Eli Glasner, Benjamin Weiss
We show the existence, over an arbitrary infinite ergodic $\mathbb{Z}$-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.
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On the denseness of distal points
该论文解答了Xu和Ye关于可数离散群的伯努利转移中远端点稠密性的问题，引入点根基概念，证明相关群类性质并构造出答案存在差异的2步幂零群。
According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maxi…
Disjointness for measurably distal group actions and applications
本研究将Berg拟不相交性推广至可数群作用，证明可测远端系统与保测系统拟不相交，给出易检验的不相交充要条件，还得到带远端权重的可数顺从群Wiener--Wintner型定理及在遍历平均、多重回复中的应用。
Let $X$ be a compact metric countable space, let $f:X\to X$ be a homeomorphism and let $E(X,f)$ be its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) $(X,f)$ is equicontinuous, (ii) $(X,f)$ is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that $(X,f)$ is distal if, and only if, $E(X,f)$ is a group.
We generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg--Zimmer tower of a measurably distal system and showing that at each step, there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.
In this article we will apply complex projective metrics to sequences of complex transfer operators generated by Young towers, countable shifts and other types of distance expanding maps (possibly time dependent) with countable degrees. We will derive a sequential Ruelle-Perron-Frobenius theorem for sequence of complex operators defined by such maps, and relying on it a variety of probabilistic limit theorems (finer than the CLT) for non-stationary processes follows.
Let $η$ be an arbitrary countable ordinal. Using results of Bourgain and Gamburd on compact systems with spectral gap we show the existence of an action of the free group on three generators $F_3$ on a compact metric space $X$, admitting an invariant probability measure $μ$, such that the resulting dynamical system $(X, μ, F_3)$ is strongly ergodic and distal of rank $η$. In particular this shows that there is a $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.
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.
We develop a model-theoretic framework for the study of distal factors of strongly ergodic, measure-preserving dynamical systems of countable groups. Our main result is that all such factors are contained in the (existential) algebraic closure of the empty set. This allows us to recover some rigidity results of Ioana and Tucker-Drob as well as prove some new ones: for example, that strongly ergodic, distal systems are coalescent and that every two such systems that are weakly equivalent are isomorphic. We also prove the existence of a universal distal, ergodic system that contains any other di…
Locally conformal expanding Actions: Markov Partition and Thermodynamic of the induced skew product
本文研究了由有限个 $C^{1+α}$ 共形局部微分同胚生成的拓扑混合局部共形半群作用，提出了满足有限像和有限环性质的可数 Markov 分割，并基于诱导映射等价于可数型子移位的塔构造，证明了可提升测度的存在性及诱导斜积的热力学形式。
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 …
We find a countable partition $P$ on\textbf{} a Lebesgue space, labeled $\{1,2,3...$\}, for any non-periodic measure preserving transformation $T$ such that $P$ generates $T$ and for the $T,P$ process, if you see an $n$ on time -1 then you only have to look at times $-n,1-n,...-1$ to know the positive integer $i$ to put at time 0. We alter that proof to extend every non-periodic $T$ to a uniform martingale (i.e. continuous $g$ function) on an infinite alphabet. If $T$ has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining question…
We study fibrations in the category of cubespaces/nilspaces. We show that a fibration of finite degree $f \colon X\rightarrow Y$ between compact ergodic gluing cubespaces (in particular nilspaces) factors as a (possibly countable) tower of compact abelian Lie group principal fiber bundles over $Y$. If the structure groups of $f$ are connected then the fibers are (uniformly) isomorphic (in a strong sense) to an inverse limit of nilmanifolds. In addition we give conditions under which the fibers of $f$ are isomorphic as subcubespaces. We introduce regionally proximal equivalence relations relati…
Disjointness with all minimal systems under group actions
本文针对可数离散群作用，给出了传递系统与所有极小系统不交的充要条件，并证明该条件蕴含无穷传递性与稠密极小点，同时建立了具有稠密 distal 点的无穷传递系统与所有极小系统不交的结论。
Return time sets and product recurrence
该研究探讨了在紧Hausdorff空间中，群作用下子集包含返回时间集的条件，并通过积递归与distality的关系回答了相关问题。
Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on $X$, and then, show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions (i) this dynamics is pointwise recurrent, (ii) $X$ is a union of $G$-minimal sets, (iii) the $G$-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous, are pairwise equivalent. Consequently, if this dynamics is distal,…
On compact extensions of tracial $W^*$-dynamical systems
本文对任意离散群作用的迹$W^*$-动力系统紧扩张进行分类，回答多个开放问题，并推导非交换Furstenberg-Zimmer型二分法及可数阿贝尔群遍历作用的Host-Kra-Ziegler塔类比。
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