Von Neumann and Birkhoff ergodic theorems for negatively curved groups

Bowen, Lewis; Nevo, Amos

doi:10.24033/asens.2267

Von Neumann and Birkhoff ergodic theorems for negatively curved groups
[Théorèmes ergodiques de von Neumann et de Birkhoff sur les groupes à courbure négative]

Bowen, Lewis ; Nevo, Amos

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1113-1147.

Résumé
Abstract

Pour tout groupe hyperbolique au sens de Gromov et pour toute action, préservant la mesure, sur un espace de probabilités, nous démontrons une inégalité maximale pour les moyennes sur des boules concentriques ou sur des anneaux sphériques concentriques de même épaisseur. Sous une hypothèse supplémentaire, valable par exemple pour les actions isométriques et proprement discontinues sur des espaces CAT( $- 1$ ), nous démontrons de plus un théorème ergodique ponctuel pour une suite de mesures de probabilités à support dans des anneaux sphériques concentriques.

We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on CAT( $- 1$ ) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

DOI : 10.24033/asens.2267

Classification : 28D15, 37A20, 20F67, 60J50.
Keywords: Negatively curved group, ergodic theorem, maximal inequality, Patterson-Sullivan measure, measurable equivalence relations, Poisson boundary.
Mot clés : Groupe à courbure négative, action de groupe, théorème ergodique, inégalité maximale, mesure de Patterson-Sullivan, relation d'équivalence mesurable, bord de Poisson.

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     title = {Von {Neumann} and {Birkhoff} ergodic theorems for negatively curved groups},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1113--1147},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Bowen, Lewis; Nevo, Amos. Von Neumann and Birkhoff ergodic theorems for negatively curved groups. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1113-1147. doi : 10.24033/asens.2267. http://www.numdam.org/articles/10.24033/asens.2267/

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