Dimensions in complex dynamics: spiders' webs and speed of escape

复杂动力学的维度：蜘蛛网和逃逸速度

基本信息

批准号：
EP/J022160/1
负责人：
Gwyneth Stallard
金额：
$ 22.78万
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：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ022160%2F1
关键词：
Dimensions complex dynamics spiders webs

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 and the Julia set.Much work in this area has focused on obtaining an understanding of the sizes of these sets and significant subsets as measured by their Hausdorff dimensions. This has led to fundamental insights into the nature of these sets with some results being completely unexpected. Most work to date has, however, focused on functions in the so called class B for which a range of powerful techniques are avaliable.Recently, however, the first estimates for dimensions of these sets for functions outside of the class B have been obtained. Moreover, the investigators have introduced new techniques to the area and shown that there are many functions outside the class B for which the escaping set has a novel 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 obtain a greater understanding of this structure.It is therefore timely to begin a programme of research investigating the size of the escaping and Julia sets for functions outside the class B. This project will focus on those functions for which the escaping set has the structure of a spider's web and also on two significant subsets of the escaping set, namely the fast escaping set which has been shown to play a key role and the slow escaping set which has only recently been introduced into the subject. The proposed research aims to build upon the new techniques recently introduced to the area in order to establish a framework which will form the foundation for future research in this area.

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，必须对逃避集合和朱莉娅集合的结构有更深入的了解。该领域的工作重点是了解这些集合的尺寸和通过其Hausdorff尺寸衡量的重要子集的理解。这导致了对这些集合本质的基本见解，一些结果是完全出乎意料的。但是，迄今为止，大多数工作都集中在所谓的B类中的功能上，该级别的一系列强大技术是可以避免的。但是，最终，已经获得了这些集合对B类函数的最初估计值。此外，研究人员已经向该地区引入了新技术，并表明B级外部有许多功能，该功能逃脱的集合具有一种新颖的结构，形成了一种描述为无限蜘蛛网的新结构。 They have further shown that the existence of certain types of spiders' webs implies several strong properties and so it is highly desirable to obtain a greater understanding of this structure.It is therefore timely to begin a programme of research investigating the size of the escaping and Julia sets for functions outside the class B. This project will focus on those functions for which the escaping set has the structure of a spider's web and also on two significant subsets of the escaping set, namely the fast逃脱的套装已显示出扮演关键角色和缓慢的逃逸套件，直到最近才引入该主题。拟议的研究旨在基于最近引入该地区的新技术，以建立一个框架，该框架将为该领域的未来研究奠定基础。