Skip to main content
SpringerLink
Log in

Möbius disjointness for models of an ergodic system and beyond

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

Abstract

Given a topological dynamical system (X, T) and an arithmetic function u: ℕ → ℂ, we study the strong MOMO property (relatively to u) which is a strong version of u-disjointness with all observable sequences in (X, T). It is proved that, given an ergodic measure-preserving system (Z, \(\mathcal{D}\), к, R),the strong MOMO propertly (relately to u) of a uniquely ergodic midel (X, T)of R yields all other uniquely ergodic midel of R to be u-disjiont. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue—Viorse and Rudin—Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak5s conjecture implies the strong MOMO property relatively to μ in all zero entropy systems; in particular, it makes μ-disjointness uniform. The absence of the strong MOMO property in positive entropy systems is discussed and it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

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. H. el Abdalaoui, M. Lemanczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, Journal of Functional Analysis 266 (2014), 284–317.

    Article  MathSciNet  Google Scholar 

  2. E. H. el Abdalaoui, M. Lemanczyk and T. de la Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, International Mathematics Research Notices 14 (2017), 4350–4368.

    MathSciNet  Google Scholar 

  3. E. H. el Abdalaoui, S. Kasjan and M. Lemanczyk, 0-1 sequences of the Thue—Morse type and Sarnak,s conjecture, Proceedings of the American Mathematical Society 144 (2016), 161–176.

    Article  MathSciNet  Google Scholar 

  4. E. H. el Abdalaoui, J. Kulaga-Przymus, M. Lemanczyk and T. de la Rue, The Chowla and the Sarnak conjectures from ergodic theory point of view, Discrete and Continuous Dynamical Systems 37 (2017), 2899–2944.

    Article  MathSciNet  Google Scholar 

  5. J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge University Press, Cambridge, 2003.

    Book  Google Scholar 

  6. J. Bourgain, Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat, Journal d'Analyse Mathématique 119 (2013), 147–163.

    Article  MathSciNet  Google Scholar 

  7. J. Bourgain, On the correlation of the Moebius function with rank-one systems, Journal d'Analyse Mathématique 120 (2013), 105–130.

    Article  MathSciNet  Google Scholar 

  8. J. Bourgain, P. Sarnak and T. Ziegler, Disjointneßs of Möbius from horocycle ßows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Developments in Mathematics, Vol. 28, Springer, New York, 2013, pp. 67–83.

    Google Scholar 

  9. J. Chaika and A. Eskin, Möbius disjointness for interval exchange transformations on three intervals, https://arxiv.org/abs/1606.02357.

  10. J.-P. Conze, T. Downarowicz and J. Serafin, Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles, Transactions of the American Mathematical Society 369 (2017), 3421–3441.

    Article  MathSciNet  Google Scholar 

  11. H. Davenport, On some infinite series involving arithmetical functions. II, Quarterly Journal of Mathematics 8 (1937), 313–320.

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. A. del Junco, Disjointness of measure-preserving transformations, minimal self-joinings and category, in Ergodic Theory and Dynamical Systems, I (College Park, Md.; 1979-80), Progress in Mathematics, Vol. 10, Birkhäuser, Boston, MA, 1981, pp. 81–89.

    MATH  Google Scholar 

  14. J.-M. Deshouillers, M. Drmota and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Mathematica 231 (2015), 83–95.

    MathSciNet  MATH  Google Scholar 

  15. T. Downarowicz and J. Serafin, Almost full entropy subshift s uncorrelated to the Möbius function, International Mathematics Research Notices, to appear, https://arxiv.org/abs/1611.02084.

  16. S. Fe re ne zi, J. Kulaga-Przymus, M. Lemanczyk and C. Mauduit, Substitutions and Möbius disjointness, in Ergodic Theory, Dynamical Systems, and the Continuing Infhience of John C. Oxtoby, Contemporary Mathematics, Vol. 678, American Mathematical Society, Providence, RI, 2016, pp. 151–173.

    Google Scholar 

  17. L. Flaminio, K. Freeze к, J. Kuiaga-Przymus and M. Lemanczyk, Approximate orthogonality of powers for ergodic affine и nip ot ent diffeomorphisms on nilmani folds, Studia Mathematica, to appear, https://arxiv.org/abs/1609.00699.

  18. N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  19. N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Analysis (2017), paper no. 19.

    MATH  Google Scholar 

  20. H. Fürstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Mathematical Systems Theory 1 (1967), 1–49.

    Article  MathSciNet  Google Scholar 

  21. B. Green, On (not) computing the Möbius function using bounded depth circuits, Combinatorics, Probability and Computing 21 (2012), 942–951.

    Article  MathSciNet  Google Scholar 

  22. W. Huang, Z. Wang and G. Zhang, Möbius disjointness for topological models of ergodic systems with discrete spectrum, https://arxiv.org/abs/1608.08289.

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

    Article  Google Scholar 

  24. T. Kamae, A topological invariant of substitution minimal sets, Journal of the Mathematical Society Japan 24 (1972), 285–306.

    Article  MathSciNet  Google Scholar 

  25. I. К át ai, A remark on a theorem of H. Daboussi, Acta Mathematica Hungarica 47 (1986), 223–225.

    Article  MathSciNet  Google Scholar 

  26. M. Keane, Generalized Morse sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 10 (1968), 335–353.

    Article  MathSciNet  Google Scholar 

  27. H. B. Keynes and D. Newton, The structure of ergodic measures for compact group extensions, Israel Journal of Mathematics 18 (1974), 363–389.

    Article  MathSciNet  Google Scholar 

  28. J. Kulaga-Przymus and M. Lemanczyk, The Möbius function and continuous extensions of rotations, Monatshefte für Mathematik 178 (2015), 553–582.

    Article  MathSciNet  Google Scholar 

  29. J. Kulaga-Przymus and M. Lemanczyk, Möbius disjointness along ergodic sequences for uniquely ergodic actions, Ergodic Theory and Dynamical Systems, to appear, https://arxiv.org/abs/1703.02347.

  30. J. Kwiatkowski, Spectral isomorphism of Morse dynamical systems, L'Académie Polonaise des Sciences. Bulletin. Série des Sciences Mathématiques 29 (1981), 105–114.

    MathSciNet  MATH  Google Scholar 

  31. J. Kwiatkowski and T. Rojek, The centralizer of Morse shifts induced by arbitrary blocks, Studia Mathematica 88 (1988), 165–181.

    Article  MathSciNet  Google Scholar 

  32. M. Lemanczyk, Toeplitz Z2-extensions, Annales de Plnstitut Henri Poincaré. Probabilités et Statistique 24 (1988), 1–43.

    MATH  Google Scholar 

  33. J. Liu and P. Sarnak, The Möbius function and distal ßows, Duke Mathematical Journal 164 (2015), 1353–1399.

    Article  MathSciNet  Google Scholar 

  34. K. Matomäki and M. RadziwiH, Multiplicative functions in short intervals, Annals of Mathematics 183 (2016), 1015–1056.

    Article  MathSciNet  Google Scholar 

  35. K. Matomäki, M. RadziwiH and T. Tao, An averaged form of Chowla?s conjecture, Algebra & Number Theory 9 (2015), 2167–2196.

    Article  MathSciNet  Google Scholar 

  36. C. Müllner, Automatic sequences fulfill the Sarnak conjecture, Duke Mathematical Journal 166 (2017), 3219–3290.

    Article  MathSciNet  Google Scholar 

  37. M. Queffélec, Substitution Dynamical Systems—Spectral Analysis, Lecture Notes in Mathematics, Vol. 1294, Springer-Verlag, Berlin, 2010.

    Chapter  Google Scholar 

  38. V. V. Ryzhikov, Bounded ergodic constructions, disjointness, and weak limits of powers, Transactions of the Moscow Mathematical Society 74 (2013), 165–171.

    Article  MathSciNet  Google Scholar 

  39. P. Sarnak, Three lectures on the Möbius function, randomness and dynamics, http://publications.ias.edu/sarnak/.

  40. T. Tao, Equivalence of the logarithmically averaged Chowla and Sarnak conjectures, in Number Theory——Diophantine Problems, Uniform Distribution and Applications, Springer, Cham, 2017, pp. 391–421.

    Google Scholar 

  41. T. Tao, Whatfs new. Blog. the Chowla Conjecture and the Sarnak conjecture, https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-thesarnak-conjecture/.

  42. W. A. Veech, Möbius orthogonality for generalized Morse—K akiit ani Bows, American Journal of Mathematics 139 (2017), 1157–1203.

    Article  MathSciNet  Google Scholar 

  43. Z. Wang, Möbius disjointness for analytic skew products, Inventiones Mathematicae 209 (2017), 175–196.

    Article  Google Scholar 

  44. B. Weiss, Strictly ergodic models for dynamical systems, Bulletin of the American Mathematical Society 13 (1985), 143–146.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rouen Normandy, LMRS UMR 60 85 CNRS, Avenue de l’Université, BP. 12, 76801, Saint Etienne du Rouvray, France

    El Houcein el Abdalaoui & Thierry de la Rue

  2. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopin street 12/18, 87-100, Toruń, Poland

    Joanna Kułaga-Przymus & Mariusz Lemánczyk

Authors
  1. El Houcein el Abdalaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joanna Kułaga-Przymus
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mariusz Lemánczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Thierry de la Rue
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mariusz Lemánczyk.

Additional information

Research supported by the special program in the framework of the Jean Morlet semester "Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combinatorics".

Research supported by Narodowe Centrum Naiuki Grant UMO-014/15/B/ST1/03736.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

el Abdalaoui, E.H., Kułaga-Przymus, J., Lemánczyk, M. et al. Möbius disjointness for models of an ergodic system and beyond. Isr. J. Math. 228, 707–751 (2018). https://doi.org/10.1007/s11856-018-1784-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-018-1784-z

Search

Navigation