Ergodic Theory and Dynamical Systems最新文献

英文中文

Deformational rigidity of integrable metrics on the torus 环上可积分度量的变形刚度

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-09 DOI: 10.1017/etds.2024.48

JOSCHA HENHEIK

It is conjectured that the only integrable metrics on the two-dimensional torus are Liouville metrics. In this paper, we study a deformative version of this conjecture: we consider integrable deformations of a non-flat Liouville metric in a conformal class and show that for a fairly large class of such deformations, the deformed metric is again Liouville. The principal idea of the argument is that the preservation of rational invariant tori in the foliation of the phase space forces a linear combination on the Fourier coefficients of the deformation to vanish. Showing that the resulting linear system is non-degenerate will then yield the claim. Since our method of proof immediately carries over to higher dimensional tori, we obtain analogous statements in this more general case. To put our results in perspective, we review existing results about integrable metrics on the torus.

有人猜想，二维环上唯一可积分的度量是Liouville度量。在本文中，我们研究了这一猜想的变形版本：我们考虑了共形类中非平坦的Liouville度量的可积分变形，并证明了对于相当大的一类这样的变形，变形后的度量又是Liouville。该论证的主要思想是，相空间对折中有理不变环的保留迫使变形的傅立叶系数上的线性组合消失。如果证明所得到的线性系统是非退化的，就可以得出这个结论。由于我们的证明方法可以立即应用到更高维度的环上，因此我们可以在这种更普遍的情况下得到类似的结论。为了使我们的结果更直观，我们回顾了有关环上可积分度量的现有结果。

引用次数: 0

Bishop–Jones’ theorem and the ergodic limit set 毕晓普-琼斯定理和遍历极限集

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-09 DOI: 10.1017/etds.2024.49

NICOLA CAVALLUCCI

For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

对于一个适当的格罗莫夫双曲度量空间和一个离散的非元素等轴群，我们定义了等轴群无穷远处极限集的一个自然子集，称为遍历极限集。这个名称的由来是，商度量空间上的大地流不变的每个遍历度量都集中在端点属于遍历极限集的大地流上。我们完善了经典的毕肖普-琼斯定理，证明了遍历极限集的堆积维度与群的临界指数重合。

引用次数: 0

Minimal zero entropy subshifts can be unrestricted along any sparse set 最小零熵子移动可以沿任意稀疏集合不受限制地进行

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-09 DOI: 10.1017/etds.2024.42

RONNIE PAVLOV

We present a streamlined proof of a result essentially presented by the author in [Some counterexamples in topological dynamics. Ergod. Th. & Dynam. Sys.28(4) (2008), 1291–1322], namely that for every set

$S = {s_1, s_2, ldots } subset mathbb {N}$

of zero Banach density and finite set A, there exists a minimal zero-entropy subshift

$(X, sigma )$

so that for every sequence

$u in A^{mathbb {Z}}$

, there is

$x_u in X$

with

$x_u(s_n) = u(n)$

for all

$n in mathbb {N}$

. Informally, minimal deterministic sequences can achieve completely arbitrary behavior upon restriction to a set of zero Banach density. As a corollary, this provides counterexamples to the polynomial Sarnak conjecture reported by Eisner [A polynomial version of Sarnak’s conjecture. C. R. Math. Acad. Sci. Paris353(7) (2015), 569–572] which are significantly more general than some recently provided by Kanigowski, Lemańczyk and Radziwiłł [Prime number theorem for analytic skew products. Ann. of Math. (2)199 (2024), 591–705] and by Lian and Shi [A counter-example for polynomial version of Sarnak’s conjecture. Adv. Math.384 (2021), Paper no. 107765] and shows that no similar result can hold under only the assumptions of minimality and zero entropy.

我们对作者在 [Some counterexamples in topological dynamics.Ergod.Th. & Dynam.Sys.28(4)(2008),1291-1322]，即对于每一个集合 $S = {s_1, s_2, ldots }和有限集 A，存在一个最小零熵子移位 $(X, sigma )$，这样对于 A^{mathbb {Z}}$ 中的每一个序列 $u ，在 X$ 中存在 $x_u ，对于 mathbb {N}$ 中的所有 $n ，具有 $x_u(s_n) = u(n)$。非正式地讲，最小确定性序列在限制到零巴纳赫密度集合时可以实现完全任意的行为。作为推论，这为艾斯纳报告的多项式萨尔纳克猜想提供了反例 [A polynomial version of Sarnak's conjecture.C. R. Math.Acad.Sci. Paris353(7) (2015)，569-572] 所报告的猜想比卡尼戈夫斯基、莱曼奇克和拉齐维乌最近提供的一些猜想要宽泛得多 [Prime number theorem for analytic skew products.(2)199 (2024), 591-705] 以及 Lian 和 Shi [A counter-example for polynomial version of Sarnak's conjecture.Adv. Math.384 (2021), Paper no.107765]，并表明仅在最小性和零熵假设条件下，类似结果不可能成立。

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引用次数: 0

Subshifts of finite symbolic rank 有限符号等级的子移位

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-09 DOI: 10.1017/etds.2024.45

SU GAO, RUIWEN LI

The definition of subshifts of finite symbolic rank is motivated by the finite rank measure-preserving transformations which have been extensively studied in ergodic theory. In this paper, we study subshifts of finite symbolic rank as essentially minimal Cantor systems. We show that minimal subshifts of finite symbolic rank have finite topological rank, and conversely, every minimal Cantor system of finite topological rank is either an odometer or conjugate to a minimal subshift of finite symbolic rank. We characterize the class of all minimal Cantor systems conjugate to a rank-

$1$

subshift and show that it is dense but not generic in the Polish space of all minimal Cantor systems. Within some different Polish coding spaces of subshifts, we also show that the rank-1 subshifts are dense but not generic. Finally, we study topological factors of minimal subshifts of finite symbolic rank. We show that every infinite odometer and every irrational rotation is the maximal equicontinuous factor of a minimal subshift of symbolic rank

$2$

, and that a subshift factor of a minimal subshift of finite symbolic rank has finite symbolic rank.

有限符号秩子转移的定义是受有限秩保量变换的启发，而有限秩保量变换在遍历理论中已被广泛研究。在本文中，我们将有限符号秩的子移动本质上视为最小康托尔系统来研究。我们证明了有限符号秩的最小子移位具有有限拓扑秩，反过来说，每个有限拓扑秩的最小康托尔系统要么是有限符号秩的最小子移位的里程表，要么是其共轭。我们描述了与秩-1$子移位共轭的所有极小康托尔系统的类别，并证明它在所有极小康托尔系统的波兰空间中是密集的，但不是泛函的。在一些不同的子移位波兰编码空间中，我们也证明了秩-1 子移位是致密的，但不是泛函。最后，我们研究了有限符号秩的最小子转移的拓扑因子。我们证明了每一个无限里程表和每一个无理旋转都是符号秩为 2$ 的最小子移位的最大等连续因子，并且有限符号秩的最小子移位的子移位因子具有有限符号秩。

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引用次数: 0

Indistinguishable asymptotic pairs and multidimensional Sturmian configurations 无差别渐近对和多维斯特尔米构型

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-31 DOI: 10.1017/etds.2024.39

SEBASTIÁN BARBIERI, SÉBASTIEN LABBÉ

Two asymptotic configurations on a full

$mathbb {Z}^d$

-shift are indistinguishable if, for every finite pattern, the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of

$mathbb {Z}^d$

. We prove that indistinguishable asymptotic pairs satisfying a ‘flip condition’ are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together, the two results provide a generalization to

$mathbb {Z}^d$

of the characterization of Sturmian sequences by their factor complexity

$n+1$

. Many open questions are raised by the current work and are listed in the introduction.

如果对于每个有限模式，每个配置中的相关出现集都重合到 $mathbb {Z}^d$ 的有限支持排列，那么在一个完整的 $mathbb {Z}^d$ 移位上的两个渐近配置是不可区分的。我们证明，满足 "翻转条件 "的不可区分渐近对的特征是它们在有限连接支持上的模式复杂性。此外，我们还证明了满足翻转条件的均匀递归不可区分渐近对是由标度为一的（内部空间的维度）切割和投影方案描述的，这些方案象征性地对应于多维斯特米安构型。这两个结果共同提供了对$mathbb {Z}^d$ Sturmian序列的因子复杂度$n+1$的概括。目前的工作提出了许多悬而未决的问题，这些问题已在引言中列出。

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引用次数: 0

Partially hyperbolic endomorphisms with expanding linear part 具有扩展线性部分的部分双曲内形变

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-28 DOI: 10.1017/etds.2024.36

MARTIN ANDERSSON, WAGNER RANTER

In this paper, we study transitivity of partially hyperbolic endomorphisms of the two torus whose action in the first homology group has two integer eigenvalues of moduli greater than one. We prove that if the Jacobian is everywhere greater than the modulus of the largest eigenvalue, then the map is robustly transitive. For this, we introduce Blichfedt’s theorem as a tool for extracting dynamical information from the action of a map in homology. We also treat the case of specially partially hyperbolic endomorphisms, for which we obtain a complete dichotomy: either the map is transitive and conjugated to its linear part, or its unstable foliation must contain an annulus which may either be wandering or periodic.

在本文中，我们研究了在第一同调群中有两个模量大于 1 的整数特征值的两个环的部分双曲内形变的传递性。我们证明，如果雅各比在任何地方都大于最大特征值的模，那么这个映射就是稳健传递的。为此，我们引入了布利赫费尔特定理，作为从同调映射作用中提取动力学信息的工具。我们还处理了特殊部分双曲内形变的情况，对这种情况我们得到了一个完整的二分法：要么该映射是传递性的并与其线性部分共轭，要么其不稳定叶形必须包含一个环面，而这个环面可能是游荡的，也可能是周期性的。

引用次数: 0

Uniform syndeticity in multiple recurrence 多发性复发的均匀联合性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-28 DOI: 10.1017/etds.2024.40

ASGAR JAMNESHAN, MINGHAO PAN

The main theorem of this paper establishes a uniform syndeticity result concerning the multiple recurrence of measure-preserving actions on probability spaces. More precisely, for any integers $d,lgeq 1$ and any $varepsilon> 0$ , we prove the existence of $delta>0$ and $Kgeq 1$ (dependent only on d, l, and $varepsilon $ ) such that the following holds: Consider a solvable group $Gamma $ of derived length l, a probability space $(X, mu )$ , and d pairwise commuting measure-preserving $Gamma $ -actions $T_1, ldots , T_d$ on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0

本文的主要定理建立了关于概率空间上保度作用的多重递归的统一联合性结果。更确切地说，对于任意整数 $d,lgeq 1$ 和任意 $varepsilon> 0$，我们证明了 $delta>0$ 和 $Kgeq 1$ 的存在（仅依赖于 d、l 和 $varepsilon $），使得以下条件成立：考虑一个派生长度为 l 的可解群 $Gamma $，一个概率空间 $(X, mu )$ ，以及在 $(X, mu )$ 上的 d 个成对的保持度量的 $Gamma $ 作用 $T_1, ldots , T_d$ 。让 E 是 X 中的一个可测集合，$mu (E) geq varepsilon $ 。那么，K 是 $$ （begin{align*}）的许多（左）平移。big{gammainGammacolon mu(T_1^{gamma^{-1}}(E)cap T_2^{gamma^{-1}}ccirc T^{gamma^{-1}}_1(E)cap cdots cap T^{gamma^{-1}}_dcirc T^{gamma^{-1}}_{d-1}}circ cdots circ T^{gamma^{-1}}_1(E))geq delta big}end{align*}$$ cover $Gamma $ 。这一结果扩展并完善了 Furstenberg 和 Katznelson 的均匀性结果。作为组合应用，我们得到了下面的均匀性结果。对于任意整数 $d,lgeq 1$ 和任意 $varepsilon> 0$，有 $delta>；0$和 $Kgeq 1$（仅依赖于 d、l 和 $varepsilon $），这样对于派生长度为 l 的所有有限可解群 G 和任何具有 $m^{otimes d}(E)geq varepsilon $ 的子集 $Esubset G^d$ （其中 m 是 G 上的均匀量），我们有 $$ begin{align*} 的 K-many（左）平移。(gin Gcolon &m^{otimes d}({(a_1,ldots,a_n)in G^dcolon & (a_1,ldots,a_n),(ga_1,a_2,ldots,a_n),ldots,(ga_1,ga_2,ldots, ga_n)in E})geq delta }end{align*}我们主要结果的证明是奥斯汀的可变遍历 Szeméredi 定理的超极限版本的结果。

引用次数: 0

Complexity of non-abelian cut-and-project sets of polytopal type I: special homogeneous Lie groups 多胞型 I 的非阿贝尔切分集的复杂性：特殊同质列支群

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-13 DOI: 10.1017/etds.2024.38

PETER KAISER

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like

$r^{{mathrm {homdim}}(G) dim (H)}$

. Further, we generalize the concept of acceptance domains to locally compact second countable groups.

本文的目的是确定在非阿贝尔情况下切投集复杂性函数的渐近增长率。在同质两步零钾烈群中的多顶型模型集的情况下，我们可以确定复杂性函数渐近地表现为 $r^{{mathrm {homdim}}(G) dim (H)}$ 。此外，我们还将接受域的概念推广到局部紧凑的第二可数群。

引用次数: 0

Co-spectral radius for countable equivalence relations 可数等价关系的共谱半径

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-10 DOI: 10.1017/etds.2024.32

MIKLÓS ABERT, MIKOLAJ FRACZYK, BENJAMIN HAYES

We define the co-spectral radius of inclusions

${mathcal S}leq {mathcal R}$

of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on

$G/H$

for inclusion

$Hleq G$

of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

我们将离散的、概率度量保全等价关系的包含${/mathcal S}leq {mathcal R}$的共谱半径定义为环境关系上生成随机漫步的采样指数。共谱半径类似于包含组$Hleq G$的$G/H$上随机游走的谱半径。为了证明这一点，我们开发了我们在另一项关于单模态随机有根树的生长的工作中使用的 2-3 方法的更通用版本。我们用这种方法证明，对于任意有界度的单模态随机有根图，都存在行走增长。我们还研究了共谱半径在超无限关系中的表现，并讨论了可以用共谱半径定义的渗滤的新临界指数。

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引用次数: 0

On invariant holonomies between centers 关于中心间的不变整体性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-08 DOI: 10.1017/etds.2024.33

RADU SAGHIN

We prove that for

$C^{1+theta }$

$theta $

-bunched, dynamically coherent partially hyperbolic diffeomorphisms, the stable and unstable holonomies between center leaves are

$C^1$

, and the derivative depends continuously on the points and on the map. Also for

$C^{1+theta }$

$theta $

-bunched partially hyperbolic diffeomorphisms, the derivative cocycle restricted to the center bundle has invariant continuous holonomies which depend continuously on the map. This generalizes previous results by Pugh, Shub, and Wilkinson; Burns and Wilkinson; Brown; Obata; Avila, Santamaria, and Viana; and Marin.

我们证明，对于$C^{1+theta }$ , $theta $ -束状、动态相干的部分双曲衍射，中心叶之间的稳定和不稳定整体性为$C^1$，导数连续依赖于点和映射。同样对于 $C^{1+theta }$ , $theta $ 组合的部分双曲衍射，限制于中心束的导数环具有不变的连续整体性，这些整体性连续地依赖于映射。这概括了 Pugh、Shub 和 Wilkinson、Burns 和 Wilkinson、Brown、Obata、Avila、Santamaria 和 Viana 以及 Marin 以前的结果。

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