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.
Highlights
Abstract. — Let M be an א0-categorical structure and assume that M has no algebraicity and has weak elimination of imaginaries
Tsankov as Fraïssé limits of a class of finite structures satisfying certain conditions and there is a close correspondence between dynamical properties of the automorphism group of the limit structure and combinatorial properties of the class
Typical examples of Fraïssé classes are the class of finite graphs, the class of finite triangle-free graphs, and the class of finite linear orders (here the limit is the countable, dense linear order without endpoints (Q,
Summary
We start by recalling some basic definitions. A signature L is a collection of relation symbols {Ri} and function symbols {Fj}, each equipped with a natural number called its arity. A structure is א0-categorical if its first-order theory has a unique countable model up to isomorphism Another characterization that will be crucial is given by the RyllNardzewski theorem: M is א0-categorical iff the diagonal action Aut(M ) M k has finitely many orbits for every k (a permutation group with this property is called oligomorphic). If L is a signature that contains only finitely many relational symbols of each arity and no functions, every homogeneous L -structure is א0-categorical. If G is any closed subgroup of the full permutation group Sym(N ) of some countable set N , one can convert N into a homogeneous structure with Aut(N ) = G by naming, for every k, each G-orbit on N k by a k-ary relation symbol. An analogous argument shows that c is contained in B and e ∈ dcleq(A ∩ B)
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