Multifractal and it's application

本课题的研究目的是使奇异点谱这一分形特性在数学上更加清晰，为其在其他领域的应用奠定有效的理论基础。我们的研究是在前面两个结果的基础上开始的：推广了Hata-Yamaguti关于Takagi函数的结果II：多项式情形，Japan J.Inust。APPL数学，13(1996)，435-463与多项式测度的多重分形谱，日本学报，73，系列A(1997)，123-125与Okada，Sekiguchi和Shiota.研究人员对课题进行了思考，并做出了有益的贡献。我们有了几个结果。我们在以下1中列举了其中的一些。利用多项式测度解决了在p元数域上展开的数字求和问题。作为上述结果的一个应用，我们得到了p进展开式的子块出现的显式表示。刻画了一类具有类自相似性质的格雷码，在上述研究过程中，我们发现了一类新的包含多项式测度的自相似测度。这些度量具有与多项式度量相同的无穷多个差分方程组的特征。我们应该为下一个主题研究这些度量的奇点谱。

The purpose of this research project is to make the singular point spectrum, which is a characteristic of fractal, clear mathematically strictly, and prepare an effective theory for the application to other fields. Our research was began based on our previous two results: A generalization of Hata-Yamaguti's results on Takagi function II: Multinomial case, Japan J. Indust. Appl. Math., 13(1996),435-463 and Multifractal spectrum of multinomial measures,Proc.Japan Acad,73,Ser.A( 1997), 123-125 with Okada, Sekiguchi and Shiota. Earch investigator considered his subject and made usefule contribution. And we had several results. We enumerate some of that in the following.1. Digital sum problem expanded in the p-adic number was solved by using the multinomial measures.2. As an application of the above result we obtained the explicit representation of subblock occurrences for the p-adic expansion.3. We characterized a kind of Gray code with a property like self-similarity.In the process of the above study we found the new class of self-similar measures, which contains the multinomial measures. These measures are characterize with the system of infintely many difference equations as same as the multinomial measures. We should investigate the singular point spectrum of thsese measures for the next subject.