Multifractal and it's application

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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.