## Research interests

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

## Teaching interests

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.

## Research Activity

### Research groups

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

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

### Externally funded projects

#### Classifying Wandering Domains

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

Co-investigator | 01/Jan/2018 | 31/Dec/2020 | 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. |

## Publications

### Journal articles

**The iterated minimum modulus and conjectures of Baker and Eremenko (2020)**

Osborne, John; Rippon, Philip and Stallard, Gwyneth

Journal d’Analyse Mathematique ((In Press))

**Eremenko points and the structure of the escaping set (2019-08)**

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)

**The MacLane class and the Eremenko-Lyubich class (2017)**

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

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

**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)

**Exotic Baker and wandering domains for Ahlfors island maps (2012-06)**

Rempe, Lasse and Rippon, Philip

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

**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)

**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)

**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)

**Non-tangential limits of slowly growing analytic functions (2008)**

Barth, Karl F. and Rippon, Philip J.

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

**Baker domains of meromorphic functions (2006-08)**

Rippon, P.J.

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

**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)

**Extensions of a theorem of Valiron (2006)**

Barth, K. F. and Rippon, Philip

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

**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)

**Asymptotic values of strongly normal functions (2005-04)**

Barth, Karl F. and Rippon, Philip J.

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

**Obituary: Irvine Noel Baker 1932-2001 (2005-04)**

Rippon, Philip

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

**Even and odd periods in continued fractions of square roots (2004-05)**

Rippon, Philip and Taylor, Harold

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

**On a problem of MacLane concerning arc tracts (2004)**

Barth, Karl F. and Rippon, Philip J.

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

**Infinitely many asymptotic values of locally univalent functions (2003)**

Barth, Karl F. and Rippon, Philip J.

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

**Asymptotic tracts of locally univalent functions (2002-03)**

Barth, Karl F. and Rippon, Philip J.

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

### Book chapters

**Exceptional values and the MacLane class (2006)**

Barth, Karl and Rippon, Philip

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

*Bergman spaces and related topics in complex analysis: proceedings of a conference in honor of Boris Korenblum’s 80th birthday*. Contemporary mathematics (404) (pp. 41-52)

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