Skip to main content
SpringerLink
Log in

Self-induced systems

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

A minimal Cantor system is said to be self-induced whenever it is conjugate to one of its induced systems. Substitution subshifts and some odometers are classical examples, and we show that these are the only examples in the equicontinuous or expansive case. Nevertheless, we exhibit a zero entropy self-induced system that is neither equicontinuous nor expansive. We also provide non-uniquely ergodic self-induced systems with infinite entropy. Moreover, we give a characterization of self-induced minimal Cantor systems in terms of substitutions on finite or infinite alphabets.

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. P. Arnoux, Un exemple de semi-conjugaison entre un échange d’intervalles et une translation sur le tore, Bull. Soc. Math. France 116 (1988), 489–500.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Arnoux and S. Ito, Pisot substitutions and Rauzy fractals Bull, Belg. Math. Soc. 8 (2001), 181–207.

    MathSciNet  MATH  Google Scholar 

  3. P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité 2n + 1, Bull. Soc. Math. France 119 (1991), 199–215.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Boshernitzan and C. R. Carroll, An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields, J. Anal. Math. 72 (1997), 21–44.

    Article  MathSciNet  MATH  Google Scholar 

  5. X. Bressaud and Y. Jullian, Interval exchange transformation extension of a substitution dynamical system, Confluentes Math. 4, (2012), no. 4, 1250005.

    MATH  Google Scholar 

  6. E. Coven, A. Dykstra, M. Keane, and M. LeMasurier, Topological conjugacy to given constant length substitution minimal systems, Indag. Math. (N.S.) 25 (2014), 646–651.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Dahl and M. Molberg, Induced subsystems associated to a Cantor minimal system, Colloq. Math. 117 (2009), 207–221.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Dartnell, F. Durand, and A. Maass, Orbit equivalence and Kakutani equivalence with Sturmian subshifts, Studia Math. 142 (2000), 25–45.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. del Junco, D. J. Rudolph, and B. Weiss, Measured topological orbit and Kakutani equivalence, Discrete Contin. Dyn Syst. Ser. S, 2 (2009), 221–238.

    Article  MathSciNet  MATH  Google Scholar 

  10. T. Downarowicz, Survey of odometers and Toeplitz flows,in Algebraic and Topological Dynamics, Amer. Math. Soc., Providence RI, 2005, pp. 7–37.

    Book  MATH  Google Scholar 

  11. T. Downarowicz and A. Maass, Finite-rank Bratteli-Vershik diagrams are expansive, Ergodic Theory Dynam. Systems 28 (2008), 739–747.

    MathSciNet  MATH  Google Scholar 

  12. F. Durand. Contributions à l’étude des suites et systèmes dynamiques substitutifs, Ph.D. thesis, Université de la Méditerranée (Aix-Marseille II), 1996.

    Google Scholar 

  13. F. Durand, A characterization of substitutive sequences using return words, Discrete Math. 179 (1998), 89–101.

    Article  MathSciNet  MATH  Google Scholar 

  14. F. Durand, Combinatorics on Bratteli diagrams and dynamical systems, in Combinatorics, Automata and Number Theory, Cambridge Univ. Press, Cambridge, 2010, pp. 324–372.

    Google Scholar 

  15. F. Durand, Decidability of uniform recurrence of morphic sequences, Internat. J. Found. Comput. Sci. 24 (2013), 123–146.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Durand, B. Host, and C. Skau, Substitutive dynamical systems, Bratteli diagrams and dimension groups, Ergodic Theory Dynam. Systems 19 (1999), 953–993.

    Article  MATH  Google Scholar 

  17. A. Dykstra and D. J. Rudolph, Any two irrational rotations are nearly continuously Kakutani equivalent, J. Anal. Math. 110 (2010), 339–384.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Ferenczi, Substitution dynamical systems on infinite alphabets, Ann. Inst. Fourier (Grenoble) 56 (2006), 2315–2343.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Ferenczi, C. Holton, and L. Q. Zamboni, Structure of three-interval exchange transformations. II. A combinatorial description of the trajectories, J. Anal. Math. 89 (2003), 239–276.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Ferenczi, C. Mauduit, and A. Nogueira, Substitution dynamical systems: algebraic characterization of eigenvalues, Ann. Sci. Ecole Norm. Sup. 29 (1996), 519–533.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Springer-Verlag, Berlin, 2002.

    MATH  Google Scholar 

  23. T. Giordano, I. Putnam, and C. Skau, Topological orbit equivalence and C* crossed products, J. Reine Angew. Math. 469 (1995), 51–111.

    MathSciNet  MATH  Google Scholar 

  24. R. Gjerde and O. Johansen, Bratteli-Vershik models for cantor minimal systems: applications to Toeplitz flows, Ergodic Theory Dynam. Systems 20 (2000), 1687–1710.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Gjerde and O. Johansen, Bratteli-Vershik models for Cantor minimal systems associated to interval exchange transformations, Math. Scand. 90 (2002), 87–100.

    Article  MathSciNet  MATH  Google Scholar 

  26. R. H. Herman, I. Putnam, and C. F. Skau, Ordered Bratteli diagrams, dimension groups and topological dynamics, Internat. J. Math. 3 (1992), 827–864.

    Article  MathSciNet  MATH  Google Scholar 

  27. Y. Jullian, An algorithm to identify automorphisms which arise from self-induced interval exchange transformations, Math. Z. 274 (2013), 33–55.

    Article  MathSciNet  MATH  Google Scholar 

  28. W. Kosek, N. Ormes, and D. J. Rudolph, Flow-orbit equivalence for minimal Cantor systems, Ergodic Theory Dynam. Systems, 28 (2008), 481–500.

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Kurka, Topological and Symbolic Dynamics, Société Mathématique de France, Paris, 2003.

    Google Scholar 

  30. D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge Univ. Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  31. X. Méla and K. Petersen, Dynamical properties of the Pascal adic transformation, Ergodic Theory Dynam. Systems 25 (2005), 227–256.

    Article  MathSciNet  MATH  Google Scholar 

  32. B. Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), 327–334.

    Article  MathSciNet  MATH  Google Scholar 

  33. B. Mossé, Reconnaissabilité des substitutions et complexité des suites automatiques, Bull. Soc. Math. France 124 (1996), 329–346.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Nogueira, Nonorientable recurrence of flows and interval exchange transformations, J. Differential Equations 70 (1987), 53–166.

    Article  MathSciNet  MATH  Google Scholar 

  35. D. S. Ornstein, D. J. Rudolph, and B. Weiss, Equivalence of measure preserving transformations, Mem. Amer. Math. Society, Providence RI, 1982.

    Google Scholar 

  36. K. Petersen, Ergodic Theory, Cambridge Univ. Press, Cambridge, 1983.

    Book  MATH  Google Scholar 

  37. M. Queffélec, Substitution Dynamical Systems–Spectral Analysis, Springer-Verlag, Berlin, 1987.

    Book  MATH  Google Scholar 

  38. G. Rauzy, Échanges d’intervalles et transformations induites, Acta Arith. 34 (1979), 315–328.

    Article  MathSciNet  MATH  Google Scholar 

  39. G. Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110 (1982), 147–178.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. R. Roychowdhury and D. J. Rudolph, Nearly continuous Kakutani equivalence of adding machines, J. Mod. Dyn. 3 (2009), 103–119.

    Article  MathSciNet  MATH  Google Scholar 

  41. W. Veech, Interval exchange transformations, J. Anal. Math. 33 (1978), 222–272.

    Article  MathSciNet  MATH  Google Scholar 

  42. S. Williams Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. Verw. Gebiete 67 (1984), 95–107.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Amiénois de Mathématiques Fondamentales et Appliquées, CNRS-UMR 7352, Université de Picardie Jules Verne, 80039, Amiens Cedex 1, France

    Fabien Durand & Samuel Petite

  2. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    Nicholas Ormes

Authors
  1. Fabien Durand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicholas Ormes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Samuel Petite
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Petite.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Durand, F., Ormes, N. & Petite, S. Self-induced systems. JAMA 135, 725–756 (2018). https://doi.org/10.1007/s11854-018-0051-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0051-x

Search

Navigation