Invariant measures on products and on the space of linear orders
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 155-176.

Let M be an 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.

Soit M une structure 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.

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Accepted:
Published online:
DOI: 10.5802/jep.180
Classification: 37A50,  03C15
Keywords: Invariant measures, uniquely ergodic, linear orders, 0 -categorical, de Finetti theorem
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
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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, Volume 9 (2022), pp. 155-176. doi : 10.5802/jep.180. https://jep.centre-mersenne.org/articles/10.5802/jep.180/

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