6 research outputs found
A classification of orbits admitting a unique invariant measure
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 -invariant and concentrated on a
single isomorphism class must be zero, or one, or continuum. Further, such an
isomorphism class admits a unique -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 .Comment: 22 pages. Small changes following referee suggestion