Classifying Wandering Domains
Lead Research Organisation:
The Open University
Department Name: Faculty of Sci, Tech, Eng & Maths (STEM)
Abstract
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.
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.
Planned Impact
The proposed research is in the area of pure mathematics and so, like most research in this area, it is likely that the main impact in the short term will be within the academic community. It is hard to judge what the longer term impacts might be but, in order to maximise the chances of impact to other areas, the results of the research will be publicised as widely as possible.
Within the academic community, the initial impact will be greatest amongst researchers working in complex dynamics. The proposed research will make a significant contribution to
our knowledge of the Fatou set (the set of stability) and complete the classification of its components. The Fatou set plays a fundamental
role in complex dynamics, so the results obtained
will be of great interest to those working in this area, particularly to those working in transcendental dynamics.
The investigators also have a record of proving general results in complex analysis in order to make progress on problems in complex dynamics. These complex analysis results can have much wider applications than to the problem in complex dynamics which provided the original motivation. For example, they have proved results which have then been used by others to make progress on problems involving differential equations. This project will undoubtedly lead to more results of this type with wider applications.
Work in complex dynamics leads to highly intricate computer pictures which attract great interest. A picture of the first spider's web discovered by the investigators appeared in an exhibition at the 2008 British Association Festival of Science in Liverpool and in an audio slideshow inspired by this event that can be seen on the BBC website. More recently, in November 2015 the OU complex dynamics group took part in a Mathematics Festival at the Science Museum in London, working with the groups at Liverpool and Imperial who together form the LMS Scheme 3 holomorphic dynamics group. The group will continue to be actively involved in such outreach activities, to make the results of their research accessible to as wide an audience as possible and to inspire others to carry out mathematical research.
More broadly, the PI has won national recognition for her work supporting women in mathematics in the UK and will continue to give talks based on her research and career to inspire early career women.
Finally, the UK is currently world-leading in the area of transcendental dynamics - this project will further establish the UK's reputation in this area and strengthen links with other research groups in Europe.
Within the academic community, the initial impact will be greatest amongst researchers working in complex dynamics. The proposed research will make a significant contribution to
our knowledge of the Fatou set (the set of stability) and complete the classification of its components. The Fatou set plays a fundamental
role in complex dynamics, so the results obtained
will be of great interest to those working in this area, particularly to those working in transcendental dynamics.
The investigators also have a record of proving general results in complex analysis in order to make progress on problems in complex dynamics. These complex analysis results can have much wider applications than to the problem in complex dynamics which provided the original motivation. For example, they have proved results which have then been used by others to make progress on problems involving differential equations. This project will undoubtedly lead to more results of this type with wider applications.
Work in complex dynamics leads to highly intricate computer pictures which attract great interest. A picture of the first spider's web discovered by the investigators appeared in an exhibition at the 2008 British Association Festival of Science in Liverpool and in an audio slideshow inspired by this event that can be seen on the BBC website. More recently, in November 2015 the OU complex dynamics group took part in a Mathematics Festival at the Science Museum in London, working with the groups at Liverpool and Imperial who together form the LMS Scheme 3 holomorphic dynamics group. The group will continue to be actively involved in such outreach activities, to make the results of their research accessible to as wide an audience as possible and to inspire others to carry out mathematical research.
More broadly, the PI has won national recognition for her work supporting women in mathematics in the UK and will continue to give talks based on her research and career to inspire early career women.
Finally, the UK is currently world-leading in the area of transcendental dynamics - this project will further establish the UK's reputation in this area and strengthen links with other research groups in Europe.
Publications
Benini A
(2021)
Classifying simply connected wandering domains
in Mathematische Annalen
Canela J
(2023)
On the basins of attraction of a one-dimensional family of root finding algorithms: from Newton to Traub
in Mathematische Zeitschrift
COMDÜHR P
(2022)
Dynamics of generalised exponential maps
in Mathematical Proceedings of the Cambridge Philosophical Society
Evdoridou V
(2018)
The topology of the set of non-escaping endpoints
Evdoridou V
(2020)
Fatou's Associates
in Arnold Mathematical Journal
Evdoridou V
(2021)
The Topology of the Set of Non-Escaping Endpoints
in International Mathematics Research Notices
Evdoridou V
(2020)
On the connectivity of the escaping set in the punctured plane
in Collectanea Mathematica
Evdoridou V
(2021)
On a Result of Hayman Concerning the Maximum Modulus Set
in Computational Methods and Function Theory
Evdoridou V
(2020)
Oscillating simply connected wandering domains
Evdoridou V
(2019)
Singularities of inner functions associated with hyperbolic maps
in Journal of Mathematical Analysis and Applications
| Description | We have shown that there are nine different classes of simply connected wandering domains and have given a new technique for constructing such wandering domains. We have constructed examples of both escaping and oscillating wandering domains in each of the nine classes. We have proved many results showing how the behaviour inside the wandering domain relates to the behaviour on the boundary - and in fact proved these results in a more general setting of non-autonomous dynamical systems, and more generally still to sequences of analytic maps. These later results are linked to results from ergodic theory. |
| Exploitation Route | This raises lots of questions about the limiting behaviour in wandering domains and on the boundary of wandering domains, in more general settings - some of these are already being studied by other authors. The work has in fact led to results in the more general setting of non-autonomous dynamical systems that could be of interest to a wider group than initially expected. There are various questions in complex analysis linked to wandering domains where our classification might be useful. |
| Sectors | Education |
| Description | Academic Collaboration |
| Organisation | University of Barcelona |
| Country | Spain |
| Sector | Academic/University |
| PI Contribution | We have contributed a knowledge of techniques from classical complex analysis, inner functions and complex dynamics |
| Collaborator Contribution | A knowledge of complex dynamics, inner functions, quasiconformal techniques |
| Impact | We have one major paper submitted for publication, with work in progress on another. |
| Start Year | 2018 |
| Description | Academic Collaboration |
| Organisation | University of Parma |
| Country | Italy |
| Sector | Academic/University |
| PI Contribution | We have contributed a knowledge of classical complex analysis, complex dynamics and inner functions |
| Collaborator Contribution | A knowledge of complex dynamics and non-autonomous dynamical systems |
| Impact | We have a major paper submitted for publication and work in progress on several further results. |
| Start Year | 2018 |