My research is mainly in two areas:

1. complex dynamics, in particular the escaping set of a transcendental meromorphic function, and its relationship to the Fatou and Julia sets of the function,

2. function theory, in particular the boundary behaviour of meromorphic functions and subharmonic functions.

Together with Professor Gwyneth Stallard, I lead the complex dynamics group at the OU, including a Research Associate, a Visiting Research Associate and two PhD students, and our activities often include other complex analysts in the Department.

Gwyneth Stallard 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 project has been funded by two EPSRC grants which together cover five years, at 0.5FTE each:

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 Research Associate, Dave Sixsmith, is funded by the following grant from the EPSRC:

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

My main teaching contributions have been to the following Open University courses: MST121 Using mathematics, MS221 Exploring mathematics, M208 Pure mathematics, M337 Complex analysis, M832 Approximation theory.

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

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

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

Co-investigator | 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. |

Nicks, Daniel; Rippon, Philip and Stallard, Gwyneth

International Mathematics Research Notices ((In Press))

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

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

Rippon, Philip and Stallard, Gwyneth

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

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

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

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

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

Osborne, John; Rippon, Philip and Stallard, Gwyneth

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

Barth, Karl F.; Rippon, Philip J. and Sixsmith, David J.

Annales Academiae Scientiarum Fennicae Mathematica, 42 (pp. 859-873)

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

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

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

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

Rippon, Philip Jonathan and Stallard, Gwyneth Mary

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

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

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

Bergweiler, Walter; Rippon, Philip and Stallard, Gwyneth

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

Rippon, Philip and Stallard, Gwyneth

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

Rippon, Philip and Stallard, Gwyneth

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

Rempe, Lasse and Rippon, Philip

Journal d'Analyse Mathematique, 117(1) (pp. 297-319)

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

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

Rippon, Philip and Stallard, Gwyneth

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

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

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

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

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

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

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

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

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

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

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

Barth, Karl F. and Rippon, Philip J.

Computational Methods and Function Theory, 8(1) (pp. 85-99)

Rippon, P.J.

Ergodic Theory and Dynamical Systems, 26(4) (pp. 1225-1233)

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

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

Barth, K. F. and Rippon, Philip

Bulletin of the London Mathematical Society, 38(5) (pp. 815-824)

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

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

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

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

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

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

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

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

Barth, Karl F. and Rippon, Philip J.

Arkiv för Matematik, 43(1) (pp. 69-84)

Rippon, Philip

Bulletin of the London Mathematical Society, 37(2) (pp. 301-315)

Rippon, Philip and Taylor, Harold

Fibonacci Quarterly, 42(2) (pp. 170-180)

Barth, Karl F. and Rippon, Philip J.

Computational Methods and Function Theory, 4 (pp. 405-418)

Barth, Karl F. and Rippon, Philip J.

Annales Academiae Scientiarum Fennicae Mathematica, 28 (pp. 303-314)

Barth, Karl F. and Rippon, Philip J.

Bulletin of the London Mathematical Society, 34 (pp. 200-204)

Barth, Karl and Rippon, Philip

In: Borichev, Alexander; Hedenmalm, Håkan and Zhu, Kehe eds.

ISBN : 821837125 | Publisher : American Mathematical Society and Bar-Ilan University | Published : Providence, RI, USA and Ramat-Gan, Israel