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Unique ergodicity and measures invariant under permutations of N Freer, Cameron


Consider dense linear orders without endpoints having underlying set the natural numbers. They admit a unique probability measure that is invariant to the logic action of S_\infty on the underlying set, called the Glasner-Weiss measure. In contrast, the Rado graph admits continuum many ergodic invariant measures (among them, the Erdős-Rényi constructions for arbitrary p such that 0 < p < 1). We characterize all isomorphism classes of structures that admit an unique invariant measure, and show that any isomorphism class admitting more than one invariant measure must admit continuum many ergodic ones. Joint work with Nathanael Ackerman, Aleksandra Kwiatkowska, and Rehana Patel.

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