Sur les espaces mesures singuliers I - Etude metrique-mesuree

Abstract

Recall Jones-Schmidt theorem that an ergodic measured equivalence relation is strongly ergodic if and only if it has no nontrivial amenable quotient. We give two new characterizations of strong ergodicity, in terms of metric-measured spaces. The first one identifies strong ergodicity with the concentration property as defined, in this (foliation) setting, by Gromov \cite{Gromov00_SQ}. The second one characterize the existence of nontrivial amenable quotients in terms of "F{\o}lner sequences" in graphings naturally associated to (the leaf space of) the equivalence relation. We also present a formalization of the concept of quasi-periodicity, based on (noncommutative) measure theory. The "singular measured spaces" appearing in the title refer to the leaf spaces of measured equivalence relations.