Ergodic Theory and Dynamical Systems最新文献

英文中文

Weighted topological pressure revisited 重访加权拓扑压力

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-05-08 DOI: 10.1017/etds.2024.35

NIMA ALIBABAEI

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Feng and Huang [Variational principle for weighted topological pressure.J. Math.Pures Appl. (9)106 (2016)，411-452] 引入了动态系统间因子映射的加权拓扑熵和压力，并建立了其变分原理。Tsukamoto [New approach to weighted topological entropy and pressure.Ergod.Th. & Dynam.Sys.43（2023），1004-1034] 对最简单情况下的这些不变式进行了完全不同的重新定义，并通过变分原理证明这两个定义是重合的。我们推广了塚本的方法，重新定义了更高维度的加权拓扑熵和压力，并证明了变分原理。我们的结果允许对仿射不变集的豪斯多夫维度进行基本计算，如自仿射海绵和驻留在任意维度欧几里得空间中的某些索菲克集。

引用次数: 0

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres 多维仿射网格中方向的泊松对相关性和嵌入角球的质量逃逸估计值

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-26 DOI: 10.1017/etds.2024.31

WOOYEON KIM, JENS MARKLOF

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded

$operatorname {SL}(d,mathbb {R})$

-horospheres in the space of affine lattices.

我们证明了在移位的自然戴奥芬汀条件下，仿射网格向量落入小圆盘的方向数矩的收敛性。此外，我们还证明，对于维度 3 及更高的任何无理平移，包括可近似的向量，对相关函数都是泊松的。分布的收敛性在斯特罗姆伯格森和第二作者的著作[周期洛伦兹气体中自由路径长度的分布及相关晶格点问题。Ann. of Math. (2)172 (2010), 1949-2033]，而扩展到时刻收敛的主要步骤是对仿射网格空间中嵌入 $operatorname {SL}(d,mathbb {R})$ -horospheres 的平均值进行质量逃逸估计。

引用次数: 0

Khintchine-type double recurrence in abelian groups 无性群中的欣钦内型双递归

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-24 DOI: 10.1017/etds.2024.29

ETHAN ACKELSBERG

We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $Gamma $ is a countable discrete abelian group, $varphi , psi in mathrm {End}(Gamma )$ , and $psi - varphi $ is an injective endomorphism with finite index image, then for any ergodic measure-preserving $Gamma $ -system $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$ , any measurable set $A in {mathcal {X}}$ , and any ${varepsilon }> 0$ , there is a syndetic set of $g in Gamma$ such that $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) > mu(A)^3 - varepsilon$ . This generalizes the main results of Ackelsberg et al [Khintchine-type recurrence for 3-point configurations. <jats:italic

我们证明了可数离散无边群的成对内定态的欣钦钦型递推定理。作为主结果的一个特例，如果 $Gamma $ 是一个可数离散无边群， $varphi , psi in mathrm {End}(Gamma )$ 、并且 $psi - varphi $ 是一个具有有限索引映像的注入式内形变，那么对于任何保全遍历度量的 $Gamma $ 系统 $( X, {mathcal {X}}, mu , (T_g)_{g in Gamma } )$, 任何可度量集合 $g in Gamma } )$, 任何可度量集合 $g - varphi $ 是一个具有有限索引映像的注入式内形变。)$，{mathcal {X}}$中的任意可测集$A，以及任意${varepsilon }>；0$, there is a syndetic set of $g in Gamma$ such that $mu ( A cap T_{varphi(g)}^{-1} A cap T_{psi(g)}^{-1} A ) > mu(A)^3 - varepsilon$ .这概括了 Ackelsberg 等人[Khintchine-type recurrence for 3-point configurations.Forum Math.Sigma10 (2022), Paper no. e107] 并基本上回答了该论文中的一个未决问题 [Question 1.12; Khintchine-type recurrence for 3-point configurations.论坛数学。Sigma10 (2022), Paper no.］对于$Gamma = {mathbb {Z}}^d$ 群，结果适用于由差值为非奇异值的矩阵给出的成对内定态。证明的关键要素是(1) 与 Bergelson 和 Shalom 共同获得的最新结果[Khintchine-type recurrence for 3-point configurations.Forum Math.Sigma10 (2022), Paper no. e107]说，相关的遍历平均值由一个与准阿芬系数（或康泽-勒格朗系数）密切相关的特征因子控制；(2) 一个扩展技巧，以还原到具有良好离散谱（关于 $varphi $ 和 $psi $）的系统；(3) 描述与具有良好离散谱的旋转系统上的准阿芬环相关的麦基群。

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引用次数: 0

Schmidt games and Cantor winning sets 施密特博弈和康托尔胜局

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-19 DOI: 10.1017/etds.2024.23

DZMITRY BADZIAHIN, STEPHEN HARRAP, EREZ NESHARIM, DAVID SIMMONS

Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of study. We survey the definitions of the most common variants and connections between them. A new game called the Cantor game is invented and helps with presenting a unifying framework. We prove surprising new results such as the coincidence of absolute winning and

$1$

Cantor winning in metric spaces, and the fact that

$1/2$

winning implies absolute winning for subsets of

$mathbb {R}$

. We also suggest a prototypical example of a Cantor winning set to show the ubiquity of such sets in metric number theory and ergodic theory.

施密特博弈和康托获胜属性给出了大的另一种概念，类似于更标准的度量和范畴概念。它们直观、灵活，而且适用于最新研究，因此成为研究的热点。我们对最常见变体的定义和它们之间的联系进行了调查。我们还发明了一种新的游戏--康托尔游戏，它有助于提出一个统一的框架。我们证明了一些令人惊奇的新结果，如在度量空间中绝对胜局和$1$ Cantor胜局的重合，以及$1/2$胜局意味着$mathbb {R}$ 子集的绝对胜局。我们还提出了一个康托胜出集的原型例子，以说明这种集在公设数论和遍历理论中无处不在。

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引用次数: 0

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors 积分向量处线性和二次形式对的渐近分布

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-17 DOI: 10.1017/etds.2024.30

JIYOUNG HAN, SEONHEE LIM, KEIVAN MALLAHI-KARAI

We study the joint distribution of values of a pair consisting of a quadratic form ${mathbf q}$ and a linear form ${mathbf l}$ over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $mathrm{SL}_3(mathbb{R})$ . Math. Ann.286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n ge 5$ , then under the assumptions that for every $(alpha , beta ) in {mathbb {R}}^2 setminus { (0,0) }$ , the form $alpha {mathbf q} + beta {mathbf l}^2$ is irrational and that the signature of the restriction of ${mathbf q}$ to the kernel of ${mathbf l}$ is $(p, n-1-p)$ </jats:inline-formu

我们研究由积分向量集合上的二次形式 ${mathbf q}$ 和线性形式 ${mathbf l}$ 组成的一对值的联合分布，这个问题由 Dani 和 Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $mathrm{SL}_3(mathbb{R})$ .Math.Ann.286 (1990), 101-128].本着埃斯金、马格里斯和莫泽斯关于奥本海姆猜想定量版的著名定理的精神，我们证明了如果 $n ge 5$ , 那么在对每一个 $(alpha , beta ) in {mathbb{R}}^2 setminus { (0,0) }$ 的假设下，形式为 $alpha {mathbf q}.+ β {mathbf l}^2$ 是无理的，并且 ${mathbf q}$ 对 ${mathbf l}$ 内核的限制的签名是 $(p, n-1-p)$ ，其中 ${3le ple n-2}$ ，在 {mathbb {Z}}^n$ 中，$|v| <；T$ , $a < {mathbf q}(v) < b$ 和 $c< {mathbf l}(v) <；d$ 在 $T to infty $ 时渐近为 $ C({mathbf q}, {mathbf l})(d-c)(b-a)T^{n-3}$ ，其中 $C({mathbf q}, {mathbf l})$ 只取决于 ${mathbf q}$ 和 ${mathbf l}$ 。Gorodnik[Oppenheim conjecture for pairs consisting of a linear form and a quadatic form.Trans.Amer.Math.Soc.356(11) (2004), 4447-4463].

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引用次数: 0

Similarities and differences between specification and non-uniform specification 规范与非统一规范的异同

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-15 DOI: 10.1017/etds.2024.28

WANSHAN LIN, XUETING TIAN, CHENWEI YU

Pavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Pavlov[Adv. Math.295 (2016), 250-270; Nonlinearity32 (2019), 2441-2466]研究了具有弱版本规范属性的动力学系统的最大熵的度量，发现本征遍历性的存在会受到间隙函数假设的影响。受此启发，我们在本文中研究了具有非均匀规范性质的动力系统。我们给出了这些系统的一些基本性质，并给出了间隙函数的假设，以确保系统具有以下五个性质：CO度量在不变度量中是密集的；对于不变度量的每个非空紧凑连通子集，其饱和集在总空间中是密集的；遍历度量在不变度量中是残差的；遍历度量是连通的；熵密集。此外，我们还将举例说明该假设是最优的。

引用次数: 0

Non-existence of a universal zero-entropy system via generic actions of almost complete growth 通过几乎完全增长的一般作用不存在普遍的零熵系统

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-12 DOI: 10.1017/etds.2024.24

GEORGII VEPREV

We prove that a generic probability measure-preserving (p.m.p.) action of a countable amenable group G has scaling entropy that cannot be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of G for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. G-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that cannot be bounded from below by a given sequence. In addition, we show an example of an amenable group that has such a lower bound for every free p.m.p. action.

我们证明，可数可合并群 G 的一般概率度量保留（p.m.p. ）作用的缩放熵不能被给定的增长率所支配。作为推论，我们得到不存在一个 G 的拓扑作用，其遍历不变度量集合与熵为零的所有遍历 p.m.p. G 系统的集合重合。我们还证明了残差有限可调和群的泛函作用具有无法通过给定序列从下往上限定的缩放熵。此外，我们还展示了一个例子，说明可亲群的每个自由 p.m.p. 作用都有这样的下限。

引用次数: 0

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms 论片断光滑圆同构不变度量的豪斯多夫维度

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-11 DOI: 10.1017/etds.2024.25

FRANK TRUJILLO

We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g. piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

我们证明，一般来说，具有无理旋转数和零平均非线性（如片断线性）的足够规则的片断光滑圆同构的唯一不变度量的 Hausdorff 维数为零。为了编码这个通用条件，我们将片断光滑同态视为区间的广义区间交换变换（GIET），并依赖于 GIET 的组合旋转数概念，这可以看作是圆同态的经典旋转数概念在 GIET 环境中的扩展。

引用次数: 0

Substreetutions and more on trees 关于树木的分街道和更多信息

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-08 DOI: 10.1017/etds.2023.108

ALEXANDRE BARAVIERA, RENAUD LEPLAIDEUR

We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the

$mathbb {F}_{2}^{+}$

-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

我们定义了有色二叉树上的一种替换概念，称之为子街垒。我们将证明，由substreetution固定的点可能是（也可能不是）几乎周期性的，因此在$mathbb {F}_{2}^{+}$ 作用下的轨道闭合可能是（也可能不是）最小的。我们研究了一个特殊的例子：我们证明了它属于极小的情况，而且极小集合中的前像数只是以指数级的速度增长，而这本可以是超指数级的增长。我们还举例说明了在其轨道上没有不变度量的周期树。我们利用我们的构造得到了双曲盘的准周期彩色倾斜。

引用次数: 0

Density of mode-locking property for quasi-periodically forced Arnold circle maps 准周期强迫阿诺德圆图的模式锁定特性密度

IF 0.9 3区数学 Q2 MATHEMATICS

Ergodic Theory and Dynamical Systems

Pub Date : 2024-04-04 DOI: 10.1017/etds.2024.27

JIAN WANG, ZHIYUAN ZHANG

We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the

$C^0$

topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).

我们证明，具有拓扑通用强迫函数的准周期强迫阿诺德圆图族的锁模区是密集的。这就严格验证了 [M. Ding, C. Grebogi and E. Ott.Ding, C. Grebogi and E. Ott.准周期强迫系统吸引子的演化：从准周期到奇异非混沌再到混沌。Phys. Rev. A 39(5) (1989)，2593-2598] 对于这类强迫函数。更一般地说，在基图的一些一般条件下，我们展示了动态强迫图（定义见 [Z. Zhang.Zhang.On topological genericity of the mode-locking phenomenon.Math.Ann.376 (2020), 707-72]）配备的拓扑比 $C^0$ 拓扑强得多，与光滑光纤映射兼容。对于准周期基底映射，我们的结果概括了 [A. Avila, J. Bochi and J. M. Matters] 中的主要结果。Avila, J. Bochi and D. Damanik.薛定谔算子的康托谱系与广义偏移产生的势[A. Avila, J. Bochi and D. Damanik.Duke Math.J.146（2009），253-280]，[J.Wang, Q. Zhou and T. Jäger.准周期强迫圆图的锁模属性.Adv. Math.348 (2019), 353-377] 和 Zhang (2020).

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