Número de ramificación y percolación de un pseudogrupo

Abstract

The branching number of a rooted tree represents the average number of branches per vertex. This number is strongly related with Bernoulli percolation process, which involves removing edges at random on the tree and whose goal is to study the nature of the resulting clusters. We extend these notions to any measurable pseudogroup of finite type on a probability space. We prove that if the branching number is equal to 1 then the pseudogroup is amenable. In fact, when the measure is harmonic, the pseudogroup is Liouville. Regarding Bernoulli percolation, we remove edges at random on the orbits and we define a critical percolation. We study the influence of the number of ends of the orbits on the critical percolation. Finally, we define the percolation relative to a Borel set on group actions on a probability space, keeping the edges whose endpoints belong to the Borel set and removing the others. We use again the number of ends in order to achieve information about the clusters.