类Gibbs测度的重分形分析

本项目主要研究类Gibbs测度的重分形分析。类Gibbs测度是Moran测度的推广。目前测度的重分形主要集中在”自由能函数”存在的情形，对其分裂为上、下极限的研究较少。本项目主要研究“自由能函数”分裂为上、下极限的情况，此间混合着上、下局部维数所确定水平集的讨论，使问题变得有趣但困难，更能揭示出重分形分析背后更深层次的现象和原因。本项目采用由浅入深的研究模式，依次考虑下述测度的重分形分析：特殊的Moran测度、一般的Moran测度、类Gibbs测度，进一步考虑补充问题--离散测度的重分形分析的反问题。这些问题涉及到分形几何、动力系统、遍历理论、概率论、随机过程、度量数论等相关学科。本项目不仅需要综合已有方法和技巧，还需要引进新的思想、方法与技巧，以及相关思想和技巧的融合。

The proposal studies the multifractal analysis of Gibbs-like measures. Gibbs-like measure is a generalization of Moran measure. So far, multifractal is mainly deal with the measures with the existence of “free energy function”. The measures whose “free energy function” split into the upper and lower limitation are not studied enough. This proposal is contributed to such measures. With the consideration of the level sets with respect to upper and lower local dimension, it becomes interesting but difficult. It will reveal the deeper reason behind multifractal and promote its development. This proposal adopts the research model going from the easy to the difficult. We will consider the multifractal analysis about a special kind of Moran measure, general Moran measure and then Gibbs-like measure in turn. As an addition, we will also consider the inverse problem of the multifractal analysis for discrete measures. They are related to fractal geometry, dynamical system, ergodic theory, probability, random process, metric number theory and so on. This proposal needs not only to synthesize the previous methods and ideas, but also need to develop some new ones and compromise the related ones.