Weierstrass型函数关联分形集的测度和维数的研究

本项目拟研究Weierstrass型函数(将Weierstrass函数中的余弦函数替换为其他周期函数而得到)所关联分形集(包括象集,图像,水平集和逆象集)的测度和维数.Weierstrass型函数是处处连续处处不可微,也是分形几何中一类重要研究对象,其分形性质在理论和应用上都极具重要性.本项目拟以欧氏空间和符号空间上的Moran集为模型,将分形和组合数学中的技巧相结合来展开研究,主要考虑被替代函数为Rademacher函数和Takagi函数(锯齿函数)两种情形.项目关键点是给出相应情形(具有重叠结构或压缩比下确界为0)Moran集的维数公式.这些问题涉及分形几何研究的核心内容,相关研究结果有助于深入认识Weierstrass型函数,从而促进分形函数的研究和发展.

This project intends to study the measure and dimension of fractal sets (including image sets, graphs, level sets and inverse image sets) associated with the Weierstrass-type function, which is obtained by replacing cosine function in the Weierstrass function with other periodic function. The Weierstrass-type function is continuous nowhere differentiable, which is an important research object of fractal geometry. The fractal properties of the Weierstrass-type function are very important in both theory and application. The project will take the Moran sets in Euclidean space and the symbolic space as models, and carry out the research by combining techniques in fractals and in Combinatorial mathematics. The main consideration is the replaced function being the Rademacher function and the Takagi function(sawtooth function). The key point is to present the dimension formula of Moran sets (with overlaps or with the infimum of compression ratios being zero). These questions involve the core of the research of fractal geometry. The related research results will be helpful to understand deeply the Weierstrass-type function, and then promote research and development of fractal functions.