Multifractal and it's application

本课题的研究目的是使分形的一个特征--奇异点谱在数学上严格地清晰化，为它在其他领域的应用提供有效的理论准备。我们的研究是基于我们以前的两个结果开始的：A generalization of Hata-Yamaguti's results on Takagi function II：Multinomial case，Japan J. Indust.应用数学：13（1996），435 -463和多项测度的多重分形谱，日本学报，73，Ser. A（1997），123-125与Okada，Sekiguchi和Shiota.一位研究者考虑了他的课题并作出了有益的贡献。我们得到了几个结果。我们在下面列举了其中的一些。1.利用多项式测度解决了在p-adic数中展开的数字和问题.作为上述结果的应用，我们得到了p-adic展开式的子块出现的显式表示.我们刻画了一类具有类似自相似性质的格雷码，在研究过程中发现了一类新的自相似测度，它包含了多项式测度。这些测度与多项式测度一样，都是用无穷多个差分方程组来刻画的。下一个课题是研究这类测度的奇异点谱。

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.