关于朱利亚集和Multibrot集的纤维与重整化
On Fibers and Renormalization of Julia Sets and Multibrot Sets
摘要：
我们继续在IMS预印本1998/12和1998/13 a中开始的关于纤维的Mandelbrot集和Multibrot集以及朱利亚集的描述。这些集合的局部连通性问题在纤维方面进行了讨论，并成为纤维的平凡性问题。在本文中，重点是在重整化和其他手术程序下的纤维的行为。我们证明了Mandelbrot集和Multibrot集的纤维的平凡性在调谐映射和其他（部分）同胚下保持不变。同样，我们证明了单临界多项式的平凡纤维的朱利亚集下的重整化和其他手术程序，如Branner-Douady同胚。我们的结论与各种应用程序的二次多项式和它的参数空间：我们确定嵌入路径内的Mandelbrot集，我们表明，彼得森定理二次朱利亚集与西格尔磁盘的有界型推广从周期1到任意周期，使他们都有平凡的纤维，从而局部连接。
英文摘要：
We continue the description of Mandelbrot and Multibrot sets and of Julia sets in terms of fibers which was begun in IMS preprints 1998/12 and 1998/13a. The question of local connectivity of these sets is discussed in terms of fibers and becomes the question of triviality of fibers. In this paper, the focus is on the behavior of fibers under renormalization and other surgery procedures. We show that triviality of fibers of Mandelbrot and Multibrot sets is preserved under tuning maps and other (partial) homeomorphisms. Similarly, we show for unicritical polynomials that triviality of fibers of Julia sets is preserved under renormalization and other surgery procedures, such as the Branner-Douady homeomorphisms. We conclude with various applications about quadratic polynomials and its parameter space: we identify embedded paths within the Mandelbrot set, and we show that Petersen's theorem about quadratic Julia sets with Siegel disks of bounded type generalizes from period one to arbitrary periods so that they all have trivial fibers and are thus locally connected.
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