Baker's conjecture and Eremenko's conjecture: a unified approach.

贝克猜想和埃列缅科猜想：统一的方法。

基本信息

批准号：
EP/H006591/1
负责人：
Gwyneth Stallard
金额：
$ 38.17万
依托单位：
The Open University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH006591%2F1
关键词：
Baker conjecture Eremenko unified approach.

项目摘要

The proposed research is in the area of complex dynamics which has experienced explosive growth in the last 25 years following the advent of computer graphics. For each meromorphic function, the complex plane is split into two fundamentally different parts - the Fatou set, where the behaviour of the iterates of the function is stable under local variation, and the Julia set, where it is chaotic. Computer pictures demonstrate that most Julia sets are highly intricate.Another key object of study is the escaping set which consists of the points that escape to infinity under iteration. This set plays a major role in complex dynamics since the Julia set is equal to the boundary of the escaping set. For polynomials, the dynamics on the escaping set are relatively simple, but for transcendental entire functions the escaping set is much more complex. In order to make progress in the area of transcendental complex dynamics it is essential to gain a greater understanding of the structure of the escaping set.One of the key questions in this area is whether all the components of the escaping set are unbounded - this is now known as Eremenko's conjecture and has attracted a great deal of interest. Much work on this question has centred on identifying functions for which the escaping set has a structure known as a Cantor bouquet of curves. The investigators have introduced new techniques to the area and shown that there are many functions for which the escaping set has a very different structure described as an infinite spider's web. They have further shown that the existence of certain types of spiders' webs implies several strong properties and so it is highly desirable to identify as many functions as possible for which the escaping set has this structure.Another question in transcendental dynamics that has attracted much interest is whether functions of small growth have no unbounded components of the Fatou set - this is now known as Baker's conjecture. The investigators have discovered a surprising connection between these two conjectures and shown that the techniques used to make progress on Baker's conjecture are precisely what is needed to show that the escaping set is a spider's web of the type mentioned above. Further, if the escaping set has this form then both Eremenko's conjecture and Baker's conjecture hold.The object of the proposed research is to to build upon these new techniques and ideas to make substantial progress on both conjectures. The work will lead to an increased understanding of the structure of the escaping set for large classes of functions, and this will enable progress to be made on many other questions in transcendental dynamics.

拟议的研究是在计算机图形出现后的过去25年中经历了爆炸性增长的复杂动态领域。对于每个混子函数，复杂的平面分为两个根本不同的部分 - FATOU集合，其中该函数的迭代行为在局部变化下是稳定的，而朱莉娅集合在混乱中。计算机图片表明，大多数朱莉娅集合都高度复杂。研究的另一个关键对象是逃脱的集合，该集合由迭代下逃到无穷大的点组成。由于朱莉娅集合等于逃脱集的边界，因此该集合在复杂动力学中起着重要作用。对于多项式，逃逸集上的动力学相对简单，但是对于先验的整个功能，逃脱集的设置要复杂得多。为了在先验复杂动力学领域取得进展，必须对逃脱集的结构有更深入的了解。该领域的关键问题之一是，逃逸集的所有组成部分是否无限 - 现在被称为Eremenko的猜想，并引起了人们的极大兴趣。在这个问题上进行了许多工作，集中在识别逃逸集的功能上，其结构称为Cantor Bouquet of Curves。研究人员已经向该地区引入了新技术，并表明逃逸集的功能有很多不同的结构，其结构被描述为无限的蜘蛛网。他们进一步表明，某些类型的蜘蛛网的存在暗示了几种强大的属性，因此，非常需要确定逃逸集具有这种结构具有这种结构的尽可能多的功能。先验动态中引起了很多兴趣的另一个问题是，小型成长的功能是否没有fatou集成的fatou集成 - 现在已经知道了fatou set的结合组件 - 现在已经毫无疑问。研究人员发现了这两个猜想之间的令人惊讶的联系，并表明用于在贝克的猜想上取得进展的技术恰恰是表明逃逸集是上述类型的蜘蛛网所需的。此外，如果逃脱的集合具有这种形式，那么Eremenko的猜想和贝克的猜想都存在。拟议的研究的目的是建立这些新技术和想法，以在这两个猜想上取得重大进展。这项工作将导致人们对大量功能的逃避设置的结构有了更多的了解，这将使在先验动力学中的许多其他问题上取得进展。