不可数超有限性与随机比率遍历定理

arXiv - MATH - Logic Pub Date : 2024-09-04 DOI:arxiv-2409.02781

Nachi Avraham-Re'em, George Peterzil

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引用次数: 0

摘要

我们证明了局部紧密群的自由作用的轨道等价关系是超无限的（\`a la Connes-Feldman-Weiss），正是当它是 "超紧密 "时。这意味着奥恩斯坦-魏斯定理的不可数版本，以及每个局部紧密群都接纳超紧密概率保存自由作用是可处理的。我们还建立了丹尼连科随机比率遍历定理的不可数版本。由此，我们推导出许多非星形伯努利作用的 "霍普夫二分法"。

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Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem

We show that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (\`a 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.

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arXiv - MATH - Logic

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