Assouad dimension of self-affine carpets
John M. Mackay
我们计算Assouad维数的自仿射地毯的贝德福德和McMullen，和Lalley和Gatzouras。我们还计算了这些地毯的共形Assouad尺寸不自相似。
We calculate the Assouad dimension of the self-affine carpets of Bedford and McMullen, and of Lalley and Gatzouras. We also calculate the conformal Assouad dimension of those carpets that are not self-similar.
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The Assouad spectrum of random self-affine carpets
本文研究随机自仿贝德福德-麦克马伦地毯，推导其几乎必然Assouad谱和拟Assouad维数，发现非共形情形下它们通常不同于盒维数或Assouad维数，并给出示例说明差异。
Assouad spectrum of Gatzouras-Lalley carpets
本文研究了Gatzouras-Lalley地毯的Assouad谱，通过建立显式分段解析函数的凹共轭公式，揭示了其在动态不变集上的新性质。
Assouad type dimensions for self-affine sponges
我们研究了自仿射海绵的Assouad和较低维度;贝德福德和麦克马伦的平面自仿射地毯的高维类似物。我们的技术涉及弱切线的麦凯和泰森以及正则性的加倍措施的背景下，研究了Bylund和Gudayol。
Inhomogeneous self-affine carpets
研究了非齐次自仿射地毯的维数论。通过Olsen，Snigireva和Fraser的工作，非齐次自相似集的维数论已经比较成熟，但是对于更一般的非共形非齐次吸引子的研究几乎没有进展。如果一个维度是可数稳定的，那么结果是直接的，所以我们专注于上，下盒尺寸和计算这些显式的大类非均匀自仿射地毯。有趣的是，我们发现，“预期公式”的上盒尺寸可以失败的自仿射设置，从而揭示了新的现象，而不是发生在简单的自相似的情况下。
We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set $X$ adheres to very strong tangential regularity by showing that any two points of $X$, which are generic with respect to a self-affine measure having simple Lyapunov spectrum, share the same collection of tangent sets.
Asymptotic order of the quantization errors for self-affine measures on Bedford-McMullen carpets
该研究针对Bedford-McMullen地毯上的自仿测度，证明了任意正阶数下上下量化系数在精确量化维数处正且有限，得出量化误差的渐近阶，且该结果与地毯水平纤维无关，丰富了分形量化理论。
Assouad-type Dimensions of Overlapping Self-affine Sets
本文研究平面内投影满足分离条件但允许任意重叠的受支配矩形自仿射集，通过符号非自治迭代函数系统与符号切片建立 Assouad 及拟 Assouad 维数公式，并证明强分离下二者也可不同。
Fractal percolation on statistically self-affine carpets
本文研究随机统计自仿射地毯的分形渗流，证明其左右连通与上下连通的临界概率相同，并导出类似自相似地毯的拓扑性质。
The Assouad dimension of randomly generated fractals
我们考虑了几种不同的模型产生随机分形，包括随机自相似集，随机自仿射地毯，分形渗流。在每个设置中，我们计算\emph{almost sure}或\emph{Baire typical} Assouad维度，并考虑一些说明性示例。我们的结果揭示了一个共同的现象，在我们所有的模型：Assouad维数的随机生成的分形一般是尽可能大，不依赖于测量理论或拓扑结构的样本空间。这与其他常用的维数概念形成鲜明对比，如Hausdorff维数或包装维数。
Assouad type dimensions for certain sponges with a weak coordinate ordering condition
最近自仿射海绵已被证明是有趣的反例，以前几个开放的问题。最近研究的一类海绵是具有弱坐标有序条件的Bedford-McMullen海绵，即具有相同收缩比的几个坐标的海绵。不能使用与常规Bedford-McMullen海绵相同的公式计算此类套件的Assouad型尺寸。我们计算的Assouad型尺寸等海绵和更一般的Lalley Gatzouras海绵与弱坐标排序条件，讨论他们的一些更微妙的细节。
On the quantization for self-affine measures on Bedford-McMullen carpets
对于Bedford-McMullen地毯上的自仿射测度，我们证明了它的量子维数的存在性，并确定了它的精确值.进一步，我们给出了相应的上、下量化系数既为正又为有限的各种充分条件。最后，我们比较了量子化维数与相应的数量来自多重分形温度函数，并表明-不同的共形系统-他们在一般情况下不一致。
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings $\{f_{ij}\}_{(i,j)\in G}$ and let $μ$ be the self-affine measure associated with $\{f_{ij}\}_{(i,j)\in G}$ and a probability vector $(p_{ij})_{(i,j)\in G}$. We study the asymptotics of the geometric mean error in the quantization for $μ$. Let $s_0$ be the Hausdorff dimension for $μ$. Assuming a separation condition for $\{f_{ij}\}_{(i,j)\in G}$, we prove that the $n$th geometric error for $μ$ is of the same order as $n^{-1/s_0}$.
Convergence order of the quantization error for self-affine measures on Lalley-Gatzouras carpets
研究自仿措施在Lalley-Gatzouras地毯上的量化误差渐进行为，证明量化系数在精确量化维度上均远离零和无穷大，扩展了之前关于Bedford-McMullen地毯上自仿措施量化的工作。
Assouad type dimensions of generalized affine fractal interpolation functions and their applications
本文研究广义仿射分形函数图像的Assouad谱与维数，建立上下界并给出显式表达式，应用于Weierstrass和Takagi函数，解决了Fraser提出的开放问题。
Let ${\cal M}_{t,v,r}(n,m)$, $2\leq m<n$, be the collection of self-affine carpets with expanding matrix $\diag(n,m)$ which are totally disconnected, possessing vacant rows and with uniform horizontal fibers. In this paper, we introduce a notion of structure tree of a metric space, and thanks to this new notion, we completely characterize when two carpets in ${\cal M}_{t,v,r}(n,m)$ are Lipschitz equivalent.
Let $F$ be a Bedford-McMullen carpet defined by independent exponents. We prove that $\overline{\dim}_B (\ell \cap F) \leq \max \lbrace \dim^* F -1,0 \rbrace$ for all lines $\ell$ not parallel to the principal axes, where $\dim^*$ is Furstenberg's star dimension (maximal dimension of a microset). We also prove several rigidity results for incommensurable Bedford-McMullen carpets, that is, carpets $F$ and $E$ such that all defining exponents are independent: Assuming various conditions, we find bounds on the dimension of the intersection of such carpets, show that self affine measures on them a…
Let $F$ be a Bedford-McMullen carpet defined by independent integer exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P (\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension (maximal dimension of microsets). This improves the state of art results on Furstenberg type slicing Theorems for affine invariant carpets.
Assouad type dimensions and homogeneity of fractals
我们调查了几个方面的Assouad维度和较低的维度，它们一起形成一个自然的“维度对”。特别是，我们计算这些尺寸为某些类的自仿射集和准自相似集，并研究它们与其他概念的尺寸，如豪斯多夫维数的关系。我们还调查了这些方面的一些基本性质，包括他们的行为工会和产品和他们的集合论的复杂性。
The Assouad dimension of a metric space determines its extremal scaling properties. The derived notion of the Assouad spectrum fixes relative scales by a scaling function to obtain interpolation behaviour between the quasi-Assouad and box-counting dimensions. While the quasi-Assouad and Assouad dimensions often coincide, they generally differ in random constructions. In this paper we consider a generalised Assouad spectrum that interpolates between the quasi-Assouad to the Assouad dimension. For common models of random fractal sets we obtain a dichotomy of its behaviour by finding a threshold …
We construct a family of planar self-affine carpets with overlaps using lower triangular matrices in a way that generalizes the original Gatzouras--Lalley carpets defined by diagonal matrices. Assuming the rectangular open set condition, Barański proved for this construction that for typical parameters, which can be explicitly checked, the inequalities between the Hausdorff, box and affinity dimension of the attractor are strict. We generalize this result to overlapping constructions, where we allow complete columns to be shifted along the horizontal axis or allow parallelograms to overlap wit…
In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure's lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all se…
The sequence property for fractal dimensions
该研究证明，对有限稳定的分形维数定义，度量空间紧子集内存在与自身维数相同的收敛点序列，部分情况下该序列可同时对应多种维数定义。
In 1984 Bedford and McMullen independently introduced a family of self-affine sets now known as \emph{Bedford-McMullen carpets}. Their work stimulated a lot of research in the areas of fractal geometry and non-conformal dynamics. In this survey article we discuss some aspects of Bedford-McMullen carpets, focusing mostly on dimension theory.
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…
The Assouad dimensions of projections of planar sets
我们考虑Assouad维度的正交投影平面集到线上。我们的研究涵盖了一般和自相似集。对于一般集合，主要结果如下：如果平面上的集合有Assouad维数$s \in [0,2]$，那么投影几乎必然至少有$\min\{1,s\}$的Assouad维数.与豪斯多夫维数的著名类似物--即\emph{Marstrand's Projection Theorem} --相比，一个显著的区别是“至少”这个词不能省略：事实上，对于许多平面自相似维数集$s < 1$，我们证明了投影的Assouad维数对于一组正测度方向可以同时达到值$s$和$1$。对于自相似集，我们的研究自然分为两种情况：当旋转群是离散的，当它是稠密的。在“离散旋转”的情况下，我们证明了以下二分法为任何给定的投影：要么Hausdorff措施是积极的豪斯多夫维数，在这种情况下，豪斯多夫和Assouad维数重合;或豪斯多夫措施是零的豪斯多夫维数，在这种情况下，Assouad维数等于1。在“稠密旋转”的情况下，我们证明了每个投影的Assouad维数等于1，假设平面集不是一个单例。作为我们结果的另一个应用，我们证明了Assouad维数不存在\emph{Falconer's Theorem}.更确切地说，Assouad维度的自相似（或自仿射）集是不是在一般几乎肯定恒定时，随机的平移向量。
On the packing dimension of box-like self-affine sets in the plane
考虑一类平面自仿射集，我们称之为“盒状”集。盒状自仿射集是仿射映射的迭代函数系统（IFS）的吸引子，其中单位正方形[0，1]^2在映射的任意组合下的像是一个边平行于轴的矩形。本类包含贝德福德-麦克马伦地毯及其由Lalley-Gatzouras、Barauburski和Feng-Wang以及许多其他集合考虑的概括。特别是，我们允许在IFS的映射有非平凡的旋转和反射组件。在矩形开集条件下，我们利用基于映射的奇异值的压力型公式，计算了它们的填充维数和计盒维数。
Positive-measure intersections of Bedford-McMullen carpets under similitudes
本文证明Bedford-McMullen地毯在相似变换下若交集有正测度，则正交部分必为对角或反对角，且等距情形下仅差平移，加强了2010年猜想。
Topology automaton and Hölder equivalence of Barański carpets
该研究将拓扑自动机方法拓展至自仿射非p.c.f.的Barański地毯，定义其拓扑自动机并验证方法有效性，得到这类分形Hölder/Lipschitz等价的通用充分条件。
Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence
本文利用大偏差速率函数给出了 Bedford-McMullen 地毯中间维数的精确公式，揭示其可数多个相变及严格凹性，并证明中间维数相等等价于均匀 Bernoulli 测度的 Hausdorff 多重分形谱相等，从而为 Lipschitz 等价性提供必要条件。
The box dimension of random box-like self-affine sets
在本文中，我们研究了两个随机模拟的盒状自仿射吸引子介绍了弗雷泽，本身的Sierpienski地毯的扩展。确定了齐次随机情形（$1$ -变量随机情形）的几乎处处盒维数，并给出了几乎必然盒维数是确定性吸引子盒维数的期望的充分条件.此外，我们发现几乎必然盒计数维数的随机递归模型（$\infty$ -变量），其中包括仿射分形渗流。
We prove that the upper box dimension of an inhomogeneous self-affine set is bounded above by the maximum of the affinity dimension and the dimension of the condensation set. In addition, we determine sufficient conditions for this upper bound to be attained, which, in part, constitutes an exploration of the capacity for the condensation set to mitigate dimension drop between the affinity dimension and the corresponding homogeneous attractor. Our work improves and unifies previous results on general inhomogeneous attractors, low-dimensional affine systems, and inhomogeneous self-affine carpets…
The connected part of a metric space $E$ is defined to be the union of non-trivial connected components of $E$. We proved that for a class of self-affine sets called slicing self-affine sponges, the connected part of $E$ either coincides with $E$, or is essentially contained in the attractor of a proper sub-IFS of an iteration of the original IFS.This generalize an early result of Huang and Rao [L. Y. Huang, H. Rao. \emph{A dimension drop phenomenon of fractal cubes}, J. Math. Anal. Appl. \textbf{497} (2021), no. 2] on a class of self-similar sets called fractal cubes. Moreover, we show that t…
We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata's theorem. First we prove that every connected attractor of an IFS is $(1/s)$-Hölder path-connected, where $s$ is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a $(1/α)$-Hölder curve for all $α>s$. At the endpoint, $α=s$, a theorem of Remes from 1998 alr…
This paper presents a general procedure based on using the method of types to calculate the box dimension of sets. The approach unifies and simplifies multiple box counting arguments. In particular, we use it to generalize the formula for the box dimension of self-affine carpets of Gatzouras-Lalley and of Barański type to their higher dimensional sponge analogues. In addition to a closed form, we also obtain a variational formula which resembles the Ledrappier-Young formula for Hausdorff dimension.
We study the \emph{upper regularity dimension} which describes the extremal local scaling behaviour of a measure and effectively quantifies the notion of \emph{doubling}. We conduct a thorough study of the upper regularity dimension, including its relationship with other concepts such as the Assouad dimension, the upper local dimension, the $L^q$-spectrum and weak tangent measures. We also compute the upper regularity dimension explicitly in a number of important contexts including self-similar measures, self-affine measures, and measures on sequences.
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