Metastability in the Furstenberg-Zimmer tower

The Furstenberg-Zimmer structure theorem for stationary random walks

We prove the following version of the Furstenberg-Zimmer structure theorem for stationary actions: Any stationary action of a locally compact second-countable group is a weakly mixing extension of a measure-preserving distal system.

Metastability and the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemeredi's Theorem

The Furstenberg-Zimmer structure theorem for $\mathbb{Z}^d$ actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show …

不可数的 Furstenberg–Zimmer 结构理论

An uncountable Furstenberg--Zimmer structure theory

本文去除相对 Furstenberg–Zimmer 结构理论中对群和空间的可数性、可分性限制，建立完全一般的理论，并联系遍历论相对分析与 Boolean topos 内部逻辑。

Kaplansky-Hilbert 模上酉群表示的分解定理与 Furstenberg-Zimmer 结构定理

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

本文建立了 Stone 代数上 Kaplansky-Hilbert 模的酉群表示分解定理，并基于算子理论将其应用于推广著名的 Furstenberg-Zimmer 结构定理至任意群在任意概率空间上的保测作用情形。

Stationary dynamical systems

Following works of Furstenberg and Nevo and Zimmer we present an outline of a theory of stationary (or m-stationary) dynamical systems for a general acting group G equipped with a probability measure m. Our purpose is two-fold: First to suggest a more abstract line of development, including a simple structure theory. Second, to point out some interesting applications; one of these is a Szemeredi type theorem for SL(2,R).

Weak containment rigidity for distal actions

We prove that if a measure distal action $α$ of a countable group $Γ$ is weakly contained in a strongly ergodic probability measure preserving action $β$ of $Γ$, then $α$ is a factor of $β$. In particular, this applies when $α$ is a compact action. As a consequence, we show that the weak equivalence class of any strongly ergodic action completely remembers the weak isomorphism class of the maximal distal factor arising in the Furstenberg-Zimmer Structure Theorem.