复幂指数映射M-J对应关系及充满J集中相似结构特性

Relationship of M-J and Similar Structure in Filled-in Julia Sets from Complex Power Exponent Mapping

目的探索一类非解析复指数映射的M-J对应关系、充满J集中相似结构的特性和充满J集内部的动力学过程.方法提出了描述该模型广义M集的路径序列、路径的比例因子序列及嵌套层次的概念.构建了充满J集的PBranch-Tree.结果揭示模型的M集参数C与相应的充满J集图形结构的对应关系,分析了充满J集内部结构的自相似性,研究了充满J集吸引周期轨道中周期点的分布、被访问的次序及吸引域中点的轨道被吸引周期轨道吸引的规律.结论非解析复幂指数映射的充满J集内部的相似结构之间有明确的对应关系,吸引周期轨道中周期点有规律地被循环访问,吸引域中点的轨道按充满J集结构的对应相似性由高层次向低层次不断递进,最后迭代到吸引周期轨道上.

The corresponding relationship of M-J, characteristics of similar structure and dynamical process in filled - in Julia sets from non - analytic power exponent mapping are explored in this paper. The conceptions of Route Sequence （Rp）, Proportional Divisor Route Sequence（PDRp）, Embeded Layer and Periodpoint Branch - Tree are constructed to describe M sets and filled - in J sets easily. The relationship between M set and its filled - in J sets and structura similarity in filled - in J sets are discussed. The distribution of periodic points of the attracting orbit in filled - in J sets is observed, and the orbits of points in the attracting basin will gradually be iterated from a high layer to a low one in their similar structure in the filled - in J set, and ultimately be attracted to the attracting orbit in it.