A classification of orbits admitting a unique invariant measure

Nathanael Ackerman; Cameron Freer; Aleksandra Kwiatkowska; Rehana Patel

doi:10.1016/j.apal.2016.08.003

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arXiv · 1412.2735

A classification of orbits admitting a unique invariant measure

Nathanael Ackerman·Cameron Freer·Aleksandra Kwiatkowska·Rehana Patel

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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 $(\mathbb{Q}, <)$.

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BibTeX RIS

Nathanael Ackerman, Cameron Freer, Aleksandra Kwiatkowska, Rehana Patel. 2014-12-08. A classification of orbits admitting a unique invariant measure. https://doi.org/10.1016/j.apal.2016.08.003

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