Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911006793400320 |
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| author | Avraham-Re'em, Nachi Peterzil, George |
| author_facet | Avraham-Re'em, Nachi Peterzil, George |
| contents | We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (à la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem Avraham-Re'em, Nachi Peterzil, George Dynamical Systems Logic 37A20, 28D15, 37A40, 03E15, 22D40, 22F10 We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (à la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions. |
| title | Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem |
| topic | Dynamical Systems Logic 37A20, 28D15, 37A40, 03E15, 22D40, 22F10 |
| url | https://arxiv.org/abs/2409.02781 |