Ergodic Theory and Dynamical Systems最新文献

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Non-concentration property of Patterson–Sullivan measures for Anosov subgroups 阿诺索夫子群的帕特森-沙利文度量的非集中特性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-18 DOI: 10.1017/etds.2024.55

DONGRYUL M. KIM, HEE OH

Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $Gamma <G$ with respect to a parabolic subgroup $P_theta $ , we prove that any $Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_theta $ . More generally, we prove that for a Zariski dense $theta $ -transverse subgroup $Gamma <G$ , any $(Gamma , psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_theta $ , provided the $psi $ -Poincaré series of $Gamma $ </jats:inlin

让 G 是一个连通的半简单实代数群。对于相对于抛物子群 $P_theta $ 的扎里斯基密集阿诺索夫子群 $Gamma <G$，我们证明任何 $Gamma $ -帕特森-沙利文度量在 $G/P_theta $ 的任何适当子变上都不带质量。更广义地说，我们证明了对于一个扎里斯基密集的 $theta $ -反子群 $Gamma <G$ ，任何 $(Gamma , psi )$ -帕特森-沙利文度量在 $G/P_theta $ 的任何适当子变上都不收取质量，只要 $Gamma $ 的 $psi $ -波因卡雷数列在其收敛的上位发散。特别是，我们的结果也适用于相对阿诺索夫子群。

引用次数: 0

Multifractal analysis of homological growth rates for hyperbolic surfaces 双曲面同调增长率的多分形分析

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-18 DOI: 10.1017/etds.2024.62

JOHANNES JAERISCH, HIROKI TAKAHASI

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

我们对双曲面上定向大地线的同调增长率进行了多分形分析。我们的主要结果提供了一个关于规定增长率的水平集的豪斯多夫维度的公式，该公式是用福氏群的广义波恩卡列指数表示的。我们采用鲍恩和系列所发展的符号动力学、遍历理论和热力学形式主义来证明维度谱的解析性。

引用次数: 0

Rigidity of flat holonomies 平面整体的刚性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-18 DOI: 10.1017/etds.2024.58

GÉRARD BESSON, GILLES COURTOIS, SA’AR HERSONSKY

We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension

$n geq 3$

with strongly

$1/4$

-pinched or relatively

$1/2$

-pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.

我们证明，在维数为 $n geq 3$、截面曲率为强 1/4$ -夹角或相对 1/2$ -夹角的封闭黎曼流形的普适盖中存在一个角层，其上沿一个角层的稳定整体性与黎曼平行传输重合，这意味着该流形与实双曲流形同构。

引用次数: 0

A recurrence-type strong Borel–Cantelli lemma for Axiom A diffeomorphisms 公理 A 差分的递推型强博雷尔-康特利定理

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-18 DOI: 10.1017/etds.2024.64

ALEJANDRO RODRIGUEZ SPONHEIMER

Let

$(X,mu ,T,d)$

be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence

$(M_k)$

that converges to

$0$

slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for

$mu $

-almost every

$xin X$

$$ begin{align*} lim_{n to infty}frac{sum_{k=1}^{n} mathbf{1}_{B_k(x)}(T^{k}x)} {sum_{k=1}^{n} mu(B_k(x))} = 1, end{align*} $$

where

$mu (B_k(x)) = M_k$

. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.

让$(X,mu ,T,d)$是一个度量保全的动力系统，对于利普齐兹连续观测值，三折相关性呈指数衰减。给定一个足够慢地收敛到 $0$ 的序列 $(M_k)$，我们会得到一个强动力学的 Borel-Cantelli 递归结果，即对于 $mu $ - 几乎每一个 $xin X$ ， $$ begin{align*}limit_{n to infty}frac{sum_{k=1}^{n}mathbf{1}_{B_k(x)}(T^{k}x)} {sum_{k=1}^{n}mu(B_k(x))} = 1, end{align*}$$ 其中 $mu (B_k(x)) = M_k$ 。我们特别指出，在某些假设条件下，这一结果对于公理 A 差分和平衡态都是成立的。

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引用次数: 0

On transversal Hölder regularity for flat Wieler solenoids 论平面维勒孤岛的横向赫尔德正则性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-10 DOI: 10.1017/etds.2024.41

RODRIGO TREVIÑO

This paper studies various aspects of inverse limits of locally expanding affine linear maps on flat branched manifolds, which I call flat Wieler solenoids. Among the aspects studied are different types of cohomologies, the rates of mixing given by the Ruelle spectrum of the hyperbolic map acting on this space, and solutions of the cohomological equation in primitive substitution subshifts for Hölder functions. The overarching theme is that considerations of

$alpha $

-Hölder regularity on Cantor sets go a long way.

本文研究了平面分支流形（我称之为平面维勒孤岛）上局部膨胀仿射线性映射的逆极限的各个方面。研究的方面包括不同类型的同调、作用于该空间的双曲映射的鲁埃尔谱给出的混合率，以及霍尔德函数的原始置换子移动中的同调方程的解。总的主题是，在康托集上考虑$alpha $ -霍尔德正则性有很长的路要走。

引用次数: 0

Symbolic dynamics for pointwise hyperbolic systems on open regions 开放区域上点双曲系统的符号动力学

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-10 DOI: 10.1017/etds.2024.47

CHUPENG WU, YUNHUA ZHOU

Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism

$f:Mrightarrow M$

on an open invariant subset

$Osubset M$

, which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

在某些条件下，我们在开放不变子集 $Osubset M$ 上为点式双曲衍射 $f:Mrightarrow M$ 构造了一个可数马尔可夫分区，它允许 Lyapunov 指数为零。从这个分区出发，我们定义了一个符号扩展，它是有限对一的，并扩展到 O 的一个子集上，该子集携带与 O 相同的有限 f 不变度量。作为一个典型应用，我们估算了 f 的闭合轨道数。

引用次数: 0

Multiplication cubes and multiplication automata 乘法立方体和乘法自动机

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-10 DOI: 10.1017/etds.2024.44

JOHAN KOPRA

We extend previously known two-dimensional multiplication tiling systems that simulate multiplication by two natural numbers p and q in base

$pq$

to higher dimensional multiplication tessellation systems. We develop the theory of these systems and link different multiplication tessellation systems with each other via macrotile operations that glue cubes in one tessellation system into larger cubes of another tessellation system. The macrotile operations yield topological conjugacies and factor maps between cellular automata performing multiplication by positive numbers in various bases.

我们将以前已知的模拟两个自然数 p 和 q 以 $pq$ 为基数相乘的二维乘法镶嵌系统扩展到高维乘法镶嵌系统。我们发展了这些系统的理论，并通过宏梯形运算将不同的乘法镶嵌系统相互连接起来，宏梯形运算可将一个镶嵌系统中的立方体粘合到另一个镶嵌系统的更大立方体中。巨型运算产生了拓扑共轭，以及在不同基数下执行正数乘法的细胞自动机之间的因子映射。

引用次数: 0

Minimal and proximal examples of -stable and -approachable shift spaces 稳定和可接近移位空间的最小和近似实例

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-10 DOI: 10.1017/etds.2024.43

MELIH EMIN CAN, JAKUB KONIECZNY, MICHAL KUPSA, DOMINIK KWIETNIAK

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein’s $bar {d}$ metric ( $bar {d}$ -approachable shift spaces). The class of $bar {d}$ -approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $bar {d}$ -approachability, together with a closely connected notion of $bar {d}$ -shadowing, was introduced by Konieczny, Kupsa, and Kwietniak [Ergod. Th. & Dynam. Sys.43(3) (2023), 943–970]. These notions were developed with the aim of significantly generalizing specification properties. Indeed, many popular variants of the specification property, including the classic one and the almost/weak specification property, ensure $bar {d}$ -approachability and $bar {d}$ -shadowing. Here, we study further properties and connections between $bar {d}$ -shadowing and <jats:inline-graphic xmlns:xlink="h

我们研究有限字母表上的移位空间，这些空间可以通过与奥恩斯坦的 $bar {d}$ 度量相连的（伪）度量意义上的有限类型的混合移位来近似（$bar {d}$ -approachable shift spaces）。$bar{d}$可接近移位类可被视为度量理论伯努利系统的拓扑类似物。$bar {d}$ -可接近性的概念，以及与之密切相关的 $bar {d}$ -阴影的概念，是由科尼茨尼、库普萨和克维特尼亚克提出的[Ergod.这些概念的提出，旨在极大地推广规范属性。事实上，规范属性的许多流行变体，包括经典变体和几乎/弱规范属性，都确保了$bar {d}$可接近性和$bar {d}$阴影。在这里，我们将进一步研究 $bar {d}$ -shadowing 和 $bar {d}$ -approachability 之间的性质和联系。我们证明$bar {d}$阴影意味着$bar {d}$稳定（这是蒂姆-奥斯汀最近提出的概念）。我们证明，对于具有$bar {d}$ -shadowing性质的投射平移空间，由$bar {d}$诱导的平移空间之间的豪斯多夫伪距${bar d}^{mathrm {H}}$与它们的不变度量简元之间的豪斯多夫距相同，都是由奥恩斯坦度量$bar {d}$诱导的度量之间的豪斯多夫距。我们证明，如果没有$bar {d}$的遮挡，这并不一定是真的（众所周知，前者的距离总是约束后者的）。我们举例说明了这些结果，包括具有 $bar {d}$ -shadowing 特性的移位空间的最小例子和近似例子。这类移位空间的存在已在前面提到的论文中公布。它表明$bar {d}$-shadowing确实概括了规范属性。

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引用次数: 0

Equilibrium measures for two-sided shift spaces via dimension theory 通过维度理论实现双面移位空间的均衡度量

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-10 DOI: 10.1017/etds.2024.46

VAUGHN CLIMENHAGA, JASON DAY

Given a two-sided shift space on a finite alphabet and a continuous potential function, we give conditions under which an equilibrium measure can be described using a construction analogous to Hausdorff measure that goes back to the work of Bowen. This construction was previously applied to smooth uniformly and partially hyperbolic systems by the first author, Pesin, and Zelerowicz. Our results here apply to all subshifts of finite type and Hölder continuous potentials, but extend beyond this setting, and we also apply them to shift spaces with synchronizing words.

给定一个有限字母表上的双面移位空间和一个连续的势函数，我们给出了一些条件，在这些条件下，可以用一种类似于豪斯多夫度量的构造来描述平衡度量，这种构造可以追溯到鲍恩的研究。第一作者、Pesin 和 Zelerowicz 以前曾将这种构造应用于光滑均匀和部分双曲系统。我们这里的结果适用于有限类型的所有子移位和荷尔德连续势，但又超出了这一范围，我们还将它们应用于具有同步词的移位空间。

引用次数: 0

Non-rigidity of partially hyperbolic abelian -actions on tori 环上部分双曲阿贝尔作用的非刚性

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-09-09 DOI: 10.1017/etds.2024.18

FEDERICO RODRIGUEZ HERTZ, ZHIREN WANG

We prove that every genuinely partially hyperbolic

$mathbb {Z}^r$

-action by toral automorphisms can be perturbed in

$C^1$

-topology, so that the resulting action is continuously conjugate, but not

$C^1$

-conjugate, to the original one.

我们证明，每一个真正的部分双曲$mathbb {Z}^r$ -作用的环自动形都可以在$C^1$ -拓扑中被扰动，这样得到的作用与原始作用是连续共轭的，但不是$C^1$ -共轭的。

引用次数: 0

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