Invariant measures on products and on the space of linear orders

TL;DR

This paper generalizes classical theorems to show that ergodic invariant measures on product spaces over certain structures are product measures, and investigates the dynamics of automorphism groups on linear orders, revealing fixed points or unique ergodicity.

Contribution

It extends classical ergodic theorems to $ ext{Aut}(M)$-invariant measures on product spaces and analyzes the automorphism group's action on linear orders, establishing conditions for fixed points or unique ergodicity.

Findings

Ergodic, invariant measures on $[0,1]^M$ are product measures.

The automorphism group action on linear orders has a fixed point or is uniquely ergodic.

Conditions for fixed points or unique ergodicity depend on transitivity of the action.

Abstract

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