Infinite stationary measures of co-compact group actions
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866918140835790848 |
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| author | Alhalimi, Mohammedsaid Hutchcroft, Tom Pan, Minghao Tamuz, Omer Zheng, Tianyi |
| author_facet | Alhalimi, Mohammedsaid Hutchcroft, Tom Pan, Minghao Tamuz, Omer Zheng, Tianyi |
| contents | Let $Γ$ be a finitely generated group, and let $μ$ be a nondegenerate, finitely supported probability measure on $Γ$. We show that every co-compact $Γ$ action on a locally compact Hausdorff space admits a nonzero $μ$-stationary Radon measure. The main ingredient of the proof is a stationary analogue of Tarski's theorem: we show that for every nonempty subset $A \subseteq Γ$ there is a $μ$-stationary, finitely additive measure on $Γ$ that assigns unit mass to $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite stationary measures of co-compact group actions Alhalimi, Mohammedsaid Hutchcroft, Tom Pan, Minghao Tamuz, Omer Zheng, Tianyi Group Theory Dynamical Systems Let $Γ$ be a finitely generated group, and let $μ$ be a nondegenerate, finitely supported probability measure on $Γ$. We show that every co-compact $Γ$ action on a locally compact Hausdorff space admits a nonzero $μ$-stationary Radon measure. The main ingredient of the proof is a stationary analogue of Tarski's theorem: we show that for every nonempty subset $A \subseteq Γ$ there is a $μ$-stationary, finitely additive measure on $Γ$ that assigns unit mass to $A$. |
| title | Infinite stationary measures of co-compact group actions |
| topic | Group Theory Dynamical Systems |
| url | https://arxiv.org/abs/2410.23600 |