摘要：
本文研究具有多极值点的单参复解析映射族f(z) =zn +cz构造非线性迭代函数系及其分形.首先分析复解析映射族f (z) =zn +cz在动力平面上的数学特性、M集及充满Julia集的几何特点;进而研究该复映射族在其M集的2个1周期参数区域上选取参数构造的迭代映射在动力平面上的动力学特性;在参数模值大于1的1周期参数区域中挑选N(N≥2)个参数,构造动力平面上的迭代映射(f(z) =zn +ciz,i=1,2,…,N);在N个迭代映射的公共吸引域内构造非线性迭代函数系;根据该复映射族在动力平面上有n-1个对称分布的1周期吸引不动点的数学特性,提出了将迭代点z随机旋转(2πj/(n-1),j={0,1,…,n-2})角度后再随机挑选迭代函数系中迭代映射的双随机迭代算法.结果表明:关于复映射族( f( z) =zn +cz,n=3,4,5,…),在M集的参数|c| >1的1周期参数区域挑选N个参数可以构造出有效的非线性迭代函数系;采用本文提出的双随机迭代算法可以在动力平面上大量生成n-1旋转对称分形.%Non-linear IFSs and their fractals are constructed by the family of the complex analytic polynomial with single parameter and multi-extreme-points f(z) =zn +cz in this paper. First,the mathematic characteristics,the M sets and the filled-in Julia sets of the mapping f(z) =zn +cz are investigated. Then,the dynamic characteristics of the iterating mappings constructed with the parameters, chosen from 2 1-cycle parameter regions in a M set,are investigated for the family of f(z) =zn +cz. Next,N(N≥2)parameters in the 1-cycle parameter region,where the module value of the parameters are greater than 1,are chosen to construct the iterating mappings (f(z) =zn +ciz,i=1,2,…,N)in the danymic plane. The non-linear iterateing function system is builted in the common attracting ba-sin of N iterating mappings. Last,according to the fact that there are n-1 attracting fixed-points symmetrically distributed in the plane for a mapping,the double random iterating method,by which the iterating point z is randomly rotated the degree of (2πj/(n-1),{j=0,1,2,…,n-2})and then is iterated by randomly choosing an iterating mapping in IFS,is presented. The result shows that about the complex mapping family (f(z) =zn +cz,n=3,4,5,…),selection of N parameters in the 1-cycle parameter region of M set, where|c| >1,can construct a valid non-linear IFS;a great number of the fractals with the n-1 rotation symmetry can be generated by the double random iterating method presented in this paper.
