Dynamical Systems
Wandering Domains in Transcendental Dynamics
University of Livepool, Thursday 5th October 2023
Important notice: Due to the UCU strike action planned for the last week of September, this meeting was rescheduled from 28th September to 5th October.
The Department of Mathematical Sciences of the University of Liverpool will hold a meeting on the recent developments on wandering domains. This meeting is organised by Dr David Martí-Pete and we acknowledge the funding from the London Mathematical Society (Scheme 9 grant "Celebating New Appointments").
This will be a hybrid meeting, and it will be possible to attend the meeting online (via Zoom). If you would like to attend the meeting (either in person or online), please contact David. There is a limited amount of funding to support the travel expenses of PhD students and early-career researchers.
Schedule
The preliminary schedule of the meeting is as follows:
11:00–12:00 Alexandre Eremenko (Purdue University)
On a conjecture of Bloch
12:00–13:00 Lunch
13:00–14:00 Lasse Rempe (University of Liverpool)
Escaping sets, wandering domains and Eremenko's Conjecture
14:00–14:30 David Martí-Pete (University of Liverpool)
Wandering dynamics of transcendental functions
14:30–15:30 Coffee break
15:30–16:00 Vasiliki Evdoridou (The Open University)
Unbounded fast escaping wandering domains
16:00–17:00 Gwyneth Stallard (The Open Unviersity)
A new type of simply connected wandering domain?
Location
Most of the talks will take place at Room MATH-210, on the second floor of the Mathematical Sciences Building, except the talks by Evdoridou and Stallard which will take place at the Room MATH-G16 (also known as the MAGIC room), which is located in the Theoretical Physics wing, at the end of the ground floor. The coffee break will take place at the Penthouse, on the sixth floor (access via the back staircase on the fifth floor).
Abstracts
Alexandre Eremenko (Purdue University)
On a conjecture of Bloch
Abstract: We prove the conjecture of Andre Bloch that for an entire function of finite order,
having finitely many simple islands over two Jordan domains with disjoint closures
implies that the order is n/2 for a some positive integer n.
Bloch derived this conjecture from a philosophical principle which he called
"Principle of Topological continuity". We discuss this principle and also construct some
counterexamples to it.
Based on the joint work with Walter Bergweiler.
Lasse Rempe (University of Liverpool)
Escaping sets, wandering domains and Eremenko's Conjecture
Abstract: Let f be a transcendental entire function - that is, a non-polynomial holomorphic self-map of the complex plane. Transcendental dynamics considers what happens to a point z under repeated application (iteration) of the function f.
Two central concepts of transcendental dynamics are the escaping set (the set of points that will tend to infinity under iteration) and wandering domains (open regions whose images under iteration are all disjoint from each other). These have been the focus of intensive research over the last several decades, and David Martí-Pete has made fundamental contributions to both. This includes (with Rempe and Waterman) a solution to Eremenko's Conjecture, a central question on the topology of the escaping set that was posed in 1989.
I will explain these concepts and results in a manner that should be accessible to a general mathematical audience. All members of the department are welcome to attend!
David Martí-Pete (University of Liverpool)
Wandering dynamics of transcendental functions
Abstract: In a joint work with Rempe and Waterman, we used approximation theory to prove that every bounded simply connected domain whose boundary equals the boundary of its unbounded complementary component is a wandering domain of a transcendental entire function. The result that I will present, from a collaboration with Evdoridou and Rempe, is a strengthening of this result where we can additionally prescribe the internal dynamics of the wanderin domain (at the cost of not prescribing exactly the wandering domain, but it being homeomorphic and arbitrarily close to the starting domain). More precisely, we prove that a transcendental entire function is conjugate on the closure of a wandering domain to the model map used in the approximation. This allows us to obtain examples of simply connected wandering domains with any type of internal dynamics whose boundary is a differentiable curve, answering a question that arose in recent work of Benini, Evdoridou, Fagella, Rippon and Stallard.
Vasiliki Evdoridou (The Open University)
Unbounded fast escaping wandering domains
Abstract: We give the first example of a transcendental entire function with a sequence of unbounded fast escaping wandering domains. Our construction uses Approximation Theory and in particular, a theorem by HÖrmander, which is used for the first time in Holomorphic Dynamics. This is joint work with A. Glücksam and L. Pardo-Simón.
Gwyneth Stallard (The Open Unviersity)
A new type of simply connected wandering domain?
Abstract: Boc Thaler showed that many bounded domains exist as wandering domains for transcendental entire functions. His results led him to ask whether the closure of a bounded simply connected wandering domain must have a connected complement. Marti-Pete, Rempe and Waterman showed that this is not the case, by constructing a Lakes of Wada type example. They then refined the original question to ask whether a bounded simply connected wandering domain can have a certain topological property. Motivated by this question, we investigate the properties that such a domain can / cannot have. This is joint work with David Marti-Pete, Phil Rippon and Dave Sixsmith.