Self-avoiding walk and its continuum'limit on fractals and geometric figures defined by conformal mappings

(1)讨论了分形上自回避游动的连续体极限，并研究了其样本路径的几何性质。样本路径的轨迹被认为是一种多类型随机构造。我们首先给出了一类多类型随机结构的‘精确Hausdorff维数’的一般定理，我们的一般定理研究了具有几乎确定的Hausdorff维度D(通常，随机结构的Hausdorff维度几乎确定)且D维Hausdorff测度为零的多类型随机结构。作为该定理的一个应用，我们考虑了d维Sierpinski垫片上的一个自回避行走模型，称为‘分支模型’。我们证明了连续统极限的存在性，并确定了其确切的Hausdorff维数。(2)我们考虑了Anisot…在二维Sierpinski地毯上有更多的ROPIC扩散，这是一个无限分叉的分形，并且表明各向同性随着我们看到的扩散的尺度变大而渐近恢复。这可以通过前Sierpinski地毯上各向异性电阻网络的各向同性恢复来说明。这种恢复各向同性的现象是独特和有趣的，因为它不会发生在欧几里德空间这样的均匀空间中，而只出现在非均匀空间中，例如分数。使用Grant-in-aid，我们购买了关于分形学、Hausdorff度量、概率论、遍历理论等的书籍，还购买了用于电子通信和撰写论文的计算机软件。Grant-in-aid还使我们能够在近距离领域与研究人员见面，讨论和收集关于使用保角映射构造的几何图形的随机构造、Hausdorff和填充度量的信息，这有助于我们获得许多关于未来研究发展的洞察和提示。较少

(1) We dealt with the continuum limits of self-avoiding walks on fractals and studied geometric properties of their sample paths. The trajectory of a sample path is regarded as a multi-type random construction. We first developed a general theorem on the 'exact Hausdorff dimension' for a wide class of multi-type random constructions.Our general theorem deals with multi-type random constructions with almost sure Hausdorff dimension D (usually, Hausdorff dimensions of random constructions are determined almost surely) and with zero D-dimensional Hausdorff measure. It determines dimension functions which give positive and finite Hausdorff measures, which we call exact Hausdorff dimensions, for a wide class of constructions.As an application of this theorem, we considered a model of self-avoiding walk called the 'branching model' on the d-dimensional Sierpinski gasket. We showed the existence of the continuum limit and then determined the exact Hausdorff dimensions.(2) We considered anisot … More ropic diffusions on the 2-dimensional Sierpinski carpet, which is an infinite-ramified fractal, and showed that the isotropy is asymptotically restored as the scale in which we see the diffusions gets larger. This can be shown in terms of restoration of isotropy of anisotropic resistance networks on the pre-Sierpinski carpet. This phenomenon of restoration of isotropy is unique and of interest in the sense that it does not happen in a homogenious space such as the Eucledian spaces, but occurs only in inhomogenious spaces such as fractals.Using Grant-in-Aid, we bought books on fractals, Hausdorff measures, probability theory, ergodic theory etc, and also computer software to be used for electronic communication and writing papers.The Grant also enabled us to meet in person researchers in close fields to discuss and collect information on random constructions, Hausdorff and Packing measures of geometic figures constructed using conformal mappings, which helped us much get insight and hints for future developments of our research. Less