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Dynamical simplices and Borel complexity of orbit equivalence

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Abstract

We prove that any divisible dynamical simplex is the set of invariant measures of some Toeplitz subshift. We apply our construction to prove that orbit equivalence of Toeplitz subshifts is Borel bireducible to the universal equivalence relation induced by a Borel action of S∞.

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References

  1. E. Akin, Good measures on Cantor space, Transactions of the American Mathematical Society 357 (2005), 2681–2722.

    Article  MathSciNet  Google Scholar 

  2. R. Camerlo and S. Gao, The completeness of the isomorphism relation for countable Boolean algebras, Transactions of the American Mathematical Society 353 (2001), 491–518.

    Article  MathSciNet  Google Scholar 

  3. H. Dahl, Cantor minimal systems and AF equivalence relations, Ph.d. thesis, Norwegian University of Science and Technology, 2008.

    Google Scholar 

  4. T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel Journal of Mathematics 74 (1991), 241–256.

    Article  MathSciNet  Google Scholar 

  5. M. Foreman, A descriptive view of ergodic theory, in Descriptive Set Theory and Dynamical Systems (Marseille-Luminy, 1996), London Mathematical Society Lecture Note Series, Vol. 277, Cambridge University Press, Cambridge, 2000, pp. 87–171.

    Article  MathSciNet  Google Scholar 

  6. H. Friedman and L. Stanley, A Borel reducibility theory for classes of countable structures, Journal of Symbolic Logic 54 (1989), 894–914.

    Article  MathSciNet  Google Scholar 

  7. B. Fuchssteiner and W. Lusky, Convex Cones, North-Holland Mathematics Studies, Vol. 56, North-Holland, Amsterdam-New York, 1981.

  8. S. Gao, S. Jackson and B. Seward, Group colorings and Bernoulli subflows, Memoirs of the American Mathematical Society 241 (2016).

    Article  MathSciNet  Google Scholar 

  9. S. Gao, Invariant Descriptive Set Theory, Pure and Applied Mathematics (Boca Raton), Vol. 293, CRC Press, Boca Raton, FL, 2009.

  10. T. Giordano, I. F. Putnam and C. F. Skau, Topological orbit equivalence and C*-crossed products, Journal für die Reine und Angewandte Mathematik 469 (1995), 51–111.

    MathSciNet  MATH  Google Scholar 

  11. E. Glasner and B. Weiss, Weak orbit equivalence of Cantor minimal systems, International Journal of Mathematics 6 (1995), 559–579.

    Article  MathSciNet  Google Scholar 

  12. G. Hjorth, Classification and Orbit Equivalence Relations, Mathematical Surveys and Monographs, Vol. 75, American Mathematical Society, Providence, RI, 2000.

  13. T. Ibarlucía and J. Melleray, Dynamical simplices and minimal homeomorphisms, Proceedings of the American Mathematical Society 145 (2017), 4981–4994.

    Article  MathSciNet  Google Scholar 

  14. B. Kaya, The complexity of topological conjugacy of pointed Cantor minimal systems, Archive for Mathematical Logic 56 (2017), 215–235.

    Article  MathSciNet  Google Scholar 

  15. A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts inMathematics, Vol. 156, Springer, New York, 1995.

  16. J. Melleray, Dynamical simplices and Fraïssé theory, Ergodic Theory and Dynamical Systems 39 (2019), 3111–3126.

    Article  MathSciNet  Google Scholar 

  17. F. Sugisaki, Toeplitz flows, ordered Bratteli diagrams and strong orbit equivalence, Ergodic Theory and Dynamical Systems 21 (2001), 1867–1881.

    Article  MathSciNet  Google Scholar 

  18. S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 67 (1984), 95–107.

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622, Villeurbanne, France

    Julien Melleray

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  1. Julien Melleray
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Correspondence to Julien Melleray.

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Melleray, J. Dynamical simplices and Borel complexity of orbit equivalence. Isr. J. Math. 236, 317–344 (2020). https://doi.org/10.1007/s11856-020-1976-1

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  • DOI: https://doi.org/10.1007/s11856-020-1976-1

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