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Nombre de branchement d’un pseudogroupe

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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.

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Authors and Affiliations

  1. Departamento de Xeometría e Topoloxía, Universidade de Santiago de Compostela, 15872, Santiago de Compostela, Espagne

    Fernando Alcalde Cuesta & María P. Fernández de Córdoba

Authors
  1. Fernando Alcalde Cuesta
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  2. María P. Fernández de Córdoba
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Correspondence to Fernando Alcalde Cuesta.

Additional information

Ce travail a été financé par le Ministère de l’Éducation et de la Science de l’Espagne MTM2004-08214. Le deuxième auteur a bénéficié d’une bourse María Barbeito de la Xunta de Galicia.

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Alcalde Cuesta, F., Fernández de Córdoba, M.P. Nombre de branchement d’un pseudogroupe. Monatsh Math 163, 389–414 (2011). https://doi.org/10.1007/s00605-010-0230-z

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