Invariant measures on products and on the space of linear orders

Colin Jahel; Todor Tsankov

doi:10.5802/jep.180

Invariant measures on products and on the space of linear orders
[Mesures invariantes sur des produits et sur l’espace des ordres totaux]

Colin Jahel ¹ ; Todor Tsankov ²

¹ Institut Camille Jordan, Université Claude Bernard Lyon 1 Université de Lyon 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
² Institut Camille Jordan, Université Claude Bernard Lyon 1 Université de Lyon 43, boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France and Institut Universitaire de France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 155-176.

Résumé
Abstract

Soit $M$ une structure $ℵ_{0}$ -catégorique sans algébricité et éliminant faiblement les imaginaires. En généralisant des théorèmes classiques de de Finetti et de Ryll-Nardzewski, nous démontrons que toute mesure $Aut (M)$ -invariante et ergodique sur ${[0, 1]}^{M}$ est une mesure produit. Nous étudions également l’action de $Aut (M)$ sur l’espace compact $LO (M)$ des ordres totaux sur $M$ . Sous l’hypothèse supplémentaire que l’action $Aut (M) ↷ M$ est transitive, nous démontrons que l’action $Aut (M) ↷ LO (M)$ soit est uniquement ergodique, soit admet un point fixe.

Let $M$ be an $ℵ_{0}$ -categorical structure and assume that $M$ has no algebraicity and has weak elimination of imaginaries. Generalizing classical theorems of de Finetti and Ryll-Nardzewski, we show that any ergodic, $Aut (M)$ -invariant measure on ${[0, 1]}^{M}$ is a product measure. We also investigate the action of $Aut (M)$ on the compact space $LO (M)$ of linear orders on $M$ . If we assume moreover that the action $Aut (M) ↷ M$ is transitive, we prove that the action $Aut (M) ↷ LO (M)$ either has a fixed point or is uniquely ergodic.

Reçu le : 2021-05-22
Accepté le : 2021-11-30
Publié le : 2021-12-06

DOI : 10.5802/jep.180

Classification : 37A50, 03C15
Keywords: Invariant measures, uniquely ergodic, linear orders, $\aleph _0$-categorical, de Finetti theorem
Mot clés : Mesures invariantes, uniquement ergodique, ordres totaux, $\aleph _0$-catégorique, théorème de de Finetti

Affiliations des auteurs :

Colin Jahel ¹ ; Todor Tsankov ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Colin Jahel; Todor Tsankov. Invariant measures on products and on the space of linear orders. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 155-176. doi : 10.5802/jep.180. https://jep.centre-mersenne.org/articles/10.5802/jep.180/

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