基于无穷图递归系统Mauldin-Williams分形测度的重分形分析

Mauldin-Williams分形测度包含Moran测度,自仿(自相似)测度和图递归测度等诸多经典分形测度,其上的重分形分析是分形几何的重要课题之一.本项目研究具有重叠结构Mauldin-Williams分形测度的重分形分析.目前尚无一般性定理能保证此分形测度满足重分形机理,也缺乏$L^q$谱的确切计算公式.我们拟利用扩张-截断技巧寻找一列具有分离条件的有限图递归系统,使得其极限(无穷图递归系统)的不变测度能够拟合原始的分形测度,然后利用非负无穷矩阵的不变测度和不变向量来研究这些图递归测度,主要考虑重分形机理成立条件以及重分形谱的计算和不正则集的刻画.本项目涉及分形几何、动力系统及几何测度论等领域,它有着广泛的发展前景,其研究能使人们更加深入的理解和发展重分形分析,使之能够更好的解决实际问题.

Mauldin Williams fractal measure includes many classical fractal measures, such as Moran measure, self-affine (self-similar) measure and graph-directed measure, the multifractal analysis on which is one of the important topics of fractal geometry. In this project, the multifractal analysis of Mauldin Williams fractal measure with overlaps is studied. At present, there is no general theorem to ensure that this fractal measure meets the multifractal formalism and there is no explicit formula for calculating $L^q$ spectra. We want to use the technique of extension-truncation to find a series of finite graph-directed systems with some separation condition such that the invariant measure of the limit system (infinite graph-directed system) can fit the original fractal measures, and then use the invariant measures and invariant vectors of nonnegative infinite matrices to study these graph-directed measures. We mainly considers the condition under which the fractal measure satisfies multifractal formalism, the calculation of multifractal spectra and the characterization of irregular sets. This project has broad prospects for development in the field of fractal geometry, dynamical system and geometric measure theory. This study will make people understand better and develop better the multifractal analysis and make it better to solve practical problems.