Invariant measures on products and on the space of linear orders

Jahel, Colin; Tsankov, Todor

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

Soit $M$ une structure ${\aleph }_0$-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 $Aut\left(M\right)$-invariante et ergodique sur ${[0,1]}^M$ est une mesure produit. Nous étudions également l’action de $Aut\left(M\right)$ sur l’espace compact $\mathrm{LO}\left(M\right)$ des ordres totaux sur $M$. Sous l’hypothèse supplémentaire que l’action $Aut\left(M\right)\curvearrowright M$ est transitive, nous démontrons que l’action $Aut\left(M\right)\curvearrowright \mathrm{LO}\left(M\right)$ soit est uniquement ergodique, soit admet un point fixe.

Let $M$ be an ${\aleph }_0$-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, $Aut\left(M\right)$-invariant measure on ${[0,1]}^M$ is a product measure. We also investigate the action of $Aut\left(M\right)$ on the compact space $\mathrm{LO}\left(M\right)$ of linear orders on $M$. If we assume moreover that the action $Aut\left(M\right)\curvearrowright M$ is transitive, we prove that the action $Aut\left(M\right)\curvearrowright \mathrm{LO}\left(M\right)$ either has a fixed point or is uniquely ergodic.

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

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