My research is in the area of complex dynamics and concerns the iteration of transcendental meromorphic functions - more details are given below.

I have a long standing interest in the issues surrounding women's careers in mathematics and chaired the London Mathematical Society's Women in Mathematics Committee from 2006 to 2015. This work was recognized by the award of an OBE in 2015. Details of the work of the committee and the activities that it organises to support women's careers can be found here http://www.lms.ac.uk/women-mathematics. I also represented the LMS on the Athena Forum http://www.athenaforum.org.uk/. I continue as a member of the LMS Good Practice Scheme Steering Committee and a member of LMS Council. I chair the School's Athena SWAN self assessment team - we were awarded a Bronze award in 2014.

My research is in the area of complex dynamics and concerns the iteration of transcendental meromorphic functions. I am particularly interested in the possible dimensions of the Julia set and in the structure of the escaping set.

Together with Professor Phil Rippon, I lead the complex dynamics group at the OU. The group currently comprises a Visiting Research Associate and two PhD students and our activities often include other complex analysts in the department.

Phil Rippon and myself are currently working on a project to investigate a surprising link that we identified between two open conjectures in complex dynamics: Baker's conjecture and Eremenko's conjecture. The work has been funded by the EPSRC by two grants which together funded our work for five years (0.5FTE each). More details on the project are given here:

- EP/H006591/1: Baker's conjecture and Eremenko's conjecture: a unified approach
- EP/K031163/1: Baker's conjecture and Eremenko's conjecture: new directions

Our former Research Associate, Dave Sixsmith, was funded by the following grant from the EPSRC:

- EP/J022160/1: Dimensions in complex dynamics: spiders' webs and speed of escape

Most of my teaching has been in the area of analysis. I am currently chair of the exam board for __M208 Pure Mathematics__ and am responsible for the MSc course __M835 Fractal Geometry__.

Name | Type | Parent Unit |
---|---|---|

Pure Maths: Analysis & Geometry Research Group | Group | Faculty of Mathematics, Computing and Technology |

Role | Start date | End date | Funding source |
---|---|---|---|

Lead | 01 Apr 2018 | 31 Mar 2021 | EPSRC EPSRC Engineering and Physical Sciences Research Council |

The main aim of the proposed research is to complete the classification of the types of dynamical behaviour that can occur within the components of the Fatou set (the set of stability) arising in the iteration of analytic functions. There is a large historical body of work classifying the dynamical behaviour within periodic Fatou components and a recent paper gives a complete description of the behaviour in multiply connected wandering domains. This project will address the outstanding case of simply connected wandering domains. There are many different types of dynamical behaviour within such domains and they have not previously been studied systematically. The development of a unifying theory will be fundamental to future work on iteration. The theory will be illustrated and informed by the construction of new examples of wandering domains. |

Eremenko's Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus (2020)

Nicks, Daniel; Rippon, Philip and Stallard, Gwyneth

International Mathematics Research Notices ((Early access))

The iterated minimum modulus and conjectures of Baker and Eremenko (2019-10)

Osborne, John W.; Rippon, Philip J. and Stallard, Gwyneth M.

Journal d’Analyse Mathematique, 139(2) (pp. 521-558)

Eremenko points and the structure of the escaping set (2019-09-01)

Rippon, Philip and Stallard, Gwyneth

Transactions of the American Mathematical Society, 372(5) (pp. 3083-3111)

Baker's conjecture for functions with real zeros (2018-07-09)

Nicks, Daniel A.; Rippon, Philip J. and Stallard, Gwyneth M.

Proceedings of the London Mathematical Society, 117(1) (pp. 100-124)

Boundaries of univalent Baker domains (2018-02-28)

Rippon, P. J. and Stallard, G. M.

Journal d'Analyse Mathematique, 134(2) (pp. 801-810)

Connectedness properties of the set where the iterates of an entire function are unbounded (2017-06-23)

Osborne, John; Rippon, Philip and Stallard, Gwyneth

Ergodic Theory and Dynamical Systems, 37(4) (pp. 1291-1307)

Permutable entire functions and multiply connected wandering domains (2016-01-10)

Benini, Anna Miriam; Rippon, Philip J. and Stallard, Gwyneth M.

Advances in Mathematics, 287 (pp. 451-462)

Annular itineraries for entire functions (2015)

Rippon, P. J. and Stallard, G. M.

Transactions of the American Mathematical Society, 367(1) (pp. 377-399)

Regularity and fast escaping points of entire functions (2014-10)

Rippon, Philip Jonathan and Stallard, Gwyneth Mary

International Mathematics Research Notices, 2014(19) (pp. 5203-5229)

A sharp growth condition for a fast escaping spider’s web (2013-09-10)

Rippon, P. J. and Stallard, G. M.

Advances in Mathematics, 244 (pp. 337-353)

Multiply connected wandering domains of entire functions (2013)

Bergweiler, Walter; Rippon, Philip and Stallard, Gwyneth

Proceedings of the London Mathematical Society, 107(6) (pp. 1261-1301)

Fast escaping points of entire functions (2012-10)

Rippon, Philip and Stallard, Gwyneth

Proceedings of the London Mathematical Society, 105(4) (pp. 787-820)

Baker's conjecture and Eremenko's conjecture for functions with negative zeros (2012-08)

Rippon, Philip and Stallard, Gwyneth

Journal d'Analyse Mathematique, 120(1) (pp. 291-309)

Boundaries of escaping Fatou components (2011-03-02)

Rippon, P. J. and Stallard, G. M.

Proceedings of the American Mathematical Society, 139(8) (pp. 2807-2820)

Slow escaping points of meromorphic functions (2011)

Rippon, Philip and Stallard, Gwyneth

Transactions of the American Mathematical Society, 363(8) (pp. 4171-4201)

Hausdorff dimensions of escaping sets of transcendental entire functions (2010-05)

Rempe, Lasse and Stallard, Gwyneth M.

Proceedings of the American Mathematical Society, 138(5) (pp. 1657-1665)

Are Devaney hairs fast escaping? (2010)

Rempe, Lasse; Rippon, Philip J. and Stallard, Gwyneth M.

Journal of Difference Equations and Applications, 16(5&6) (pp. 739-762)

The growth rate of an entire function and the Hausdorff dimension of its Julia set (2009-12)

Bergweiler, Walter; Karpińska, Boguslawa and Stallard, Gwyneth M.

Journal of the London Mathematical Society, 80(3) (pp. 680-698)

Functions of small growth with no unbounded Fatou components (2009-05)

Rippon, P. J. and Stallard, G. M.

Journal d’Analyse Mathematique, 108(1) (pp. 61-86)

Escaping points of entire functions of small growth (2009)

Rippon, P. J. and Stallard, G. M.

Mathematische Zeitschrift, 261(3) (pp. 557-570)

Dynamics of meromorphic functions with direct or logarithmic singularities (2008)

Bergweiler, W.; Rippon, Philip J. and Stallard, Gwyneth M.

Proceedings of the London Mathematical Society, 97(2) (pp. 368-400)

On multiply connected wandering domains of meromorphic functions (2008)

Rippon, P. J. and Stallard, G. M.

Journal of the London Mathematical Society, 77(2) (pp. 405-423)

Dimensions of Julia sets of meromorphic functions with finitely many poles (2006-04)

Rippon, P.J. and Stallard, G.M.

Ergodic Theory and Dynamical Systems, 26(2) (pp. 525-538)

Singularities of meromorphic functions with Baker domains (2006)

Rippon, P. J. and Stallard, G. M.

Mathematical Proceedings of the Cambridge Philosophical Society, 141(2) (pp. 371-382)

Escaping points of meromorphic functions with a finite number of poles (2005-07)

Rippon, P. J. and Stallard, G. M.

Journal d’Analyse Mathematique, 96 (pp. 225-245)

Dimensions of Julia sets of meromorphic functions (2005-06)

Rippon, P.J. and Stallard, G.M.

Journal of the London Mathematical Society, 71(3) (pp. 669-683)

On questions of Fatou and Eremenko (2005-04-01)

Rippon, P.J. and Stallard, G.M.

Proceedings of the American Mathematical Society, 133(4) (pp. 1119-1126)

Dimensions of Julia sets of hyperbolic entire functions (2004-03)

Stallard, Gwyneth M.

Bulletin of the London Mathematical Society, 36(2) (pp. 263-270)

Dimensions of Julia sets of hyperbolic meromorphic functions (2001-11-28)

Stallard, Gwyneth M.

Bulletin of the London Mathematical Society, 33(6) (pp. 689-694)