中文摘要：

3x＋1推广函数T（x）的不动点性质及存在区域分析是分形中的一个重要研究问题.T（x）是结构复杂的超越函数,其在复平面上的不动点难于求解,不动点性质难于估计,这成为进一步研究T（x）动力系统的一个障碍.首先通过T（x）的拓扑不变性,给出了T（x）在复平面上存在不动点的构造性证明,分析了不动点的存在区域及其性质.根据存在区域,给出了T（x）的不动点在复平面上的分布.通过不动点的分布,提出了一种求T（x）不动点的数值算法.找到了T（x）在复平面上的多个收敛域,并绘制了收敛域处的分形图形.数值实验结果表明,本文算法正确、简捷.

英文摘要：

For generalized 3x＋1 function T（x）,the feature of fixed points and their existence-domain analysis is an important problem in fractal.T（x） is a complex transcendental function and its fixed point in C-plane is hardly to solve.Meanwhile,the feature of fixed point is difficult to analyze.All these become an obstacle for the further study of T（x） dynamic system.In this paper,Because of the topological invariance of T（x）,we constructively proved its fixed point in C-plane firstly.Then we give the analysis for the existence domain of fixed points as well as their feature.Based on the existence domain of T（x） fixed points,we estimated their distributions in C-plane.So we put forward a numerical algorithm for solving the fixed points of T（x） by analyzed the basis of distributions.Furthermore,we obtained some convergence domains of T（x） in C-plane and drew fractal image of these domains.The result of numerical experiment shows that the algorithm in this paper is correct and easy to implement.