Fast escaping points of entire functions
Fast escaping points of entire functions
复制标题
DOI：
10.1112/plms/pds001
复制
发表时间：
2010-09
影响因子：
1.8
通讯作者：
P. Rippon;G. Stallard
中科院分区：
文献类型：
--
作者：
P. Rippon;G. Stallard
设f是超越整函数，A(F)表示在迭代过程中“尽可能快”逃逸到无穷远的点集。通过将A(F)写成闭集的可数并，称为A(F)的水平，我们对这一集合的结构有了新的理解。例如，我们证明了如果U是A(F)中的Fatou分支，则∂U⊂A(F)，这导致了关于A(F)的重要的新结果和对已有结果的相当大的改进。特别地，我们研究了A(F)及其每一级具有“无限蜘蛛网”结构的函数。我们证明了存在许多这样的函数，并且它们具有许多强的动力学性质。这种新结构在Baker关于Fatou集的分支的猜想和Eremenko的关于转义集的分支的猜想之间提供了意想不到的联系。
Let f be a transcendental entire function and let A(f) denote the set of points that escape to infinity ‘as fast as possible’ under iteration. By writing A(f) as a countable union of closed sets, called ‘levels’ of A(f), we obtain a new understanding of the structure of this set. For example, we show that if U is a Fatou component in A(f), then ∂U⊂A(f) and this leads to significant new results and considerable improvements to existing results about A(f). In particular, we study functions for which A(f), and each of its levels, has the structure of an ‘infinite spider's web’. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.