A classification of orbits admitting a unique invariant measure

TL;DR

This paper classifies the number of ergodic invariant measures on the space of countable structures, showing they are either zero, one, or continuum, and characterizes when a structure admits a unique such measure.

Contribution

It establishes a complete classification of invariant measures on countable structures and links uniqueness to high homogeneity and reducts of the rational order.

Findings

Number of ergodic invariant measures is 0, 1, or continuum.

Unique invariant measure occurs only for highly homogeneous structures.

Structures with a unique measure are interdefinable with reducts of the rational linear order.

Abstract

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\infty$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_\infty$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q}, <)$.