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

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(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