Title

Invariant measures on the space of linear orders on an aleph_0-categorical structure

Abstract

Let M be an aleph_0-categorical structure and denote by LO(M) the compact space of linear orders on M. We investigate the probability measures on LO(M) invariant under the natural action of the automorphism group of M and prove, under rather general model-theoretic assumptions, that either M has a definable linear order or LO(M) carries a unique invariant measure (which can be easily and explicitly described). For many structures M, the space LO(M) is the universal minimal flow of the group Aut(M) and our work is in part motivated by a general unique ergodicity question of Angel, Kechris, and Lyons in topological dynamics. This is joint work with Colin Jahel.

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