Communications in Mathematical Physics

, Volume 372, Issue 2, pp 367–383 | Cite as

Existence and Non-existence of Length Averages for Foliations

  • Yushi Nakano
  • Tomoo YokoyamaEmail author


Since the pioneering work of Ghys et al., it has been known that several methods of dynamical systems theory can be adopted to study of foliations. Our aim in this paper is to investigate complexity of foliations, by generalising existence problem of time averages in dynamical systems theory to foliations: It has recently been realised that a positive Lebesgue measure set of points without time averages only appears for complicated dynamical systems, such as dynamical systems with heteroclinic connections or homoclinic tangencies. In this paper, we introduce the concept of length averages to singular foliations, and attempt to collect interesting examples with/without length averages. In particular, we demonstrate that length averages exist everywhere for any codimension one \(\mathscr {C}^1 \) orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi-minimal sets of the foliation, which is in strong contrast to time averages of surface flows.



We are deeply grateful to anonymous reviewers for many suggestions, all of which substantially improved the paper.


  1. 1.
    Alves, J.F., Bonatti, C., Viana, M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. In: The Theory of Chaotic Attractors, pp. 443–490 (2000)CrossRefGoogle Scholar
  2. 2.
    Alves, J.F.: SRB measures for non-hyperbolic systems with multidimensional expansion. Ann. Sci. Ecol. Norm. Supér. 33, 1–32 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barreira, L., Schmeling, J.: Sets of “non-typical” points have full topological entropy and full Hausdorff dimension. Isr. J. Math. 116(1), 29–70 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. In: The Theory of Chaotic Attractors, pp. 55–76 (1975)CrossRefGoogle Scholar
  5. 5.
    Candel, A., Conlon, L.: Foliations I: Graduate Studies in Mathematics, vol. 23. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  6. 6.
    Candel, A., Conlon, L.: Foliations II: Graduate Studies in Mathematics, vol. 60. American Mathematical Society, Providence (2003)zbMATHGoogle Scholar
  7. 7.
    Cantwell, J., Conlon, L.: Endsets of exceptional leaves: a theorem of G. Duminy. In: Foliations: Geometry and Dynamics, pp. 225–261 (2002)Google Scholar
  8. 8.
    Deroin, B., Kleptsyn, V., Navas, A.: On the ergodic theory of free group actions by real-analytic circle diffeomorphisms. Invent. Math. 212(3), 731–779 (2018)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dold, A.: Lectures on Algebraic Topology, Classics in Mathematics, Springer, Berlin (Reprint of the 1972 edition) (1995)CrossRefGoogle Scholar
  10. 10.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, New York (2004)zbMATHGoogle Scholar
  11. 11.
    Ghys, E., Langevin, R., Walczak, P.: Entropie géométrique des feuilletages. Acta Math. 160(1), 105–142 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gutierrez, C.: Smoothing continuous flows on two-manifolds and recurrences. Ergod. Theory Dyn. Syst. 6(1), 17–44 (1986)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hector, G., Hirsch, U.: Introduction to the geometry of foliations. Part A. Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. In: Aspects of Mathematics, vol. 1, Friedr. Vieweg & Sohn, Braunschweig (1986)Google Scholar
  14. 14.
    Hector, G., Hirsch, U.: Introduction to the geometry of foliations. Part B. Foliations of codimension one. In: Aspects of Mathematics, vol. 3, Friedr. Vieweg & Sohn, Braunschweig (1987)Google Scholar
  15. 15.
    Hirsch, M.: A stable analytic foliation with only exceptional minimal sets. In: Manning, A. (ed.) Lecture Notes in Mathematics, vol. 468. Springer, Berlin (1975)Google Scholar
  16. 16.
    Hofbauer, F., Keller, G.: Quadratic maps without asymptotic measure. Commun. Math. Phys. 127(2), 319–337 (1990)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  18. 18.
    Keane, M.: Interval exchange transformations. Math. Z. 141(1), 25–31 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Keynes, H.B., Newton, D.: A “minimal”, non-uniquely ergodic interval exchange transformation. Math. Z. 148(2), 101–105 (1976)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kiriki, S., Soma, T.: Takens’ last problem and existence of non-trivial wandering domains. Adv. Math. 306, 524–588 (2017)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kubarski, J.: About Stefan’s definition of a foliation with singularities: a reduction of the axioms. Bull. Soc. Math. Fr. 118(4), 391–394 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Labouriau, I.S., Rodrigues, A.A.: On Takens’ last problem: tangencies and time averages near heteroclinic networks. Nonlinearity 30(5), 1876 (2017)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Markley, N.G.: On the number of recurrent orbit closures. In: Proceedings of the American Mathematical Society, pp. 413–416 (1970)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Masur, H.: Interval exchange transformations and measured foliations. Ann. Math. 115(1), 169–200 (1982)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mayer, A.: Trajectories on the closed orientable surfaces. Rec. Math. 12(54), 71–84 (1943)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Nikolaev, I., Zhuzhoma, E.: Flows on 2-Dimensional Manifolds. Lecture Notes in Mathematics, vol. 1705. Springer, Berlin (1999)CrossRefGoogle Scholar
  27. 27.
    Ruelle, D.: Resonances for Axiom A Flows. Institut des Hautes Etudes Scientifiques (1986)Google Scholar
  28. 28.
    Ruelle, D.: Historical behaviour in smooth dynamical systems. In: Global Analysis of Dynamical Systems, pp. 63–66 (2001)Google Scholar
  29. 29.
    Sacksteder, R.: On the existence of exceptional leaves in foliations of co-dimension one. In: Annales de l’institut Fourier, pp. 221–225 (1964)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc. 73(6), 747–817 (1967)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stefan, P.: Accessibility and foliations with singularities. Bull. Am. Math. Soc. 80(6), 1142–1145 (1974)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 180, 171–188 (1973)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Takens, F.: Heteroclinic attractors: time averages and moduli of topological conjugacy. Bol. Soc. Bras. Mat. Bull. Braz. Math. Soc. 25(1), 107–120 (1994)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Takens, F.: Orbits with historic behaviour, or non-existence of averages. Nonlinearity 21(3), T33–T36 (2008)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Tamura, I.: Topology of foliations: an introduction. In: Translations of Mathematical Monographs, vol. 97, American Mathematical Society, Providence (Translated from the 1976 Japanese edition and with an afterword by Kiki Hudson, with a foreword by Takashi Tsuboi) (1992)Google Scholar
  36. 36.
    Thompson, D.: The irregular set for maps with the specification property has full topological pressure. Dyn. Syst. 25(1), 25–51 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Veech, W.A.: Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 1, 201–242 (1982)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Viana, M.: Complutense. Rev. Mat. Complut. 19(1), 7–100 (2006)MathSciNetCrossRefGoogle Scholar
  39. 39.
    von Koch, H.: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Ark. Mat. Astron. Fys. 1, 681–704 (1904)zbMATHGoogle Scholar
  40. 40.
    Walczak, P.: Dynamics of Foliations, Groups and Pseudogroups, vol. 64. Birkhäuser, Boston (2012)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTokai UniversityHiratukaJapan
  2. 2.Department of MathematicsKyoto University of Education/JST PrestoFushimi-kuJapan

Personalised recommendations