论文标题
随机二次多项式的朱莉娅集的总脱节性
Total disconnectedness of Julia sets of random quadratic polynomials
论文作者
论文摘要
对于一系列复杂参数$ \ {c_n \} $，我们考虑函数的组成$ f_ {c_n}（z）= z^2 + c_n $，这是经典Quadratic Dynamical System的非自主版本。朱莉娅（Julia）和法图（Fatou）集合的定义自然而然地概括为这种设置。我们回答了Brück，Büger和Reitz提出的一个问题，如果将$ C_N $随机从大磁盘中随机选择，那么朱莉娅设置的朱莉娅是否总是完全断开连接。我们的证明很容易被概括，以回答有关随机朱莉娅集合的典型连通性的许多其他相关问题。实际上，我们证明了一个比磁盘更大的套装家族的说法，特别是如果一个人从曼德布罗特（Mandelbrot）的主要有心脏中随机选择$ c_n $，那么朱莉娅（Julia）套装几乎仍然总是完全断开连接的。
For a sequence of complex parameters $\{c_n\}$ we consider the compositions of functions $f_{c_n} (z) = z^2 + c_n$, which is the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disk. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just disks, in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.
