An uncountable Furstenberg–Zimmer structure theory

Jamneshan, Asgar

doi:10.1017/etds.2022.43

Cited by 11 publications

(16 citation statements)

References 54 publications

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“…Homogeneous extensions are very closely related to the notions of isometric extensions and compact extensions, which play a fundamental role in the Furstenberg-Zimmer structure theory [7,29,30] of measure-preserving systems, as well as subsequent refinements of that theory, such as is found in the work of Host and Kra [12]. See [14] for a discussion of these relationships in both the countable and uncountable settings. It is common in the literature to reduce to the corefree case when k∈K kLk −1 = {1}, by quotienting out by the normal core k∈K kLk −1 , but we will not need to use the corefree property here.…”

Section: And One Can Then Define the Notion Of Two Cncprbmentioning

confidence: 99%

An uncountable Mackey–Zimmer theorem

Jamneshan

Tao

2022

Studia Math.

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The Mackey-Zimmer theorem classifies ergodic group extensions X of a measure-preserving system Y by a compact group K, by showing that such extensions are isomorphic to a group skew-product X ≡ Y ⋊ρ H for some closed subgroup H of K. An analogous theorem is also available for ergodic homogeneous extensions X of Y , namely that they are isomorphic to a homogeneous skew-product Y ⋊ρ H/M . These theorems have many uses in ergodic theory, for instance playing a key role in the Host-Kra structural theory of characteristic factors of measure-preserving systems.The existing proofs of the Mackey-Zimmer theorem require various "countability", "separability", or "metrizability" hypotheses on the group Γ that acts on the system, the base space Y , and the group K used to perform the extension. In this paper we generalize the Mackey-Zimmer theorem to "uncountable" settings in which these hypotheses are omitted, at the cost of making the notion of a measure-preserving system and a group extension more abstract. However, this abstraction is partially counteracted by the use of a "canonical model" for abstract measure-preserving systems developed in a companion paper. In subsequent work we will apply this theorem to also obtain uncountable versions of the Host-Kra structural theory.

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Section: And One Can Then Define the Notion Of Two Cncprbmentioning

confidence: 99%

An uncountable Mackey–Zimmer theorem

Jamneshan

Tao

2022

Studia Math.

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“…Some foundational aspects arising in the ergodic theory of uncountable groups and inseparable spaces was systematically investigated by the fourth author and Tao in [28,30,29] and by the fourth author in [27]. For example, in the area of multiple recurrence, the tool of disintegration of measures is used extensively.…”

Section: Introduction a Famous And Deep Theorem Of Szemerédimentioning

confidence: 99%

“…One of the major challenges is then to find a suitable alternative framework in which we find viable replacements for tools such as disintegration of measures which can help to meaningfully adapt the arguments from the countable setting. For further details, we refer the interested reader to [28,30,29,27].…”

Section: Introduction a Famous And Deep Theorem Of Szemerédimentioning

confidence: 99%

An uncountable ergodic Roth theorem and applications

Durcik

Greenfeld

Iseli

et al. 2022

DCDS

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<p style='text-indent:20px;'>We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for <inline-formula><tex-math id="M1">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>-systems for uniformly amenable groups <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. As a special case, we obtain this uniformity over all <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-systems, and our result seems to be novel already in this case. Our uncountable Roth theorem is crucial in the proof of both of these results.</p>

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“…The goal is to capture a wider range of structures and phenomena in probability theory and related fields such as statistics and information theory. On one hand, there is interest in moving measure theory beyond the need of countability, for example in the work of Jamneshan and Tao [15], and in their work with others in expanding ergodic theory in that direction [12][13][14]16].…”

Section: Introductionmentioning

confidence: 99%

“…By the definition of quotient σ-algebra, it suffices to show that q −1 (q(A)) is a measurable subset of X. But now by (12), q −1 (q(A)) = A, which is measurable. So q(A) is a measurable set, and hence h is a measurable kernel.…”

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confidence: 99%

A category-theoretic proof of the ergodic decomposition theorem

Moss¹,

Perrone²

2022

Preprint

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The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular,• ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories);• the invariant σ-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels.In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much simpler than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

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An uncountable Furstenberg–Zimmer structure theory

Cited by 11 publications

References 54 publications

An uncountable Mackey–Zimmer theorem

An uncountable Mackey–Zimmer theorem

An uncountable ergodic Roth theorem and applications

A category-theoretic proof of the ergodic decomposition theorem

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