重叠结构自相似集的若干问题研究

具有重叠结构的自相似集是近期分形几何领域研究的热点问题。本项目拟采用代数和几何结合的方法对具有重叠结构自相似集的局部性质进行研究。具有重叠结构自相似集F中的点如果用N个（依赖于相似压缩变换的个数）符号所生成的符号空间中的点来编码，U表示只有唯一位置码与之对应点的集合，M表示至少有两个位置码与之对应点的集合。我们拟对集合U和M进行研究，主要包括在这两类集合的代数刻画、分形维数；这两类集合为自相似集的充要条件；分析集合U、M 和自相似集F基数之间的关系；在自相似集F勒贝格测度几乎处处每个点都有不可数个位置码条件下，研究点对应位置码的增长率等。这些问题与实数非整数基表示、符号动力系统密切相关。因此本项目的研究工作将更深刻地认识具有重叠结构的分形集，同时也有助于实数非整数基表示论、符号动力系统中相关问题的深入研究，促进分支学科之间交叉与渗透，丰富它们的理论，发现新方法和应用。

Self-similar sets with overlaps have been extensively investigated recently in fractal study. This project will investigate the local qualities of self-similar sets with overlaps from both algebraic and geometrical views. Let U and M denote the sets having unique code and multiple codes, respectively. Our investigation will concentrate on the sets U and M, includeing computing their Hausdorff dimensions.based on their algebraic characterizations; giving a equivalent characterization for the sets to be self-similar sets; analyzing the cardinality relationships among sets U, M and F; exploring the growth rate of points, if a.e. points of F has a continuum of distinct beta-expansions. These problems are closely related to representations of real numbers in non-integer bases and symbolic dynamics. Consequently, this project will enhance intercross and penetration in these subjects, enrich their theoretics, find new methods and applications, which will give a more deep understanding on the self-similar sets with overlaps and make a more deep study on the related problems in representation of real numbers and symbolic dynamics.