Monatshefte für Mathematik

, Volume 163, Issue 4, pp 389–414 | Cite as

Nombre de branchement d’un pseudogroupe

  • Fernando Alcalde Cuesta
  • María P. Fernández de Córdoba
Article

Abstract

Lyons has defined an average number of branches per vertex of an infinite locally finite rooted tree. This number has an important role in several probabilistic processes such as random walk and percolation. In this paper, we extend the notion of branching number to any measurable graphed pseudogroup of finite type acting on a probability space. We prove that such a pseudogroup is Liouvillian (i.e. almost every orbit does not admit non-constant bounded harmonic functions) if its branching number is equal to 1. In order to prove that this actually generalizes results of C. Series and V. Kaimanovich on equivalence relations with polynomial and subexponential growth, we describe an example of minimal lamination whose holonomy pseudogroup has exponential growth and branching number equal to 1.

Keywords

Graphed pseudogroup Lamination Branching number Liouville property Amenability 

Mathematics Subject Classification (2000)

05C25 37A20 37C85 43A07 58H05 (57R30 · 60J10 · 60J80) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alcalde Cuesta, F.: Moyennes harmoniques, À paraître dans Ann. Fac. Sci. ToulouseGoogle Scholar
  2. 2.
    Anantharaman-Delaroche, C., Renault, J.: Amenable groupoids, Monographies de L’Enseignement Mathématique 36, L’Enseignement Mathématique, Genève (2000)Google Scholar
  3. 3.
    Avez A.: Entropie des groupes de type fini. C. R. Acad. Sci. Paris 275, 1363–1366 (1972)MathSciNetMATHGoogle Scholar
  4. 4.
    Avez, A.: Croissance des groupes de type fini et fonctions harmoniques. In: Théorie Ergodique (Rennes, 1973/1974). Lectures Notes in Math., vol. 532, pp. 35–49. Springer, Berlin (1976)Google Scholar
  5. 5.
    Blanc, E.: Examples of mixed minimal foliated spaces, Non-published paper (2002)Google Scholar
  6. 6.
    Connes, A.: Sur la théorie non commutative de l’intégration. In: Algèbres d’opérateurs (Les Plans-sur-Bex, 1978). Lecture Notes in Math., vol. 725, pp. 19–143. Springer, Berlin (1979)Google Scholar
  7. 7.
    Connes A., Feldman J., Weiss B.: An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam. Syst. 1, 431–450 (1981)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Derriennic Y.: Quelques applications du théorème ergodique sous-additif. Astérisque 74, 183–201 (1980)MathSciNetMATHGoogle Scholar
  9. 9.
    Derriennic, Y.: Entropie, théorèmes limites et marches aléatoires. In: Probability Measures on Groups VIII (Oberwolfach, 1985). Lectures Notes in Math., vol. 1210, pp. 241–284. Springer, Berlin (1986)Google Scholar
  10. 10.
    Feldman J., Moore C.: Ergodic equivalence relations, cohomology and Von Neumann algebras. I Trans. Am. Math. Soc. 234, 289–324 (1977)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gaboriau D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Garnett L.: Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51, 285–311 (1983)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ghys E.: Topologie des feuilles génériques. Ann. Math. 141, 387–422 (1995)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ghys E.: Laminations par surfaces de Riemann. Panor. Synth. 8, 49–95 (1999)MathSciNetGoogle Scholar
  15. 15.
    Ghys E., de la Harpe P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Birkhäuser, Boston (1990)MATHGoogle Scholar
  16. 16.
    Ghys E., Sergiescu V.: Stabilité et conjugaison différentiable pour certains feuilletages. Topology 19, 179–197 (1980)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Goodman S.E., Plante J.F.: Holonomy and averaging in foliated sets. J. Differ. Geom. 14, 401–407 (1979)MathSciNetMATHGoogle Scholar
  18. 18.
    Gromov M.: Groups of polynomial growth and expanding maps. Publ. Math. IHES 53, 53–78 (1981)MathSciNetMATHGoogle Scholar
  19. 19.
    Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, vol. 2 (Sussex, 1991). London Math. Soc., Lecture Note Ser., vol. 182. Cambridge University Press, Cambridge (1993)Google Scholar
  20. 20.
    Haefliger A.: Groupoïdes d’holonomie et classifiants, in Transversal structure of foliations (Toulouse, 1982). Astérisque 116, 70–97 (1984)MathSciNetGoogle Scholar
  21. 21.
    Haefliger, A.: Foliations and compactly generated pseudogroups. In: Foliations: Geometry and Dynamics (Warsaw, 2000), pp. 275–295. World Scientific Publishing, River Edge (2002)Google Scholar
  22. 22.
    Kaimanovich V.A.: Brownian motion on foliations: entropy, invariant measures, mixing. Funct. Anal. Appl. 22, 326–328 (1988)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kaimanovich V.A.: Hausdorff dimension of the harmonic measure on trees. Ergodic Theory Dynam. Syst. 18, 631–660 (1998)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kaimanovich V.A., Vershik A.M.: Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457–490 (1983)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kaimanovich V.A., Woess W.: Boundary and entropy of space homogeneous Markov chains. Ann. Probab. 30, 323–363 (2002)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Levitt G.: On the cost of generating an equivalence relation. Ergodic Theory Dynam. Syst. 15, 1173–1181 (1995)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lozano Rojo Á.: The Cayley foliated space of a graphed pseudogroup. Publ. de la RSME 10, 267–272 (2006)MathSciNetGoogle Scholar
  28. 28.
    Lyons R.: Random walks and percolation on trees. Ann. Probab. 18, 931–958 (1990)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Lyons R., Pemantle R., Peres Y.: Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure. Ergodic Theory Dynam. Syst. 15, 593–619 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lyons, R., Peres, Y.: Probability on trees and networks, Draft, version of 22 November (2004)Google Scholar
  31. 31.
    Paulin F.: Propriétés asymptotiques des relations d’équivalences mesurées discrètes. Markov Process. Relat. Fields 5, 163–200 (1999)MathSciNetMATHGoogle Scholar
  32. 32.
    Peres, Y.: Probability on trees: an introductory climb. In: Lectures on probability theory and statistics (Saint-Flour, 1997). Lecture Notes in Math., vol. 1717, pp. 193–280. Springer, Berlin (1999)Google Scholar
  33. 33.
    Samuélidès M.: Tout feuilletage à croissance polynomiale est hyperfini. J. Funct. Anal. 34, 363–369 (1979)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Series C.: Foliations of polynomial growth are hyperfinite. Israel J. Math. 34, 245–258 (1979)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Thurston W.: Non cobordant foliations of S 3. Bull. Am. Math. Soc. 78, 511–514 (1972)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Virág B.: On the speed of random walks on graphs. Ann. Probab. 28, 379–394 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Fernando Alcalde Cuesta
    • 1
  • María P. Fernández de Córdoba
    • 1
  1. 1.Departamento de Xeometría e TopoloxíaUniversidade de Santiago de CompostelaSantiago de CompostelaEspagne

Personalised recommendations