三维自仿tile的拓扑性质和分形集的参数化

本项目拟研究分形几何中的两个拓扑问题：三维自仿tile的拓扑性质和分形集的参数化问题。. 我们考虑一类由共线数字集和整扩张矩阵生成的三维自仿tile.在前期工作中，我们证明了当此类自仿tile有14个邻居时，它的边界与二维球面同胚。我们拟研究此类自仿tile与三维球同胚的问题。此外，本项目也计划研究由非共线数字集生成的三维自仿tile的拓扑结构。. 关于分形集的参数化，我们研究图递归集和拟共形集的最佳参数化。对其他分形集，如自仿集，我们研究其保测参数化(勒贝格测度vs波雷尔测度)。我们从不变集的自仿测度出发，可以构造GIFS上的图递归测度，需要选择合适的概率权重来满足保测性。

This project is concerned with the topological properties of three dimensional self-affine tiles and measure preserving parametrization of fractal sets. We consider a class of three dimensional self-affine tiles generated by collinear digit set and expanding integer matrix. In our previous work, we show that the boundary of such a tile is homeomorphic to a 2-sphere whenever its set of neighbors contains 14 elements. We will show such kind of tiles with 14 neighbors are homeomorphic to three dimensional ball. In addition, we plan to study the topological structure of a three dimensional self-affine tile generated by non-collinear digit set.. Considering the parametrization of fractal sets, we will study the optimal parametrizations of GIFS and conformal IFS. As for other fractal sets such as self-affine sets, we consider measure preserving parametrizations (Lebesgue measure and Borel measure). By a self-affine measure on the invariant set we construct different graph-directed iterated measures on a GIFS. We have to choose a proper probability weight to get the measure preserving property.