Statistically self-similar fractal sets
Konstantin Igudesman
本文定义了统计自相似集，并利用K.J.Falconer的方法求出了统计自相似集的豪斯多夫维数.
In the present paper we define statistically self-similar sets, and, using a modification of method described K.J.Falconer find a Hausdorff dimension of a statistically self-similar set.
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Let $(X,\mathscr{B}, μ,T,d)$ be a measure-preserving dynamical system with exponentially mixing property, and let $μ$ be an Ahlfors $s$-regular probability measure. The dynamical covering problem concerns the set $E(x)$ of points which are covered by the orbits of $x\in X$ infinitely many times. We prove that the Hausdorff dimension of the intersection of $E(x)$ and any regular fractal $G$ equals $\dim_{\rm H}G+α-s$, where $α=\dim_{\rm H}E(x)$ $μ$--a.e. Moreover, we obtain the packing dimension of $E(x)\cap G$ and an estimate for $\dim_{\rm H}(E(x)\cap G)$ for any analytic set $G$.
Dimension conservation for self-similar sets and fractal percolation
我们介绍了一种技术，利用分形渗流的投影特性，建立确定性自相似集的部分尺寸守恒的结果。例如，假设$K$是具有Hausdorff维数$\dim_H K >1$的$\mathbb{R}^2$的自相似子集，使得底层相似性的旋转分量生成完整旋转群。然后，对于所有$ε>0$，将$π_θ$写为在$θ$方向上投影到线$L_θ$上，对于正勒贝格测度的$x \in L_θ$集合，对于除了Hausdorff维数为0的集合中的方向之外的所有方向$θ$，截面的豪斯多夫维数满足$\dim_H (K\cap π_θ^{-1}x)> \dim_H K - 1 - ε$。对于一类自相似集，我们在所有方向上得到了类似的结论，只是用较低的盒维数代替了截面的豪斯多夫维数.我们得到了关于Mandelbrot渗流集的截面维数的类似不等式。
We study geometric rigidity of a class of fractals, which is slightly larger than the collection of self-conformal sets. Namely, using a new method, we shall prove that a set of this class is contained in a smooth submanifold or is totally spread out.
Projections of fractal percolations
本文研究了随机Cantor集$E\subset \mathbb{R}^2 $的径向投影、正交投影和距离集，称之为Mandelbrot逾渗或逾渗分形。证明了如下命题几乎必然成立：若$E$的豪斯多夫维数大于1，则到\textbf{every}直线的正交投影、以\textbf{every}为中心的径向投影以及到\textbf{every}点的距离集都包含区间.
In this paper, we prove that many fractal sets generated by the associated dynamical systems only contain irrationals. As an application, we explicitly construct some overlapping self-similar sets which only consist of irrationals.
The cookie-cutter-like set is defined as the limit set of a sequence of classical cookie-cutter mappings. For this cookie-cutter set it is shown that the topological pressure function exists, and that the fractal dimensions such as the Hausdorff dimension, the packing dimension and the box-counting dimension are all equal to the unique zero $h$ of the pressure function. Moreover, it is shown that the $h$-dimensional Hausdorff measure and the $h$-dimensional packing measure are finite and positive.
This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.
Fractal curvatures and Minkowski content of self-conformal sets
对于自相似分形，Minkowski容度和分形曲率被引入作为其平行集的几何特征的适当限制，即，均匀的薄分形涂层。对于某些自共形集，已知存在曲面和Minkowski容度。共形迭代函数系统是比相似性更灵活的模型。这一工作统一并推广了这些结果到R^d中的一般自共形集。我们证明了平行集的重标体积，表面积和曲率收敛于极限的Cesaro平均意义。分形Lipschitz-Killing曲率方向措施本地化这些限制对组成的一个基点的分形和它的任何正常方向。有一个积分公式。我们只假设一阶几何的流行的开集条件，并删除大量的几何假设。对于曲率，我们还假设规则的欧氏距离函数的分形，如果周围的维数超过3，和一个统一的可积性条件，以限制曲率的“重叠集”。一个有限的逆证明了可积性条件是尖锐的。我们讨论更简单的充分条件。主要工具来自于遍历论。独立的兴趣是一个乘法遍历定理：失真，有多少共形迭代函数系统偏离其线性化，收敛。
Perturbation of self-similar sets and some regular configurations and comparison of fractals
我们考虑两个点集之间的距离，这是修改的Hausdorff度量，并应用它们来描述一些分形，如$δ$ -拟自相似集，和其他一些几何概念，在欧几里得空间，如平铺与准原型和模式与准图案。对于满足开集条件的$δ$ -拟自相似集，我们得到了与P.A. P.莫兰。本文试图从分形的“形式”的角度来探讨分形，并提出一些相关的问题。最后，我们试图探讨一个问题--近似晶体的非晶态固体表现出什么性质和行为？
Analysis of non-linear fractal functions on PCF self-similar sets
本文利用Edelstein压缩建立PCF自相似集上非线性分形插值函数的一般构造框架，分析其能量、法向导数与图像盒维数界，并通过Sierpinski垫、von-Koch曲线及多组数值示例验证。
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$μ$ Cantor sets. Analogues of the Lyapunov functions and features are given for asymptotic behaviors of fractal differential equations. Stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presente…
Micromeasure distributions and applications for conformally generated fractals
本文研究了自共形分形上有限型子移位的Gibbs测度的标度性及其在Falconer距离集问题和投影维数中的应用。我们的分析包括双曲朱利亚集，Schottky群的极限集和图定向自相似集。
Strict Hölder equivalence of self-similar sets
本文研究分形集的严格Hölder等价问题，完全刻画了两类自相似集（完全断开的分形立方体与满足强分离条件的双分支自相似集）的严格Hölder等价性。
Quantifying inhomogeneity in fractal sets
非均匀分形集是指在不同点上表现出不同标度行为的分形集。一个集合的Assouad维数是一个找到覆盖该集合的“最困难的位置和尺度”的量，它与盒维数的差异可以被认为是该集合不均匀程度的第一级总体度量。对于下一个层次的分析，我们开发了一个定量的理论不均匀性，考虑的措施，周围的一组点，该集表现出一定程度的不均匀性在一定的规模。对于一组例子，一个家庭的$(\times m, \times n)$不变的子集的2-环面，我们表明，这个数量满足大偏差原则。我们比较这个家庭的成员，演示率函数如何让我们更深入地了解他们的不均匀性。
Fractal curvature measures and Minkowski content for one-dimensional self-conformal sets
证明了满足开集条件的一维共形迭代函数系的不变集的分形曲率测度存在的充要条件是其对应的几何势函数是非格的.在非格情形下，我们得到了Minkowski容度的存在性，并证明了分形曲率测度是$δ$ -共形测度的常数倍，其中$δ$表示不变集的Minkowski维数.对于第一个分形曲率测度，这个常数因子与不变集的Minkowski容度一致。在格的情况下，我们给出了充分条件的Minkowski内容的不变集存在，对比的事实，Minkowski内容的自相似格分形从来没有存在。然而，每一个满足开集条件的自相似集都有一个Minkowski可测的$\mathcal{C}^{1+α}$同构像。无论是在晶格和非晶格的情况下，平均版本的分形曲率措施总是存在的。
Topological, metric and fractal properties of one family of self-similar sets
本文研究了一类自相似集的拓扑、度量和分形性质，证明其为Cantorval，并计算了其测度和边界维度。
Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-Hölder-$α$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations g…
Sets of beta-expansions and the Hausdorff Measure of Slices through Fractals
我们研究自然措施集的β-扩张和切片通过自相似集。在β-展开式的背景下，这些结果使我们能够更好地理解随机β-变换的最大熵的度量，并重新解释Lindenstrauss，Peres和Schlag关于等分布的一个结果。这些应用程序中的每一个都与伯努利卷积的研究有关。在分形设置，这使我们能够了解如何分解豪斯多夫措施切片，导致条件下，几乎每一个切片通过自相似集有积极的豪斯多夫措施，推广长期已知的结果几乎处处的价值观的豪斯多夫维数。
From fractal groups to fractal sets
本文是对群、半群及其作用的自相似性的一些观点和概念的综述，并给出了一些证明。它试图将这些概念与更熟悉的概念联系起来，如分形、自相似集和可重正化的动力系统。特别是，它提出了一个似是而非的定义什么是“分形组”应该是，并给出了许多这样的群体的例子。一类特别有趣的例子，来自单值群的迭代分支覆盖，或等价地从伽罗瓦群的迭代多项式，提出。这类包含有趣的群从代数的角度来看（恰好非可解群，群的中间增长，分支群，.），并且同时群的几何是明显的，因为群的极限自然地与覆叠映射的朱利亚集一致。在综述中，本文讨论了有限状态变换、群和语言的增长、群的极限空间、双曲空间和群、动力系统、Heck型算子、C^*-代数、随机矩阵、遍历定理和非交换变换的熵。自相似的群体就像一条自然的线，贯穿了这些看似不同的主题。
A note on Diophantine-type fractals for α-Lüroth systems
给出了某些实数集合用α-Lüroth展开式描述的豪斯多夫维数.
A compact set $E\subset {\Bbb R}^d$ is said to be arithmetically thick if there exists a positive integer $n$ so that the $n$-fold arithmetic sum of $E$ has non-empty interior. We prove the arithmetic thickness of $E$, if $E$ is uniformly non-flat, in the sense that there exists $ε_0>0$ such that for $x\in E$ and $0<r\leq {\rm diam}(E)$, $E\cap B(x,r)$ never stays $ε_0r$-close to a hyperplane in ${\Bbb R}^d$. Moreover, we prove the arithmetic thickness for several classes of fractal sets, including self-similar sets, self-conformal sets in ${\Bbb R}^d$ (with $d\geq 2$) and self-affine sets in …
This is a book to be published in 2020 by Cambridge University Press (Tracts in Mathematics Series). It focuses on the Assouad dimension of sets and measures in Euclidean space, as well as many variants on the Assouad dimension, including the lower dimension and Assouad spectrum. It considers these notions in the setting of fractal geometry and topics covered include: weak tangents, self-similar sets, self-affine sets, Mandelbrot percolation, limit sets of Kleinian groups, orthogonal projections, distance sets, and Kakeya sets. It also considers applications of these ideas to other areas, incl…
Conjugacies provided by fractal transformations I : Conjugate measures, Hilbert spaces, orthogonal expansions, and flows, on self-referential spaces
定理和明确的例子被用来显示如何自相似集（一般意义）之间的转换可能是连续的几乎处处相对于固定措施的集合，并可用于进行众所周知的流量和谱分析从熟悉的设置到新的。这项工作的重点是一些令人惊讶的应用程序，包括（一）我们所谓的分形傅里叶分析，其中的图形的基础功能是康托集，是不连续的一个可数密集的一组点，但有很好的近似性能;（二）勒贝格测度保持流动，多边形层，其波前是分形。其核心思想是利用分形变换提供定义在迭代函数系统吸引子上的希尔伯特空间之间的酉变换。一些例子涉及到工作的奥克斯托比和乌拉姆关于遍历流的地区所界定的多边形。
Large intersection classes on fractals
研究了一类共形迭代函数系的极限集，引入了极限集的子集类，并证明了该类在可数交下是闭的，且该类中的所有集合都具有大的豪斯多夫维数.我们展示了它在遍历论、丢番图逼近和复动力学中的一些应用。
Measuring Fractal Dimension using Discrete Global Grid Systems
本研究提出使用离散全球网格系统作为覆盖集计算地理空间矢量数据的分形维数，在合成数据上误差小于1%，并验证了其理论有效性与实用优势。
In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The…
General fractals represented by $\mathcal{F}$-limit sets of compression maps
本文提出利用集值压缩映射将一般非齐次分形表示为固定参数化空间中的 $\mathcal{F}$-极限集，证明了其计算复杂度与序列无关，并引入“一致覆盖条件”估计极限集的 Hausdorff 维数下界。
We consider self-similar sets in three-dimensional Euclidean space related to a regular tetrahedron. Sierpi${\rm \acute{n}}$ski tetrahedron is one such self-similar set. In this paper, we study the whole family of those sets. Our motivation is to obtain three-dimensional analogues of the fractal $n$-gons. In particular, we focus on the geometric properties of those sets from a viewpoint of ``imaginary cube''. An imaginary cube is a set $A$ for which there is some cube $C$ such that the projections of $A$ in the directions of the faces of $C$ equal these projections of $C$. It is already known …
Dimensions of some fractals defined via the semigroup generated by 2 and 3
我们计算的Hausdorff和Minkowski维数的符号空间$Σ_m=\{0,...,m-1\}^\N$的子集是不变的乘整数。结果适用于集合$\{x\in Σ_m: \forall\, k, \ x_k x_{2k}... x_{n k}=0\}$，其中$n\ge 3$。我们证明，对于这样的集合，豪斯多夫维度和闵可夫斯基维度通常不同。
Box dimension of generic Hölder level sets
本文研究一般Hölder水平集的盒维数，扩展了之前关于Hausdorff维数的研究，发现下盒维数与Hausdorff维数行为相似，而上盒维数则不同，并通过对Sierpiński三角形的估计说明差异。
Fractal Transformations in 2 and 3 Dimensions
我们提出了一些工作有关的分形变换的掩蔽迭代函数系统，并演示如何众所周知的算法生成分形变换可以修改这些系统。我们还表明，这些算法同样适用于三维数据集时，并提出了一些可能的应用特殊效果和建模。
Fractal Models for Normal Subgroups of Schottky Groups
对于具有至少两个生成元的自由群$\F_d$的正规子群$N$，我们引入了$N$关于与$\F_d$相关联的图向马尔可夫系统$Φ$的径向极限集$\Lr(N,Φ)$.这些集示出提供分形模型的径向极限集的正规子群的Kleinian群的肖特基型。我们的主要结果表明，如果$Φ$是对称的和线性的，那么我们有$\dim_{H}(\Lr(N,Φ))=\dim_{H} \Lr(\F_d,Φ))$当且仅当商群$\F_d /N$是顺从的，其中$\dim_{H}$表示豪斯多夫维数。这将Brooks关于Kleinian群的正规子群的一个结果推广到一个大类的分形集。此外，我们表明，如果$\F_d /N$是不服从，然后$\dim_{H}(\Lr(N,Φ))>\dim_{H}(\Lr(\F_d,Φ))/2$，这扩展了由Falk和Stratmann和Roblin的结果。
In this article, we introduce a notion of size for sets called thickness that can be used to guarantee that two Cantor sets intersect (the Gap Lemma), and show a connection among Thickness, Schmidt Games and Patterns. We work mostly in the real line, but we also introduce the topic in higher dimensions.
Hausdorff dimension of intersections between the Jarník sets and Diophantine fractals
研究了Jarník集与迪奥菲内分数集合交集的Hausdorff维度，证明其为满维，并指出无理数指数大于2且部分商有界的集合维度为1。
Projections of self-similar and related fractals: a survey of recent developments
近年来，人们对理解许多具体分形集和测度的投影产生了很大的兴趣，并取得了很大的进展。总的目标是能够超越一般的结果，如马斯特兰德定理，并量化每个投影的大小-或至少在一些非常小的集合之外的每个投影。本文调查了其中的一些结果和技术，以获得它们，重点是线性投影的平面自相似集和措施。
Fractal analysis for sets of non-differentiability of Minkowski's question mark function
在本文中，我们研究了各种分形几何方面的闵可夫斯基问号功能$Q.$我们表明，单位间隔可以写为工会的三个集$Λ_{0}:=\{x:Q'(x)=0\}$，$Λ_{\infty}:=\{x:Q'(x)=\infty\}$，和$Λ_{\sim}:=\{x:Q'(x)$不存在和$Q'(x)\not=\infty\}.$的主要结果是，Hausdorff维数的这些集是相关的以下方式。$\dim_{H}(ν_{F})<\dim_{H}(Λ_{\sim})= \dim_{H} (Λ_{\infty}) = \dim_{H} (\mathcal{L}(h_{\mathrm{top}}))<\dim_{H}(Λ_{0})=1.$这里，$\mathcal{L}(h_{\mathrm{top}})$是指在Farey映射$F,$的拓扑熵$h_{\mathrm{top}}=\log2$处的Stern-Brocot多重分形分解的水平集，而$\dim_{H}(ν_{F})$表示与$F.$相关的动力系统的最大熵测度的Hausdorff维数。
In AC electrical systems, the power depends on the real power (P) due to resistive elements and the reactive power (Q) due to the inductive and capacitive elements, which are commonly studied by using phasor and scalar methods. Thus, this paper focuses on applying the fractal Julia sets to observe the topological properties related to the signs and magnitudes of the real and reactive powers consumed or supplied by an electrical circuit. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Julia sets…
In a power system, the load demand considers two components such as the real power (P) because of resistive elements, and the reactive power (Q) because inductive or capacitive elements. This paper presents a graphical representation of the electric power demand based on the topological properties of the Julia Sets, with the purpose of observing the different graphic patterns and relationship with the hourly load consumptions. An algorithm that iterates complex numbers related to power is used to represent each fractal diagram of the load demand. The results show some representative patterns r…
Additive and geometric transversality of fractal sets in the integers
本文在整数离散背景下定义对应于环面 $\times r$ 不变集的分形集，证明同时不变集分类、交集与和集横截性及迭代和集维数结果。
Measure and dimension theory of permeable sets and its applications to fractals
本文研究可渗透集合与 Lebesgue 测度及多种维数概念的关系，证明维数小于 d-1 的子集通常可渗透，并利用 Nagata 维数刻画自相似集的可渗透性。
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