论文标题
在微观方面，随机分形的尺寸和较低的维
On microsets, Assouad dimension and lower dimension of random fractals, and Furstenberg's homogeneity
论文作者
论文摘要
我们研究了随机构造的分形的微型集合，在本文中被称为Galton-Watson分形。这是一个将Mandelbrot Percolation推广的模型，其中Galton-Watson树（其后代分布不一定是二项式）被预测到$ \ Mathbb {r}^d $，这是由来自相似性图的迭代函数系统（IFS）产生的编码图。我们表明，对于这样一个随机的分形$ e $，每当底层IF满足开放式条件时，几乎可以肯定的是，$ e $的Assouad尺寸是$ \ text {supp} \ left（e \ weft（e \ right）$的最大Hausdorff尺寸介于两者之间的每个值都是$ e $的某些微问题的Hausdorff尺寸。为了获得上述内容，我们首先分析了（确定性）集合集合的微观集合与该集合的适当编码树子树的某些限制之间的关系。还采用了必要的调整，以获取对Furstenberg同质性属性的一些见解。我们定义了比同质性较弱的属性，并表明在$ \ mathbb {r} $中，其Hausdorff尺寸小于1，它等同于弱分离条件。
We study the collection of microsets of randomly constructed fractals, which in this paper, are referred to as Galton-Watson fractals. This is a model that generalizes Mandelbrot percolation, where Galton-Watson trees (whose offspring distribution is not necessarily binomial) are projected to $\mathbb{R}^d$ by a coding map which arises from an iterated function system (IFS) of similarity maps. We show that for such a random fractal $E$, whenever the underlying IFS satisfies the open set condition, almost surely the Assouad dimension of $E$ is the maximal Hausdorff dimension of a set in $\text{supp}\left(E\right)$, the lower dimension is the smallest Hausdorff dimension of a set in $\text{supp}\left(E\right)$, and every value in between is the Hausdorff dimension of some microset of $E$. In order to obtain the above, we first analyze the relation between the collection of microsets of a (deterministic) set, and certain limits of subtrees of an appropriate coding tree for that set. The results of this analysis are also applied, with the required adjustments, to gain some insights on Furstenberg's homogeneity property. We define a weaker property than homogeneity and show that for self-homothetic sets in $\mathbb{R}$ whose Hausdorff dimension is smaller than 1, it is equivalent to the weak separation condition.
