Indagationes Mathematicae-New Series最新文献

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IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2024.02.001

In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.

在具有二元符号动力学的旋转中，临界曲线是参数的位置，在该位置上，定义符号动力学的分区边界通过规定的迭代次数和符号行程相连。我们研究了这些曲线在参数空间中的算术和几何特性。

引用次数: 0

Semicocycle discontinuities for substitutions and reverse-reading automata 替换和反读自动机的半循环不连续

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.05.003

In this article we define the semigroup associated to a primitive substitution. We use it to construct a minimal automaton which generates a substitution sequence $u$ in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the semicocycle discontinuities of $u$ .

在本文中，我们定义了与原始替换相关的半群。我们用它来构造一个最小自动机，以反向阅读的方式生成一个替换序列 u。我们证明，在替换具有重合性的情况下，这种自动机可以完全描述 u 的半周期不连续性。

引用次数: 0

Flow views and infinite interval exchange transformations for recognizable substitutions 可识别替换的流动视图和无限区间交换变换

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2024.07.004

A flow view is the graph of a measurable conjugacy $Φ$ between a substitution or S-adic subshift $(Σ, σ, μ)$ and an exchange of infinitely many intervals in $([0, 1], F, m)$ , where $m$ is Lebesgue measure. A natural refining sequence of partitions of $Σ$ is transferred to $([0, 1], m)$ using a canonical addressing scheme, a fixed dual substitution $S_{*}$ , and a shift-invariant probability measure $μ$ . On the flow view, $τ \in Σ$ is shown horizontally at a height of $Φ (τ)$ using colored unit intervals to represent the letters.

The infinite interval exchange transformation $F$ is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that $Φ$ is self-similar. We discuss why the spectral type of $Φ \in L^{2} (Σ, μ),$ is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

流视图是替换或 S-adic 子移位与无穷多个区间的交换之间的可测共轭图，其中是 Lebesgue 度量。使用一个典型寻址方案、一个固定的对偶置换，以及一个移位不变的概率度量，可以将的分区的一个自然精炼序列转移到。在流动视图中，用彩色单位间隔表示字母，水平高度为。

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引用次数: 0

Automorphism groups of random substitution subshifts 随机替换子移的自同构群

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.08.006

We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.

我们证明，对于一类合适的随机置换，其相应的子移位的自变群包含一个无限简单子群和一个全移位自变群的副本。因此，它们是可数的、不可门的和非剩余有限的。为了证明这一点，我们引入了随机置换的洗牌和广义洗牌的概念，以及随机置换的可识别性的局部版本。如果没有可识别性，我们就需要一个更精细的可识别词概念，以便理解它们的自动变形。我们证明，一个可识别词的存在往往足以将全移位的自形群嵌入随机置换子移位的自形群中。

引用次数: 0

Inter-model sets in Rd are model sets Rd中的模型间集是模型集

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.06.007

We show that any union of finitely many shifted model sets from a given cut-and-project scheme is a model set in some modified cut-and-project scheme. Restricting to direct space $R^{d}$ , we show that any inter-model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original windows.

我们证明，给定切分方案中有限多个移位模型集的任何联合都是某个修正切分方案中的模型集。限于直接空间 Rd，我们证明任何模型间集合都是某个具有第二可数内部空间的修正切分方案中的模型集合。在这两种情况下，修正的切分-投影方案中的窗口都继承了原始窗口的拓扑和度量理论性质。

引用次数: 0

Aperiodic order: Papers in honour of Uwe Grimm 非周期性秩序：纪念乌韦-格林的论文

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2024.07.005

引用次数: 0

Correlations of the Thue–Morse sequence Thue–Morse序列的相关性

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.02.001

The pair correlations of the Thue–Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue–Morse sequence with general real weights are effectively determined by a single value of the balanced 2-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue–Morse sequence vanish, and that, for any even $n$ , the $n$ -point correlations of the balanced Thue–Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue–Morse system. We finish by showing how the correlations of the Thue–Morse system with general real weights can be derived from the balanced 2-point correlations.

本文重新探讨了 Thue-Morse 序列和系统的成对相关性，重点是各种手段的渐近结果。首先，我们证明了具有一般实权重的 Thue-Morse 序列的所有高阶相关性都是由平衡 2 点相关性的单一值有效决定的。因此，我们证明了平衡 Thue-Morse 序列的所有奇阶相关性都消失了，而且对于任何偶数 n，平衡 Thue-Morse 序列的 n 点相关性的均值为零，它们的绝对值也是零，并可提升到任意正幂次。所有这些结果也适用于整个图伊-莫尔斯系统。最后，我们将展示如何从平衡 2 点相关性推导出具有一般实权重的 Thue-Morse 系统的相关性。

引用次数: 0

Catalan numbers as discrepancies for a family of substitutions on infinite alphabets 加泰罗尼亚语数字作为无穷大字母替换族的差异

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.06.010

In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function is exponential (and guided by the inflation factor), the behaviour of the second-order term is strikingly different. For the finite setting, it is known that the second term is also exponential or exponential times a polynomial. We exhibit a large family of examples where the second term is at least exponential in $n$ divided by half-integer powers of $n$ , where $n$ is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.

在这项工作中，我们考虑了无限字母表上的一类替换，并证明它们表现出一种增长行为，而有限字母表上的替换是不可能出现这种增长行为的。虽然在这两种情况下，瓦片计数函数的前导项都是指数型的（并由膨胀因子引导），但二阶项的行为却截然不同。对于有限设置，已知第二阶项也是指数或指数乘以多项式。我们展示了一大类例子，其中第二项至少是 n 的指数除以 n 的半整数幂，其中 n 是替换步数。特别是，我们用加泰罗尼亚数的线性组合为这种差异提供了一个标识。

引用次数: 0

Directional ergodicity, weak mixing and mixing for Zd- and Rd-actions 定向遍历性，弱混合和混合Zd-和<m

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.06.006

<div>For a measure preserving <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>- or <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-action <math><mi>T</mi></math>, on a Lebesgue probability space <math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></math>, and a linear subspace <math><mrow><mi>L</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math>, we define notions of direction <math><mi>L</mi></math> ergodicity, weak mixing, and strong mixing. For <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions, it is clear that these direction <math><mi>L</mi></math> properties should correspond to the same properties for the restriction of <math><mi>T</mi></math> to <math><mi>L</mi></math>. But since an arbitrary <math><mrow><mi>L</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math> does not necessarily correspond to a nontrivial subgroup of <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>, a different approach is needed for <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions. In this case, we define direction <math><mi>L</mi></math> ergodicity, weak mixing, and mixing in terms of the restriction of the unit suspension <math><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></math> to <math><mi>L</mi></math>, but also restricted to the subspace of <math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mover><mrow><mi>μ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></mrow></mrow></math> perpendicular to the suspension direction. For <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions, we show (as is more or less clear for <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>) that these directional properties are spectral properties. For weak mixing <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>- and <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions, we show that directional ergodicity is equivalent to directional weak mixing. For ergodic <math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions <math><mi>T</mi></math>, we explore the relationship between direction <math><mi>L</mi></math> properties as defined via unit suspensions and embeddings of <math><mi>T</mi></math> in <math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math>-actions. Finally,

对于 Lebesgue 概率空间 (X,μ) 上的保度 Zd 或 Rd 作用 T 和线性子空间 L⊆Rd，我们定义了方向 L 的遍历性、弱混合和强混合的概念。但由于任意的 L⊆Rd 不一定对应于 Zd 的一个非难子群，因此需要对 Zd 作用采用不同的方法。在这种情况下，我们用单位悬浮 T˜对 L 的限制来定义方向 L 的遍历性、弱混合和混合，但也限制在垂直于悬浮方向的 L2(X˜,μ˜) 子空间。对于 Zd-作用，我们证明（对于 Rd 或多或少是清楚的）这些方向特性是光谱特性。对于弱混合 Zd- 和 Rd-作用，我们证明了方向遍历性等同于方向弱混合。对于遍历 Zd-作用 T，我们探讨了通过单位悬浮定义的方向 L 特性与 T 在 Rd-作用中的嵌入之间的关系。最后，我们确定了非遍历和非弱混合方向的可能集合的结构，并讨论了通性问题。

引用次数: 0

The (reflected) Eberlein convolution of measures 度量的（反射）艾伯林卷积

IF 0.5 4区数学 Q3 MATHEMATICS

Indagationes Mathematicae-New Series

Pub Date : 2024-09-01 DOI: 10.1016/j.indag.2023.10.005

In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution.

在本文中，我们研究了度量的埃伯林卷积的性质，并引入了它的反射版本。对于函数，我们证明反射埃伯林卷积可视为平移不变的函数值内积。我们研究了它的正则特性，并证明了它在合适的函数集合上的存在性。对于平移有界的度量，我们证明了（反射）艾伯林卷积总是沿着给定序列的子序列存在，并且是一种弱几乎周期性的可傅里叶变换度量。我们证明，如果两个度量中的一个是平均几乎周期性的，那么（反射）艾伯林卷积就是强几乎周期性的。此外，如果其中一个度量是常模几乎周期性的，那么（反射的）艾伯林卷积也是常模几乎周期性的。

引用次数: 0

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