Classifying Wandering Domains

对漂移域进行分类

基本信息

批准号：
EP/R010560/1
负责人：
Gwyneth Stallard
金额：
$ 49.81万
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：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR010560%2F1
关键词：
Classifying Wandering Domains

项目摘要

The proposed research is in the field of complex dynamics which has experienced explosive growth in the last 30 years, with the advent of computer graphics demonstrating the highly intricate nature of the sets involved, and with the introduction of many deep techniques from complex analysis.This research project will seek to achieve the ambitious objective of completing the classification of the different types of dynamical behaviour that can occur within the components of the Fatou set. (The Fatou set of an analytic function is the set where the behaviour of the iterates of the function is stable.) A complete classification of Fatou components exists for rational functions and this is fundamental to work on rational dynamics. Until recently, such a classification for transcendental entire functions seemed out of reach.The background to the project is that a complete classification of periodic Fatou components for rational functions was given by the founders of complex dynamics, Fatou and Julia, nearly 100 years ago. For many years it was unknown whether there are other Fatou components which never map into a periodic component - such components are known as wandering domains. One of the most famous results in complex dynamics is Sullivan's `no wandering domains theorem' published in the Annals in 1982, which shows that, for rational functions, all Fatou components are eventually periodic. A major difference between rational dynamics and transcendental dynamics is that wandering domains can exist for transcendental functions. There is currently no general description of the dynamical behaviour inside wandering domains.The first example of a wandering domain was given by Baker in 1976. His example was multiply connected and he later showed that many of the basic geometric properties of this example hold for all multiply connected wandering domains. In a recent major paper, the investigators together with Walter Bergweiler gave a remarkably complete description of the dynamical behaviour in multiply connected wandering domains. This holds out the prospect, for the first time, that it might be possible to give a complete description of the dynamical behaviour in simply connected wandering domains. The purpose of this project is to bring this prospect to fruition, thus giving a complete classification of Fatou components.There are many types of simply connected wandering domains, and so the task of giving a complete description of the dynamical behaviour in such domains will be much more challenging than for multiply connected wandering domains. We will begin by analysing examples of wandering domains that can be thought of as escaping versions of the different types of periodic Fatou components, and using a range of techniques to construct new examples. By studying the limiting behaviour of the hyperbolic distance between pairs of points in such domains we hope to produce a classification of the different types of behaviour that are possible. This should enable us to identify sequences of absorbing domains inside the wandering domains, within which the dynamics behave in a specified way. We will then address the more challenging task of constructing new examples of wandering domains that do not escape and classifying such domains. Our classification should provide new insight into major open problems in complex dynamics which we will explore in the latter part of the project. In particular, it could lead to a resolution of the question as to whether commuting analytic functions have equal Julia sets. This would be a key step towards addressing the fundamental question as to which pairs of analytic functions commute.

拟议的研究是在过去30年中经历了爆炸性增长的复杂动态领域，随着计算机图形的出现，表明所涉及的集合的高度错综复杂的性质，并且从复杂分析中引入了许多深入的技术。这项研究项目将寻求实现雄心勃勃的目标，以完成不同类型的动态行为的分类，这些动态行为可能会在组合中发生疲劳的组合中发生。（分析函数的FATOU集合是该函数迭代术的行为稳定的集合。）FATOU组件的完整分类是有理功能的，这对于有理动力学的工作至关重要。直到最近，这种对整个功能的分类似乎都遥不可及。该项目的背景是，将近100年前的复杂动力学，Fatou和Julia的创始人对有理功能的定期FATOU组件进行了完整的分类。多年来，尚不清楚是否还有其他FATOU组件永远不会映射到周期性的组件中 - 此类组件被称为流浪域。复杂动力学中最著名的结果之一是沙利文（Sullivan）在1982年在《 Annals》上发表的“无流浪域定理”，这表明，对于理性功能，所有FATOU组件最终都是周期性的。理性动力学和先验动力学之间的主要区别在于，可以为先验功能而存在流浪域。当前尚无对流浪域内动态行为的一般描述。贝克在1976年给出了一个流浪域的第一个示例。他的示例是乘数乘以连接的，后来他表明，此示例的许多基本几何特性都符合所有倍数连接的徘徊域。在最近的一篇主要论文中，研究人员与沃尔特·伯格威勒（Walter Bergweiler）一起对多连接的流浪域中的动态行为进行了非常完整的描述。这首先使前景首次可以对简单连接的流浪域中的动态行为进行完整描述。该项目的目的是使这一前景实现，从而对FATOU组件进行完整的分类。有许多类型的简单连接的徘徊域，因此，对此类域中的动态行为进行完整描述的任务将比对于乘以徘徊的徘徊域而更具挑战性。我们将首先分析流浪域的示例，这些示例可以被认为是不同类型的周期性FATOU组件的逃避版本，并使用一系列技术来构建新示例。通过研究此类域中积分对之间双曲线距离的限制行为，我们希望对可能的不同类型的行为产生分类。这应该使我们能够确定流浪域内吸收域的序列，在该域中，动态以指定的方式行为。然后，我们将解决构建不逃脱和分类此类域的新示例的新示例的更具挑战性的任务。我们的分类应该为复杂动态的主要开放问题提供新的见解，我们将在项目的后半部分探索。特别是，这可能导致问题解决通勤分析功能是否具有相等的朱莉娅集合。这将是解决哪一对分析功能通勤的基本问题的关键步骤。