紧可度量化空间的加权分划、双曲性和Ahlfors正则共形维数
Weighted partition of a compact metrizable space, its hyperbolicity and Ahlfors regular conformal dimension
摘要：
紧度量空间的连续划分出现在许多不同的数学领域，如自相似集的构造，与双曲动力系统相关的马尔可夫分割，与加倍度量空间相关的并元立方体。它们的共同特征是将空间划分为有限数量的子集，然后将每个子集划分为有限块，并反复重复这个过程。在本文中，我们推广这样的连续划分，并称之为分区。给定一个分区，我们考虑一个权函数的概念，它为分区的每一块分配一个“大小”。直觉上，我们相信一个划分和一个权函数应该提供一个“几何”和“分析”的空间，我们的利益。我们将分三个部分探讨这一想法。在第一部分中，度量化的权函数，即存在一个度量“适应”一个给定的权函数，被证明是等价的Gromov双曲的图形与权函数。在第二部分中，我们证明了双Lipschitz等价性、Ahlfors正则性、体积加倍性和拟对称性等概念与权函数的某些性质是等价的。特别地，我们发现在权函数的世界中，准对称性和体积加倍性质是同一个概念。第三部分给出了紧度量空间的Ahlfors正则共形维数的一个特征，即与分划相关联的p-能量的临界指数p和与度量相对应的权函数。
英文摘要：
Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partitions associated with hyperbolic dynamical systems, dyadic cubes associated with a doubling metric space. The common feature in these is to divide a space into a finite number of subsets, then divide each subset into finite pieces and repeat this process again and again. In this paper we generalize such successive divisions and call them partitions. Given a partition, we consider the notion of a weight function assigning a "size" to each piece of the partition. Intuitively we believe that a partition and a weight function should provide a "geometry" and an "analysis" on the space of our interest. We are going to pursue this idea in three parts. In the first part, the metrizability of a weight function, i.e. the existence of a metric "adapted to" a given weight function, is shown to be equivalent to the Gromov hyperbolicity of the graph associated with the weight function. In the second part, the notions like bi-Lipschitz equivalence, Ahlfors regularity, the volume doubling property and quasisymmetry will be shown to be equivalent to certain properties of weight functions. In particular, we find that quasisymmetry and the volume doubling property are the same notion in the world of weight functions. In the third part, a characterization of the Ahlfors regular conformal dimension of a compact metric space is given as the critical index p of p-energies associated with the partition and the weight function corresponding to the metric.
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