Skip to main content
SpringerLink
Log in

Uniformly distributed orbits in \(\mathbb {T}^d\) and singular substitution dynamical systems

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We find sufficient conditions for the singularity of a substitution \(\mathbb {Z}\)-action’s spectrum, which generalize a result of Bufetov and Solomyak, and we also obtain a similar statement for a collection of substitution \(\mathbb {R}\)-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector \(\textbf{v}\in \mathbb {Q}^d\), we find necessary and sufficient conditions for the orbit of \(\omega \textbf{v}\) to be uniformly distributed modulo 1 for almost every \(\omega \in \mathbb {R}\). We use our results to find new examples of singular substitution \(\mathbb {Z}\)- and \(\mathbb {R}\)-actions.

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. Baake, M., Frank, N.P., Grimm, U., Robinson, E.A., Jr.: Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction. Stud. Math. 247(2), 109–154 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baake, M., Gähler, F.: Pair correlations of aperiodic inflation rules via renormalisation: some interesting examples. Topol. Appl. 205, 4–27 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baake, M., Gähler, F., Mañibo, N.: Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction. Commun. Math. Phys. 370(2), 591–635 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baake, M., Grimm, U.: In: Aperiodic Order. A Mathematical Invitation, vol. 1. Cambridge University Press, Cambridge (2013)

  5. Baake, Michael, Grimm, Uwe, Mañibo, Neil: Spectral analysis of a family of binary inflation rules. Lett. Math. Phys. 108(8), 1783–1805 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baake, M., Haynes, A., Lenz, D.: Averaging almost periodic functions along exponential sequences. In: Aperiodic Order. Volume 2: Crystallography and Almost Periodicity, vol. 166, pp. 343–362. Cambridge University Press, Cambridge (2017)

    Chapter  Google Scholar 

  7. Barge, M., Diamond, B.: Coincidence for substitutions of Pisot type. Bull. Soc. Math. France 130(4), 619–626 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bartlett, A.: Spectral theory of \(\mathbb{Z} ^d\) substitutions. Ergodic Theory Dyn. Syst. 38(4), 1289–1341 (2018)

    Article  MATH  Google Scholar 

  9. Berend, D., Radin, C.: Are there chaotic tilings? Commun. Math. Phys. 152(2), 215–219 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berlinkov, A., Solomyak, B.: Singular substitutions of constant length. Ergodic Theory Dyn. Syst. 39(9), 2384–2402 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boyd, D.W.: Kronecker’s theorem and Lehmer’s problem for polynomials in several variables. J. Number Theory 13(1), 116–121 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bufetov, A.I., Solomyak, B.: On the modulus of continuity for spectral measures in substitution dynamics. Adv. Math. 260, 84–129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bufetov, A.I., Solomyak, B.: On singular substitution \(\mathbb{Z}\)-actions, 2020. arXiv:2003.11287

  14. Bufetov, A.I., Solomyak, B.: A spectral cocycle for substitution systems and translation flows. J. Anal. Math. 141, 165–205 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chan, L., Grimm, U..: Substitution-based sequences with absolutely continuous diffraction. J. Phys. Conf. Series 809 (2017)

  16. Clark, A., Sadun, L.: When size matters: subshifts and their related tiling spaces. Ergodic Theory Dyn. Syst. 23(4), 1043–1057 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dekking, F.M.: The spectrum of dynamical systems arising from substitutions of constant length. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41(3), 221–239 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dekking, F.M., Keane, M.: Mixing properties of substitutions. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 42(1), 23–33 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  19. Einsiedler, M., Ward, T.: Ergodic Theory with a View Towards Number Theory. Graduate Texts in Mathematics, vol. 259. Springer-Verlag, London (2011)

    Book  MATH  Google Scholar 

  20. Everest, G., van der Poorten, A.J., Shparlinski, I., Ward, T.: Recurrence Sequences. American Mathematical Society, Providence, RI (2003)

    Book  MATH  Google Scholar 

  21. Evertse, J.-H.: On sums of \(S\)-units and linear recurrences. Compos. Math. 53(2), 225–244 (1984)

    MathSciNet  MATH  Google Scholar 

  22. Ferenczi, S., Mauduit, C., Nogueira, A.: Substitution dynamical systems: algebraic characterization of eigenvalues. Annal. Sci. de l’École Normale Supérieure 29(4), 519–533 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fogg, N.P.: Substitutions in Dynamics, Arithmetics and Combinatorics. In: V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel (Eds.) Lecture Notes in Mathematics, vol. 1794. Springer-Verlag, Berlin (2002)

  24. Natalie Priebe Frank: Substitution sequences in \(\mathbb{Z} ^d\) with a non-simple Lebesgue component in the spectrum. Ergodic Theory Dyn. Syst. 23(2), 519–532 (2003)

    MATH  Google Scholar 

  25. Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31(2), 457–469 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hoffman, K., Kunze, R.: Linear Algebra, 2nd edn. Prentice-Hall Inc, Englewood Cliffs (1971)

    MATH  Google Scholar 

  27. Hollander, M., Solomyak, B.: Two-symbol Pisot substitutions have pure discrete spectrum. Ergodic Theory Dyn. Syst. 23(2), 533–540 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Host, B.: Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable. Ergodic Theory Dyn. Syst. 6(4), 529–540 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Host, B.: Représentation géométrique des substitutions sur 2 lettres. Unpublished manuscript (1992)

  30. Host, B.: Some results of uniform distribution in the multidimensional torus. Ergodic Theory Dyn. Syst. 20(2), 439–452 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Wiley, New York (1974)

    MATH  Google Scholar 

  32. Mañibo, N.: Lyapunov exponents for binary substitutions of constant length. J. Math. Phys. 58(11), 113504 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Meiri, D.: Entropy and uniform distribution of orbits in \(\mathbb{T} ^d\). Israel J. Math. 105(1), 155–183 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  35. Pushkin, L.N.: Borel normal vectors on a manifold in \(\textbf{R} ^n\). Theory Probab. Appl. 36(2), 391–395 (1992)

    Article  Google Scholar 

  36. Queffélec, M.: Substitution Dynamical Systems—Spectral Analysis. Lecture Notes in Mathematics, vol. 1294, 2nd edn. Springer-Verlag, Berlin (2010)

    Book  MATH  Google Scholar 

  37. Shorey, T.N., Tijdeman, R.: Exponential Diophantine Equations. Cambridge Tracts in Mathematics, vol. 87. Cambridge University Press, Cambridge (1986)

    Book  MATH  Google Scholar 

  38. Solomyak, B.: Dynamics of self-similar tilings. Ergodic Theory Dyn. Syst. 17(3), 695–738 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. van der Alf Poorten, J.: Some problems of recurrent interest. In: Topics in Classical Number Theory. Colloquia mathematica Societatis Janos Bolyai, 34 (Budapest, 1981), vol. 2, pp. 1265–1294. North-Holland, Amsterdam (1984)

  40. Viana, M.: Lectures on Lyapunov Exponents Cambridge. Studies in Advanced Mathematics, vol. 145. Cambridge University Press, Cambridge (2014)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author is grateful to Boris Solomyak for many helpful ideas, suggestions and comments. This research is a part of the author’s master’s thesis (in preparation) at the Bar-Ilan University under the direction of B. Solomyak and was supported in part by the Israel Science Foundation grant 911/19 (PI B. Solomyak).

Author information

Authors and Affiliations

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

    Rotem Yaari

Authors
  1. Rotem Yaari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rotem Yaari.

Additional information

Communicated by H. Bruin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yaari, R. Uniformly distributed orbits in \(\mathbb {T}^d\) and singular substitution dynamical systems. Monatsh Math 201, 289–306 (2023). https://doi.org/10.1007/s00605-023-01829-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-023-01829-y

Keywords

Mathematics Subject Classification

Search

Navigation