分形几何中的嵌入问题

Embedding, especially which preserves some geometric structures, is an important means in the study of geometry and topology. This project is planed to study the affine embedding and biLipschitz embedding in fractal geometry, the former preserves linear structure of Euclidean space and the latter preserves metric structure with some distortion. We want to study the affine embedding problem of self-similar sets and a related conjecture of Furstenberg, and investigate the difference of geometric structure caused by the difference algebraic properties of the ratios. This problem comes from Furstenberg's study of times2 and times3 dynamical systems. Furstenberg conjectured that the intersection of any affine image of a 2-invariant set and a 3-invariant set must be small in the sense that the dimension of the intersection does not greater than the generic value given by Mastrand theorem. We also want to study the biLipschitz embedding problem and its relationship with biLipschitz equivalence problem. BiLipschitz equivalence problem is an important topic in fractal geometry and it is closely related to algebraic number theory. We hope to obtain some general results on biLipschitz embedding problem and then promote the research of biLipschitz equivalence problem.

嵌入是研究几何、拓扑的重要途径，特别是保持某种几何结构的嵌入。本项目计划研究分形几何中的仿射嵌入与双Lipschitz嵌入，前者保持所在欧氏空间的线性结构，后者则在一定扭曲下保持了度量。我们将研究自相似集的仿射嵌入问题及相关的Furstenberg猜测，探讨压缩比代数性质的不同导致的几何结构的差异。这个问题源于Furstenberg对乘2和乘3动力系统的研究。Furstenberg猜测，乘2不变集的仿射像与乘3不变集的交一定会比较小，其维数不超过Mastrand定理给出的典型值。我们还将研究分形集的双Lipschitz嵌入问题，及其与双Lipschitz等价问题的关系。双Lipschitz等价问题是分形几何的重要课题，与代数数论也有密切的联系。我们希望在双Lipschitz嵌入问题上得到一些普遍的结果，以此来推动相对困难的双Lipschitz等价的研究。

嵌入是研究几何、拓扑的重要途径，特别是保持某种几何结构的嵌入。本项目主要关注分形几何中的仿射嵌入与双Lipschitz嵌入，前者保持所在欧氏空间的线性结构，后者则在一定扭曲下保持了度量。通过对嵌入问题以及与之相关的各种问题的研究，我们可以了解相应分形集的几何结构，并且能够比较不同分形集的几何结构的异同之处。.我们主要研究了非齐次自相似集的仿射嵌入问题，利用多重无理旋转的结果，对于部分情况证明了压缩比对数可公度猜测；研究了分形集分离结构与gap序列的关系，证明了gap序列的渐进公式；研究了Sierpinski垫片有理方向的截集，并得到了盒维数的相关结果；研究了齐性集的双Lipschitz嵌入问题和有向图集的C1嵌入问题；研究了与几何不变量相关的各种几何结构问题；并且将这些研究中所得的技巧和方法应用到复杂网络等问题的研究中。