关于唯一遍历性问题的一些进展

Colin Jahel
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The notion of unique ergodicity relating to a group was introduced by Angel, Kechris and Lyons. They also ask the following question which is the main focus of the thesis: Let G be an amenable Polish group with metrizable universal minimal flow, is G uniquely ergodic? Note that unique ergodicity is an interesting notion only for relatively large groups, as it is proved in the last chapter of this thesis that locally compact non compact Polish groups are never uniquely ergodic. This result is joint work with Andy Zucker. The thesis includes proofs of unique ergodicity of groups with interesting universal minimal flows, namely the automorphism group of the semigeneric directed graph and the automorphism group of the \n$2$\n -graph. It also includes a theorem stating that under some hypothesis on a \n$\\omega $\n -categorical structure M, the logic action of \n${\\mathrm {Aut}}(M)$\n on \n${\\mathrm {LO}}(M)$\n , the compact space of linear orders on M, is uniquely ergodic. 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引用次数: 1

摘要

本文是动力学、概率论和模型理论的交叉研究。它侧重于适应性概念的专业化:独特的遍历性。设G是一个波兰群,即一个拓扑群,其拓扑是可分的且完全可度量的。我们称G流为G在紧化空间上的作用。如果每个轨道都是密集的,那么g流就是最小的。Ellis的一个著名定理指出,任何波兰群G都存在唯一的普遍极小流,我们将其记为${\mathrm {M}}(G)$。这意味着对于任何最小G流X,存在一个从${\ mathm {M}}(G)$到X的满射G映射,如果每个G流都允许一个不变的概率测度,则G是可服从的,如果每个最小流都允许一个唯一的不变概率测度,则G是唯一遍历的。与群体相关的独特遍历性概念是由Angel、Kechris和Lyons提出的。他们还提出了以下问题,这也是论文的主要焦点:假设G是一个具有可度量的普遍最小流的可服从的波兰群,G是唯一遍历的吗?注意,唯一遍历性是一个有趣的概念,只有对于相对较大的群,因为它证明了在本论文的最后一章,局部紧非紧波兰群从来没有唯一遍历。这个结果是与Andy Zucker共同完成的。本文给出了具有有趣的泛最小流群的唯一遍历性的证明,即半同向图的自同构群和$2$ -图的自同构群。它还包括一个定理,证明在一个$\ -范畴结构M上的某个假设下,${\mathrm {Aut}}(M)$对${\mathrm {LO}}(M)$的逻辑作用是唯一遍历的,即M上线性阶的紧化空间。这意味着如果群的普遍最小流恰好是线性排序空间,则群具有唯一遍历性。它也可以用来证明${\mathrm {Aut}}(M)$对${\mathrm {LO}}(M)$的作用不是最小的一些群的不可适应性。这个结果是与Todor Tsankov共同完成的。最后,本文还证明了在所涉及的泛最小流是可度量的假设下,唯一遍历性在群扩展下是稳定的。这个结果是与Andy Zucker共同完成的。摘要由Colin Jahel准备。电子邮件:cjahel@andrew.cmu.edu URL: http://math.univ-lyon1.fr/~jahel/doc/These.pdf
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Some Progress on the Unique Ergodicity Problem
Abstract This thesis is at the intersection of dynamics, probability and model theory. It focuses on a specialization of the notion of amenability: unique ergodicity. Let G be a Polish group, i.e., a topological group whose topology is separable and completely metrizable. We call a G-flow the action of G on a compact space. A G-flow is said to be minimal if every orbit is dense. A famous theorem of Ellis states that any Polish group G admits a unique universal minimal flow that we denote ${\mathrm {M}}(G)$ . This means that for any minimal G-flow X there is a surjective G-map from ${\mathrm {M}}(G)$ to X. G is said to be amenable if every G-flow admits an invariant probability measure, and uniquely ergodic if every minimal flow admits a unique invariant probability measure. The notion of unique ergodicity relating to a group was introduced by Angel, Kechris and Lyons. They also ask the following question which is the main focus of the thesis: Let G be an amenable Polish group with metrizable universal minimal flow, is G uniquely ergodic? Note that unique ergodicity is an interesting notion only for relatively large groups, as it is proved in the last chapter of this thesis that locally compact non compact Polish groups are never uniquely ergodic. This result is joint work with Andy Zucker. The thesis includes proofs of unique ergodicity of groups with interesting universal minimal flows, namely the automorphism group of the semigeneric directed graph and the automorphism group of the $2$ -graph. It also includes a theorem stating that under some hypothesis on a $\omega $ -categorical structure M, the logic action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ , the compact space of linear orders on M, is uniquely ergodic. This implies unique ergodicity for the group if its universal minimal flow happens to be the space of linear orderings. It can also be used to prove non-amenability of some groups for which the action of ${\mathrm {Aut}}(M)$ on ${\mathrm {LO}}(M)$ is not minimal. This result is joint work with Todor Tsankov. Finally, the thesis also presents a proof that under the assumption that the universal minimal flows involved are metrizable, unique ergodicity is stable under group extensions. This result is joint work with Andy Zucker. Abstract prepared by Colin Jahel. E-mail: cjahel@andrew.cmu.edu URL: http://math.univ-lyon1.fr/~jahel/doc/These.pdf
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POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
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