分形集在拟对称映射下的变形

This project is plan to study the distortion of fractals by quasisymmetric mappings on Eulidean spaces. Our study lies on the overlapping region between fractal geometry and the theory of quasisymmetric mapping. On the one hand, we can use the theory of quasisymmetric mapping to describe the property of fractals, on the other hand, we can also use the technique in fractal geometry to study the quasisymmetric mappings. Our study consists of three issues. The first is the distortion of dimension by quasisemmetric mappings, including the study of quasisymmetrically minimal sets on the line and the study of determining the conformal dimension of some specific fractals on the plane. The second is the study of metric properties of fractals in some specific quasisymmetric equivalent classes. We want to conclude from this study that by quasisymmetric mappings what metric properties are preserved and what metric properties are distored. The third is the study of properties of all quasisymmetric mappings which preserve dimension, by making use of theory of fractal geometry. The issues studies in this project have been a subject of interest within fractal geometry and the theory of quasisymmetric mappings in the past a few years. Our study puts emphisis on the fusion of the ideas and the methods of the two subject. It is important to the undstanding of the geometric structure of complicated fractals and the development of the theory of quasisymmetric mappings.

本项目计划研究拟对称映射所导致的欧式空间中分形集的变形，这是一个分形几何和拟对称映射的交叉课题，一方面可以利用拟对称映射的理论刻画分形集的性质，另一方面也可以利用分形几何的手段来研究拟对称映射。 具体研究内容包括紧密联系的三个方面。之一是拟对称映射对维数的改变，包括全面系统地研究直线上的拟对称极小集，以及计算平面上特定分形集的共形维数。之二是在整体上研究特定分形集的拟对称等价类的度量性质，从而得出拟对称映射能保持何种度量性质以及它能改变何种度量性质。之三是利用分形几何研究保持维数不变的拟对称映射的性质。 本项目所涉及的研究是分形几何和拟对称映射相交叉的一个新的方向，是近年来国内外相关方向的一个研究热点。我们的研究更强调融合分形几何与拟对称映射这两门不同学科的思想和方法，它对于理解复杂分形集的结构、进一步丰富和发展拟对称映射的相关理论都具有十分重要的意义。

我们主要研究了拟对称映射和分形集的相关问题。拟对称映射研究的一个重要内容是研究那些在拟对称映射下不变的性质，即拟对称不变量。我们利用定义集合间隔序列方法，获得了一个新的d-维拟对称映射的不变量，深化了 Lipschitz 不变量研究中类似的结果。.关于分形集我们研究了两类集合，即齐次集和 Moran 集。齐次集是 Ahlfors-David 正则集的推广，其定义来自于测度意义下的齐次、一致完全性以及加倍性。该类分形包含所有的（拟）Ahlfors-David 正则集和许多非正则的 Moran 集。我们以 Moran 集为工具，研究了这类齐次分形的维数、双 Lipschitz 嵌入和拟 Lipschitz 等价。对于欧式空间的 Moran 集，在适当的条件下，我们得到了它们的拟 Ahlfor s-David 正则性和它们的拟 Lipschitz 等价性。