Zalcman引理在随机迭代函数族动力系统中的应用

黄小杰, 刘芝秀

数学学报 ›› 2020, Vol. 63 ›› Issue (5) : 531-536.

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PDF(482 KB)
数学学报 ›› 2020, Vol. 63 ›› Issue (5) : 531-536. DOI: 10.12386/A2020sxxb0044
论文

Zalcman引理在随机迭代函数族动力系统中的应用

    黄小杰1,2, 刘芝秀1
作者信息 +

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

    Xiao Jie HUANG1,2, Zhi Xiu LIU1
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文章历史 +

摘要

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={fi|fi为C(C)上的非线性解析函数,iM},其中M为非空指标集,ΣM={(j1j2,…,jn,…)|jiMi ∈ N},若对任意的指标序列σ=(j1j2,…,jn,…)∈ ΣM,迭代序列{Wσn=fjn º fjn-1 º … ºfj1z)|n ∈ N}在点z处正规,则函数族F本身在点z处正规.

Abstract

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= {fi|fi is a nonlinear analytic function on C (C), iM}, where M is non empty index set, ΣM = {(j1, j2, …, jn, …)|jiM, i ∈ N}, if for any index sequence σ = (j1, j2, …, jn, …) ∈ ΣM, the iterative sequence {Wσn = fjn º fjn-1 º … º fj1(z)|n ∈ N} is normal at point z, then the function family F is also normal at point z.

关键词

随机动力系统 / 半群动力系统 / Julia集 / 排斥周期点 / 正规族

Key words

random dynamics / dynamics of semigroups / Julia set / repulsive periodic point / normal family

引用本文

导出引用
黄小杰, 刘芝秀. Zalcman引理在随机迭代函数族动力系统中的应用. 数学学报, 2020, 63(5): 531-536 https://doi.org/10.12386/A2020sxxb0044
Xiao Jie HUANG, Zhi Xiu LIU. An Application of Zalcman Lemma in Dynamical Systems of Random Iterated Function Families. Acta Mathematica Sinica, Chinese Series, 2020, 63(5): 531-536 https://doi.org/10.12386/A2020sxxb0044

参考文献

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基金

国家自然科学基金资助项目(11671091,11731003);江西省教育厅科技项目(GJJ180944,GJJ190963)

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