poincaré. 21 (1985), 177–186.

    MathSciNet  MATH  Google Scholar 

  16. S. Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), 35–65.

    Article  MathSciNet  Google Scholar 

  17. M. Foreman, Models for measure preserving transformations, Topol. Appl. 157 (2010), 1404–1414.

    Article  MathSciNet  Google Scholar 

  18. M. Foreman and B. Weiss, An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. 6 (2004), 277–292.

    Article  MathSciNet  Google Scholar 

  19. N. A. Friedman and J. L. King, Rankone lightly mixing, Israel J. Math. 73 (1991), 281–288.

    Article  MathSciNet  Google Scholar 

  20. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation, Math. Systems Theory 1 (1967), 1–50.

    Article  MathSciNet  Google Scholar 

  21. S. Gao and A. Hill, A model for rank one measure preserving transformations, Topology Appl. 174 (2014), 25–40.

    Article  MathSciNet  Google Scholar 

  22. S. Gao and A. Hill, Topological isomorphism for rank-1 systems, J. Anal. Math. 128 (2016), 1–49.

    Article  MathSciNet  Google Scholar 

  23. S. Gao and A. Hill, Bounded rank-one transformations, J. Anal. Math. 129 (2016), 341–365.

    Article  MathSciNet  Google Scholar 

  24. S. Gao and A. Hill, Disjointness between bounded rank-one transformations, arXiv:1601.04119 [math.DS].

  25. E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. Math. 78 (1992), 131–142.

    Article  MathSciNet  Google Scholar 

  26. A. Hill, Centralizers of rank-one homeomorphisms, Ergodic Theory Dynam. Systems 34 (2014), 543–556.

    Article  MathSciNet  Google Scholar 

  27. A. Hill, The inverse problem for canonically bounded rank-one transformations, in Ergodic Theory, Dynamical Systems, and the Continuing Influence of John C. Oxtoby, American Mathematical Society, Providence, RI, 2016, pp. 219–229.

    Chapter  Google Scholar 

  28. S. A. Kalikow, Twofold mixing implies threefold mixing for rank one transformations, Ergodic Theory Dynam. Systems 4 (1984), 237–259.

    Article  MathSciNet  Google Scholar 

  29. A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.

    Book  Google Scholar 

  30. J. Kellerhals, N. Monod and M. Rørdam, Non-supramenable groups acting on locally compact spaces, Doc. Math. 18 (2013). 1597–1626.

    MathSciNet  MATH  Google Scholar 

  31. F. W. Levi, Ordered groups, Proc. Indian Acad. Sci. A16 (1942), 256–263.

    Article  MathSciNet  Google Scholar 

  32. H. Matui and M. Rørdam, Universal properties of group actions on locally compact spaces, J. Funct. Anal. 268 (2015), 3601–3648.

    Article  MathSciNet  Google Scholar 

  33. D. J. Rudolph, Fundamentals of Measurable Dynamics, The Clarendon Press, Oxford University Press, New York, 1990.

    MATH  Google Scholar 

  34. V. V. Ryzhikov, Minimal self-joinings, bounded constructions, and weak closure of ergodic actions, preprint, arXiv:1212.2602 [math.DS].

  35. C. E. Silva, Invitation to Ergodic Theory, American Mathematical Society, Providence, RI, 2008.

    MATH  Google Scholar 

  36. B. Weiss, Monotileable amenable groups, in Topology, Ergodic Theory, Real Algebraic Geometry, American Mathematical Society, Providence, RI, 2001, pp. 257–262.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, 47 Nauki Ave., Kharkov, 61164, Ukraine

    Alexandre I. Danilenko

Authors
  1. Alexandre I. Danilenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexandre I. Danilenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Danilenko, A.I. Rank-one actions, their (C, F)-models and constructions with bounded parameters. JAMA 139, 697–749 (2019). https://doi.org/10.1007/s11854-023-0075-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-023-0075-8

Search

Navigation