分形集的Lipschitz等价问题

The problem of Lipschitz equivalent of the fractals, is to reseach the Lipschitz equivalence of the fractals and their subsets. This is an important problem in the reseach of fractals, but it is very complicated. For the study of Lipschitz equivalence of self-similar set in complete metric space, we introduce a new definition which is s-structure, and show that some classical fractals, besides of self-similar sets satisfying the strong separation condition, have s-structure. And we analysis the main results abouts Lipschitz equivalence and BPI equivalence. To study the Lipschitz equivalence of the other fractals, we will do further study on s-structure, besides of its geometrical properties. we want to find a larger Lipschitz equivalence class. Furthermore, we want to find the Lipschitz equivalence of the general fractals ,such as general self-similar sets, fractals with general Moran structure, graph-directed sets, and so on. For finding the relation between BPI equivalence and Lipschitz equivalence, we will study the properties of the two equivalence on general fractals, besides of self-similar sets with open set condition. And then we want to apply our results to calculate the Hausdorff measure of some fractals.

分形集的Lipschitz等价问题,旨在探讨分形集与其子集间Lipchitz等价性,是分形几何研究的中心问题，但这个问题复杂而困难一直难有突破.为讨论完备度量空间中自相似集的Lipschitz 等价问题, 我们引入了新集类s-结构集,分析了Lipschitz等价与BPI等价的相关论著,已经可以得到一些很好的结论.为了得到更一般分形集的Lipschitz等价性,我们将对s-结构集做进一步的研究，弄清楚该结构的几何特征，以及最本质的结构性质，从而找到与建立更大的等价类.进一步,考虑更一般的自相似集与Moran结构上的Lipschitz等价问题，并力求找到图递归集上的Lipschitz等价映射. 争取建立各种Ahlfors-David正则集之间的Lipschitz等价映射. 通过在更一般的分形集上研究，争取找到Lipschitz等价与BPI等价之间更多的关联,以期能运用于自相似集的测度计算.

分形集的双Lipschitz等价与嵌入问题,都意在探讨分形集与其子集间双Lipchitz等价性,是分形几何研究的中心问题，但因其复杂困难而难有突破.为讨论完备度量空间中分形集的Lipschitz等价问题,我们曾引入并分析了s-结构集对于Lipschitz等价的意义,得到了一些很好的结论.对于更一般的分形集上的Lipschitz嵌入问题,我们一直在努力尝试与探索,但因其结构复杂性,在目前的工具下难以得到突破,于是新的工具和方法变得尤为必要.我们主要从经典分形集的间隔序列,C^1-嵌入和Assouad极小性三方面着力.满足强分离条件的自共形集与凸递归集作为经典的分形集,有着重要的研究意义,我们得到了其间隔序列与分形维数之间的关系,并完全刻画了其间隔序列的构成与性质.同时,我们在一些凸递归集上得到了仿射嵌入与C^1-嵌入的关系.另外,我们还研究了拟Lipschitz映射下Moran集的Assouad维数的变化.