PhD Studentship - Julia Sets of Hyperbolic Meromorphic Functions

Research theme: Analysis, geometry and dynamical systems

How to apply: uom.link/pgr-apply-2425

No. of positions: 1

This 3.5 year PhD project is fully funded for home students; the successful candidate will receive an annual tax free stipend set at the UKRI rate (£20,780 for 2025/26) and tuition fees will be paid. We expect the stipend to increase each year.

The goal of this project is to develop new results in the area of transcendental dynamics. The following describes a specific project, but this could be tailored to some degree to the prior experience of the successful applicant.

In the study of dynamical systems, hyperbolic systems within a given class show a particularly regular type of behaviour, and are typically the first class of systems one wishes to understand. We are interested in the setting of one-dimensional holomorphic dynamics (the iteration of functions of one complex variable). For polynomials and rational functions, hyperbolic functions are well-understood: Such a function is uniformly expanding on a neighbourhood of its Julia set, and the remaining part of the plane consists of the basins of finitely many attracting periodic cycles. Hyperbolicity can also be characterised by the property that all critical points belong to attracting basins.

Bergweiler, Fagella and Rempe ([BFR], Comm. Math. Helvetici 2015) studied bounded Fatou components of hyperbolic transcendental entire functions and showed that these are always bounded by simple closed curves. They also showed that, under certain conditions, the full Julia set is locally connected. Also, Rempe ([R], Acta Math. 2009) showed that the Julia set of a hyperbolic transcendental entire function can be described as the quotient of a certain simpler map in the same parameter space.

Hyperbolic transcendental meromorphic functions (which are allowed to have poles) have also been studied, for example by Rippon and Stallard (Proceedings of the American Mathematical Society, 1999). However, their dynamics remains significantly less explored than that of their entire counterparts. The goal of this project is to investigate Julia sets of hyperbolic transcendental meromorphic functions. In particular, it will consider the following questions.

1. Are simply-connected bounded Fatou components of hyperbolic meromorphic functions bounded by simple closed curves (in analogy with [BFR])?
2. Suppose that the Julia set of a hyperbolic meromorphic functions is connected, and that each component of any attracting basin contains only finitely many critical points, counting multiplicity. Is the Julia set locally connected (again in analogy with [BFR])?
3. Under which conditions can the Julia set of a hyperbolic meromorphic function be expressed as a quotient of a simpler function in the same parameter space, in analogy with [R]?

The project will begin by considering an explicit family of transcendental meromorphic functions, such as the family of tangent maps, and then generalising the insights gained through this study to more general classes of functions

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in mathematics or a related discipline. Good background knowledge in complex analysis is essential; prior knowledge in the area of dynamical systems is desirable.

To apply, please contact the main supervisor, Prof Lasse Rempe - lasse.rempe@manchester.ac.uk. Please include details of your current level of study, academic background and any relevant experience and include a paragraph about your motivation to study this PhD project.

Advert information