Journal d'Analyse Mathématique

, Volume 109, Issue 1, pp 1–31

Minimality and unique ergodicity for adic transformations

  • Sebastien Ferenczi
  • Albert M. Fisher
  • Marina Talet


We study the relationship between minimality and unique ergodicity for adic transformations. We show that three is the smallest alphabet size for a unimodular “adic counterexample”, an adic transformation which is minimal but not uniquely ergodic. We construct a specific family of counterexamples built from (3 × 3) nonnegative integer matrix sequences, while showing that no such (2 × 2) sequence is possible. We also consider (2 × 2) counterexamples without the unimodular restriction, describing two families of such maps.

Though primitivity of the matrix sequence associated to the transformation implies minimality, the converse is false, as shown by a further example: an adic transformation with (2 × 2) stationary nonprimitive matrix, which is both minimal and uniquely ergodic.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AF01]
    P. Arnoux and A. M. Fisher, The scenery flow for geometric structures on the torus: the linear setting, Chinese Ann. of Math. 4 (2001), 427–470.CrossRefMathSciNetGoogle Scholar
  2. [AF05]
    P. Arnoux and A.M. Fisher, Anosov families, renormalization and nonstationary subshifts, Ergodic Theory Dynam. Systems 25 (2005), 661–709.MATHCrossRefMathSciNetGoogle Scholar
  3. [BKMS09]
    S. Bezuglyi, J. Kwiatkowski, K. Medynets and B. Solomyak, Invariant measures on stationary Bratteli diagrams, Ergodic Theory Dynam. Systems (2009), to appear.Google Scholar
  4. [BM77]
    R. Bowen and B. Marcus, Unique ergodicity for horocycle foliations, Israel J. Math. 26 (1977), 43–67.MATHCrossRefMathSciNetGoogle Scholar
  5. [Cha69]
    R.V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22 (1969), 559–562.MATHCrossRefMathSciNetGoogle Scholar
  6. [Fer95]
    S. Ferenczi, Les transformations de Chacon: combinatoire, structure géométrique, lien aves les syst`emes de complexité 2n + 1, Bull. Soc. Math. France 123 (1995), 272–292.MathSciNetGoogle Scholar
  7. [Fer97]
    S. Ferenczi, Systems of finite rank, Colloq. Math. 73 (1997), 35–65.MATHMathSciNetGoogle Scholar
  8. [Fer02]
    S. Ferenczi, Substitutions and symbolic dynamical systems, in Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Math. 1794, Springer, Berlin, 2002, pp. 101–142.Google Scholar
  9. [FZ06]
    S. Ferenczi and L. Zamboni, Eigenvalues and simplicity of 4-interval exchanges, preprint 2009,
  10. [Fis]
    A. M. Fisher, Abelian differentials, interval exchanges, and adic transformations, in preparation, 2008.Google Scholar
  11. [Fis92]
    A. M. Fisher, Integer Cantor sets and an order-two ergodic theorem, Ergodic Theory Dynam. Systems 13 (1992), 45–64.Google Scholar
  12. [Fis08]
    A. M. Fisher, Nonstationary mixing and the unique ergodicity of adic transformations, Stochastics and Dynamics, 2009, to appear.Google Scholar
  13. [Fur61]
    H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J.Math. 83 (1961), 573–601.MATHCrossRefMathSciNetGoogle Scholar
  14. [Fur73]
    H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics, Lecture Notes in Math. 318, Springer-Verlag, Berlin, 1973, pp. 95–115.CrossRefGoogle Scholar
  15. [Hos00]
    B. Host, Substitution subshifts and Bratteli diagrams, in Topics in Symbolic Dynamics and Applications, Cambridge Univ. Press, Cambridge, 2000, pp. 35–55.Google Scholar
  16. [Jew70]
    R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19 (1970), 717–729.MATHMathSciNetGoogle Scholar
  17. [Kea75]
    M. Keane, Interval exchange transformations, Math. Z. 141 (1975), 25–31.MATHCrossRefMathSciNetGoogle Scholar
  18. [Kea77]
    M. Keane, Non-ergodic interval exchange transformations, Israel J. Math. 26 (1977), 188–196.MATHCrossRefMathSciNetGoogle Scholar
  19. [KN76]
    H. B. Keynes and D. Newton, A “minimal”, non-uniquely ergodic interval exchange transformation, Math. Z. 148 (1976), 101–105.MATHCrossRefMathSciNetGoogle Scholar
  20. [Kri70]
    W. Krieger, On unique ergodicity, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability Theory, Univ. California press, Berkeley, Calif., 1972, 327–346.Google Scholar
  21. [LM95]
    D. Lind and B. Marcus, Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.Google Scholar
  22. [Liv87]
    A. N. Livshits, On the spectra of adic transformations of Markov compacta, Russian Math. Surveys 42 (1987), 222–223.MATHCrossRefMathSciNetGoogle Scholar
  23. [Liv88]
    A. N. Livshits, A sufficient condition for weak mixing of substitutions and stationary adic transformations, Math. Notes 44 (1988), 920–925.MATHMathSciNetGoogle Scholar
  24. [LV92]
    A. N. Livshits and A.M. Vershik, Adic models of ergodic transformations, spectral theory, substitutions, and related topics, Adv. Soviet Math. 9 (1992), 185–204.MathSciNetGoogle Scholar
  25. [LV85]
    A. A. Lodkin and A. M. Vershik, Approximation for actions of amenable groups and transversal automorphisms, in Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Math. 1132, Springer-Verlag, New York, 1985, pp. 331–346.CrossRefGoogle Scholar
  26. [Mas82]
    H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), 169–200.CrossRefMathSciNetGoogle Scholar
  27. [MP05]
    X. Mela and K. Petersen, Dynamical properties of the Pascal adic transformation, Ergodic Theory Dynam. Systems 25 (2005), 227–256.MATHCrossRefMathSciNetGoogle Scholar
  28. [Mos92]
    B. Mossé, Puissances de mots et reconnaissabilité des points fixes d’une substitution, Theoret. Comput. Sci. 99 (1992), 327–334.MATHCrossRefMathSciNetGoogle Scholar
  29. [Mos96]
    B. Mossé, Reconnaissabilité des substitutions et complexité des suites automatiques, Bull. Soc. Math. France 124 (1996), 329–346.MATHMathSciNetGoogle Scholar
  30. [Vee78]
    W. A. Veech, Interval exchange transformations, J. Analyse Math. 33 (1978), 222–272.MATHCrossRefMathSciNetGoogle Scholar
  31. [Vee82]
    W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), 201–242.CrossRefMathSciNetGoogle Scholar
  32. [Ver89]
    A.M. Vershik, A new model of the ergodic transformations, in Dynamical Systems and Ergodic Theory, Banach Center Publications 23, PWN-Polish Scientific Publishers, Warsaw, 1989, pp. 381–384.Google Scholar
  33. [Ver81]
    A. M. Vershik, Uniform algebraic approximation of shift and multiplication operators, Soviet Math. Dokl. 24 (1981), 101–103.Google Scholar
  34. [Ver94]
    A.M. Vershik, Locally transversal symbolic dynamics, Algebra i Analiz 6 (1994), 94–106.MathSciNetGoogle Scholar
  35. [Ver95a]
    A. M. Vershik, The adic realizations of the ergodic actions with the homeomorphisms of the Markov compact and the ordered Bratteli diagrams, Teor. Predstav. Din. Systemy Kombin. i Algoritm Metody. I (1995), 120–126; translated in J. Math. Sci. (New York) 87 (1997), 4054–4058.Google Scholar
  36. [Ver95b]
    A. M. Vershik, Locally transversal symbolic dynamics, St. Petersburg Math J. 6 (1995), 529–540.MathSciNetGoogle Scholar
  37. [Via06]
    M. Viana, Dynamics of interval exchange transformations and Teichmüller flows, Lecture notes, July 6, 2008,
  38. [Wal82]
    P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York/Berlin, 1982.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2009

Authors and Affiliations

  • Sebastien Ferenczi
    • 1
  • Albert M. Fisher
    • 2
  • Marina Talet
    • 3
  1. 1.Institut de Mathématiques de Luminy (UPR 9016)Marseille Cedex 9France
  2. 2.Dept Mat IME-USPSão Paulo SPBrazil
  3. 3.C.M.I. Université de Provence LATPMarseille Cedex 13France

Personalised recommendations