A classification of orbits admitting a unique invariant measure |
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Authors: | Nathanael Ackerman Cameron Freer Aleksandra Kwiatkowska Rehana Patel |
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Affiliation: | 1. Department of Mathematics, Harvard University, Cambridge, MA 02138, USA;2. Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;3. Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany;4. Instytut Matematyczny, Uniwersytet Wroc?awski, pl. Grunwaldzki 2/4, 50-384 Wroc?aw, Poland;5. Franklin W. Olin College of Engineering, Needham, MA 02492, USA |
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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 -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 . |
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Keywords: | 03C98 37L40 60G09 20B27 Invariant measure High homogeneity Unique ergodicity |
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