Espaces mesurés singuliers fortement ergodiques (Étude métrique–mesurée)

Pichot, Mikaël

Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 1-43.

D’après le théorème de Jones-Schmidt, une relation d’équivalence ergodique est fortement ergodique si et seulement si elle ne possède pas de quotient moyennable non trivial. Nous donnons dans cet article deux nouvelles caractérisations de l’ergodicité forte, en termes d’espaces métriques-mesurés. La première identifie ergodicité forte et concentration de la mesure (définie dans ce cadre dans [22]). La seconde caractérise l’existence de quotients moyennables non triviaux par la présence de « suites de Følner évanescentes » dans les structures de graphes associées aux relations d’équivalence.

Nous présentons également une formalisation du concept de quasi-périodicité, reposant sur la théorie de la mesure. Les « espaces mesurés singuliers » apparaissant dans le titre font référence aux espaces de classes d’une relation d’équivalence mesurée.

We recall Jones-Schmidt’s theorem which shows that an ergodic measured equivalence relation is strongly ergodic if and only if it has no nontrivial amenable quotient. In this paper, 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 (foliated) setting, by Gromov [22]. The second one characterizes the existence of nontrivial amenable quotients in terms of “vanishing Følner sequences” in graphs naturally associated to (the leaf space of) the equivalence relation.

We also present a formalization of the concept of quasi-periodicity, based on measure theory. The “singular measured spaces” appearing in the title refer to the leaf spaces of measured equivalence relations.

Classification : 37A20, 50C25
Mot clés : relations d’équivalence mesurées, espaces singuliers, ergodicité forte, concentration.
Keywords: Measured equivalence relations, singular spaces, strong ergodicity, concentration.

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