论文标题
朱莉娅的随机指数图
Julia sets of random exponential maps
论文作者
论文摘要
对于正实数的序列$（λ_n）$，我们考虑指数函数$ f_ {λ_n}（z）=λ_ne^z $和组成$ f_n = f_n = f_ {λ_n} \ Circ f_ {λ_{λ_{λ_{n-1}}} \ Circ ...对于这样一个非自主家庭，我们可以定义Fatou和Julia的设置，类似于通常的自主迭代案例。本文档的目的是研究朱莉娅集合如何取决于序列$（λ_n）$。在其他结果中，我们证明了为随机序列$ \ {λ_n\} $的朱莉娅设置，该设置是从$ \ frac {1} {e} $的附近统一选择的，是概率$ 1 $的整个平面。我们还证明了$ \ frac {1} {e} + \ frac {1} {n^p} $是$ p <\ frac {1} {2} $的整个飞机，并给出一个序列$ \ \ \ \ {λ_n\} $的示例，以$ 0 $ 0 $ 0的fate fate corterate fat fate contere，非空。
For a sequence $(λ_n)$ of positive real numbers we consider the exponential functions $f_{λ_n} (z) = λ_n e^z$ and the compositions $F_n = f_{λ_n} \circ f_{λ_{n-1}} \circ ... \circ f_{λ_1}$. For such a non-autonomous family we can define the Fatou and Julia sets analogously to the usual case of autonomous iteration. The aim of this document is to study how the Julia set depends on the sequence $(λ_n)$. Among other results, we prove the Julia set for a random sequence $\{λ_n \}$, chosen uniformly from a neighbourhood of $\frac{1}{e}$, is the whole plane with probability $1$. We also prove the Julia set for $\frac{1}{e} + \frac{1}{n^p}$ is the whole plane for $p < \frac{1}{2}$, and give an example of a sequence $\{λ_n \} $ for which the iterates of $0$ converge to infinity starting from any index, but the Fatou set is non-empty.
