Baker's conjecture and Eremenko's conjecture: new directions

贝克猜想和埃雷门科猜想：新方向

基本信息

批准号：
EP/K031163/1
负责人：
Gwyneth Stallard
金额：
$ 28.54万
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：
2014
资助国家：
英国
起止时间：
2014 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK031163%2F1
关键词：
Baker conjecture Eremenko new directions

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 complicated. In order to make progress in the area of transcendental complex dynamics it is essential to gain a greater understanding of the structure of these fundamental sets.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. 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 discovered a surprising connection between these two conjectures and showed that, for a large class of functions, both Baker's conjecture and Eremenko's conjecture hold, with the escaping set having a novel structure described as an infinite spider's web. This connection was discovered by considering the so called `fast escaping set' of points that escape to infinity faster than the iterated maximum modulus. This set is now known to play a key role in transcendental dynamics and all previous work on Baker's conjecture has focused on points in this set. The investigators have recently shown, however, that, in order to solve Baker's conjecture, it is necessary to consider points that escape to infinity more slowly. One of the aims of this project is to consider the set of points that escape to infinity faster than the iterated minimum modulus. The proposal to consider this set is highly novel and has the potential to transform our understanding of the structure of the escaping set. When considering points that escape to infinity at slower rates, it is necessary to introduce completely new techniques in order to demonstrate that the escaping set has the structure of a spider's web. The investigators have recently shown that, in some situations, this can be achieved by using a variety of techniques from complex analysis to prove that the images of certain curves wind many times round the origin. The object of the proposed research is to build upon these new techniques and ideas to make substantial progress on both conjectures. Moreover, it may be possible to show that one of the conclusions of Baker's conjecture holds much more generally than was envisaged when the conjecture was made. The work will lead to new results of general interest in both complex dynamics and complex analysis and to new interactions between the two areas.

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。计算机图片证明了大多数朱莉娅集合都非常复杂。研究的另一个关键对象是逃避集合，该集合包括迭代下逃到无限的点。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，逃脱集的集合要复杂得多。为了在先验复杂动力学领域取得进展，必须对这些基本集的结构有更深入的了解。该领域的关键问题之一是，逃脱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关注。 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 discovered a surprising connection between these two conjectures and showed that, for a large class of functions, both Baker's conjecture and Eremenko's conjecture hold, with the escaping set having a novel structure described as an infinite spider's web.通过考虑所谓的“快速逃脱”点的点，发现该连接比迭代的最大模量更快地逃脱到无穷大。现在，众所周知，该集合在先验动力学中起着关键作用，而贝克猜想的所有先前工作都集中在该集合中的积分上。然而，研究人员最近表明，为了解决贝克的猜想，有必要考虑更慢地逃到无穷大的点。该项目的目的之一是考虑一组比迭代的最小模量更快地逃到无穷大的点。考虑这套集合的建议是高度新颖的，并且有可能改变我们对逃脱集合结构的理解。当考虑以较慢的速度逃到无穷大的点时，有必要引入全新的技术，以证明逃逸集具有蜘蛛网的结构。研究人员最近表明，在某些情况下，这可以通过使用复杂分析的各种技术来证明某些曲线的图像多次绕过来源。拟议的研究的目的是建立在这些新技术和思想的基础上，以在两个猜想上取得重大进展。此外，有可能证明贝克猜想的结论之一比做出猜想时所设想的要多得多。这项工作将导致对复杂动态和复杂分析的普遍兴趣的新结果，以及两个领域之间的新相互作用。