关于加倍测度与拟对称映射若干问题的研究

Some questions on doubling measures and quasisymmetric mappings will be considered. We shall investigate three classes of questions on the basic theory of doubling measures as follows: The first is to give some description for doubling measures on a metric space, for example, we shall obtain the sufficient and necessary condition for a measure on a given doubling metric space to be doubling. The second is on the extension of doubling measures, for example, we shall obtain some conditions for a closed domain D in Euclidean space, such that every doubling measure on D can be extended to be doubling on the whole space. The third is on fat and thin sets for doubling measures in higher dimensional Euclidean spaces or more general metric spaces. For the basic theory of quasisymmetric mappings we shall consider the question of polar factorization of quasisymmetric mappings and the question of extension of quasisymmetric mappings defined on a connected subset of an Euclidean space, where the polar factorization asks that if every quasisymmetric mapping on a Hilbert space can be decomposed as the composition of a δ-monotone mapping and a bi-Lipschitz mapping？ For the interplay of doubling measures and quasisymmetric mappings our question is that if a straight line in an Euclidean space is an infinite set for the Jacobians of δ-monotone mappings? Through the study of these questions, we expect to pubilish more than eighteen papers of higher quality.

我们拟研究加倍测度与拟对称映射的几个问题。关于加倍测度的基础理论，我们将研究以下三类问题。第一类问题是描述具体空间上的加倍测度，例如，对于给定的加倍空间，我们将给出一个测度是加倍测度的充要条件。第二类是加倍测度的延拓问题，例如，我们将给出欧氏空间中的闭域上的加倍测度可以延拓为整个空间上的加倍测度的条件。第三类问题是高维欧氏空间或一般度量空间上加倍测度的胖集和瘦集的研究。关于拟对称映射的基础理论，我们拟研究拟对称映射的极因子分解问题及欧氏空间的连通子集上的拟对称映射的延拓问题。这里拟对称映射的极因子分解问题是指：欧氏空间上的拟对称映射是否可分解为δ-单调映射与双Lipschitz映射的复合？关于加倍测度与拟对称映射的交叉研究，我们拟研究的具体问题是：欧氏空间中的直线是不是δ-单调映射的Jacobi的无穷集？通过这些问题的研究，我们预期发表18篇以上高水平论文。

研究与维数、加倍测度以及拟对称映射相关的数学问题。具体来说，第一，研究了乘积集的维数性质。证明了乘积集的维数可达到相应估计式所允许的任意值。第二，研究了一些常见的分形集上的加倍测度。刻画了自仿地毯上具有加倍性质的自仿测度以及均匀Cantor集上所有加倍测度。第三，研究直线上的拟对称极小集。给出了拟对称极小的Cantor集的很一般的条件，证明了Hausdorff维数为1的数字限制集是拟对称极小的。第四，证明了Assouad维数的中间值可达性。第五，研究了加倍测度意义下的胖集与瘦集。证明了n个均匀Cantor集的乘积集是胖的当且仅当每个因子是胖的。此外，我们还证明了n维欧氏空间中Hausdorff维数小于n-1的集是Isotropic加倍测度的瘦集。第六，给出了一类分形方块的拓扑分类并证明了这个分类与其Lipschitz等价分类是一致的。