Zalcman引理在随机迭代函数族动力系统中的应用

本文根据Schwick的思想，利用Zalcman引理讨论了随机迭代函数族动力系统，指出了函数族随机迭代动力系统的Fatou集和函数族衍生半群动力系统的Fatou集定义差别明显但却等价.并获得了如下正规定则，设F={f_i|f_i为C（C）上的非线性解析函数，i ∈ M}，其中M为非空指标集，Σ_M={（j₁，j₂，…，j_n，…）|j_i ∈ M，i ∈ N}，若对任意的指标序列σ=（j₁，j₂，…，j_n，…）∈ Σ_M，迭代序列{W_σⁿ=f_jn º f_jn-1 º … ºf_j1（z）|n ∈ N}在点z处正规，则函数族F本身在点z处正规.

An Application of Zalcman Lemma in Dynamical Systems of Random Iterated Function Families

In this paper, it studies the properties of dynamical systems about random iterated of function family based on the idea of Schwick and using the Zalcman lemma. It points out that the Fatou set's definition of random dynamics and the Fatou set's definitions of dynamics of semigroups of a function family are obviously different but equivalent. Furthermore, the following normal criterion is obtained. Let F= {f_i|f_i is a nonlinear analytic function on C (C), i ∈ M}, where M is non empty index set, Σ_M = {(j₁, j₂, …, j_n, …)|j_i ∈ M, i ∈ N}, if for any index sequence σ = (j₁, j₂, …, j_n, …) ∈ Σ_M, the iterative sequence {W_σⁿ = f_jn º f_jn-1 º … º f_j1(z)|n ∈ N} is normal at point z, then the function family F is also normal at point z.