##### Organizing Committee

- Matthew Baker
*Georgia Institute of Technology* - Rob Benedetto
*Amherst College* - Charles Favre
*Ecole Polytechnique* - Kevin Pilgrim
*Indiana University* - Juan Rivera-Letelier
*Pontificia Universidad Catolica de Chile*

##### Abstract

This workshop will bring together researchers working in classical complex dynamics and in the newer area of p-adic (nonarchimedean) dynamics. It will promote interactions between the two groups by highlighting the similarities and differences between complex and p-adic dynamics.

In particular, it will address Berkovich space, whose introduction has greatly enhanced the exchange of ideas between complex and p-adic dynamics.

#### Confirmed Speakers & Participants

- Jackie Anderson
- Matthieu Arfeux
- Cecile Armana
- Matthew Baker
- Tatiana Bandman
- Fabrizio Barroero
- Eric Bedford
- Rob Benedetto
- Anupam Bhatnagar
- Araceli Bonifant
- Andrew Bridy
- Xavier Buff
- Alexandru Buium
- Zoe Chatzidakis
- Arnaud Chéritat
- Mark Comerford
- Carlos D'Andrea
- Laura DeMarco
- Kevin Doerksen
- Dzmitry Dudko
- Adam Epstein
- Xander Faber
- Youssef Fares
- Charles Favre
- Tanya Firsova
- Joanna Furno
- Derek Garton
- Thomas Gauthier
- Dragos Ghioca
- William Gignac
- Walter Gubler
- Liang-Chung Hsia
- Benjamin Hutz
- Zair Ibragimov
- Patrick Ingram
- Hiroyuki Inou
- Rafe Jones
- Mattias Jonsson
- Jeremy Kahn
- Shu Kawaguchi
- Tomoki Kawahira
- Kiran Kedlaya
- Sarah Koch
- Par Kurlberg
- ChongGyu Lee
- Alon Levy
- Huaibin Li
- Jan-Li Lin
- Karl-Olof Lindahl
- Sijun Liu
- Jonathan Lubin
- Mikhail Lyubich
- Mahdi Majidi-Zolbanin
- Michelle Manes
- Nikita Miasnikov
- John Milnor
- Khoa Nguyen
- Yûsuke Okuyama
- Frank Palladino
- Donghoon Park
- Carsten Petersen
- Kevin Pilgrim
- Jorge Pineiro
- Lukas Pottmeyer
- Remus Radu
- Juan Rivera-Letelier
- Bastien Rossetti
- Robert Rumely
- Adriana Salerno
- Zachary Scherr
- Dierk Schleicher
- Nikita Selinger
- Thomas Sharland
- Mitsuhiro Shishikura
- Joseph Silverman
- Lucien Szpiro
- Raluca Tanase
- Bianca Thompson
- Eugenio Trucco
- Thomas Tucker
- Eva Uhre
- Bianca Viray
- Paul Vojta
- Xiaoguang Wang
- Chi-Hao Wang
- Kazuhiko Yamaki
- Xinyi Yuan
- Ilies Zidane
- Michael Zieve

#### Workshop Schedule

##### Monday, February 13, 2012

Time | Event | Location | Materials |
---|---|---|---|

8:30 - 8:55 | Registration | ||

8:55 - 9:00 | Welcome - Jeff Brock, Deputy Director, ICERM | 11th Floor Lecture Hall | |

9:00 - 9:45 | Geometry of Feigenbaum Julia sets - Mikhail Lyubich, SUNY | 11th Floor Lecture Hall | |

10:00 - 10:30 | Coffee/Tea Break | 11th Floor Collaborative Space | |

10:30 - 11:15 | Geometric Limits and Renormalization - Jeremy Kahn, Brown University | 11th Floor Lecture Hall | |

11:30 - 12:15 | Special curves and post‐critically finite maps in the moduli space - Laura DeMarco, University of Illinois | 11th Floor Lecture Hall | |

12:30 - 2:30 | Break for Lunch Free Time | ||

2:30 - 3:15 | Quantitative Equidistribution in Non-Archimedean and Complex Dynamics - Yusuke Okuyama, Kyoto Institute of Technology | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 4:45 | Strong bifurcation loci of maximal Hausdorff dimension - Thomas Gauthier, Université de Toulouse III (Paul Sabatier) | 11th Floor Lecture Hall | |

5:00 - 6:30 | Welcome Reception |

##### Tuesday, February 14, 2012

Time | Event | Location | Materials |
---|---|---|---|

9:00 - 9:45 | The Berkovich Ramification Locus -- Structure and Applications - Xander Faber, University of Hawaii | 11th Floor Lecture Hall | |

10:00 - 10:30 | Coffee/Tea Break | 11th Floor Collaborative Space | |

10:30 - 11:15 | Wandering Domains in Spherically Complete Fields - Eugenio Trucco, Centro de Estudios Científicos de Santiago | 11th Floor Lecture Hall | |

11:30 - 12:15 | On the non-archimedean Calabi--Yau Theorem - Xinyi Yuan, Princeton University | 11th Floor Lecture Hall | |

12:30 - 2:30 | Break for Lunch Free Time | ||

2:30 - 3:15 | Canonical measures on subvarieties of abelian varieties - Walter Gubler, Universität Regensburg | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 4:45 | Singular metrics and the Calabi-Yau theorem in non-Archimedan geometry - Mattias Jonsson, University of Michigan | 11th Floor Lecture Hall |

##### Wednesday, February 15, 2012

##### Thursday, February 16, 2012

##### Friday, February 17, 2012

#### Tutorial Week Schedule

##### Thursday, February 9, 2012

Time | Event | Location | Materials |
---|---|---|---|

10:30 - 11:50 | Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University | 11th Floor Lecture Hall | |

2:30 - 3:30 | Denseness of repelling cycles in the Julia set and its parameter analogue - Tomoko Kawahira, Tomoki Kawahira, Nagoya University | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 5:00 | P-adic parametrization and the cyclic case of the dynamical Mordell-Lang conjecture - Thomas Tucker, University of Rochester | 11th Floor Lecture Hall |

##### Friday, February 10, 2012

Time | Event | Location | Materials |
---|---|---|---|

11:00 - 12:00 | Computational Working Group | 11th Floor Lecture Hall | |

2:30 - 3:30 | Dynamical Manin-Mumford conjectures - Thomas Tucker, University of Rockester | 11th Floor Lecture Hall | |

3:30 - 4:00 | Coffee/Tea Break | 11th Floor Collaborative Space | |

4:00 - 5:00 | Transversality for postcritically finite rational maps - Xavier Buff, Université de Toulouse III -Paul Sabatier | 11th Floor Lecture Hall |

#### Problems

##### Problem 1: Potential Theory and Invariant Measures.

In the past decade, a number of researchers have developed a robust potential theory on Berkovich spaces and used it to construct an invariant measure associated to each rational map acting on Berkovich projective line, which is in some respects analog to the maximal entropy measure in complex dynamics. It is known to be ergodic and supported on the Julia set, but many fundamental questions remain.

##### Problem 2: Equidistribution Theorems.

Various special sets of points (periodic, preperiodic, small height, Galois conjugates, etc.) have been shown to be equidistributed with respect to the maximal entropy measure in the Riemann sphere and/or to the analog measure on Berkovich projective line. These equidistribution results have had a number of important applications. A focus of the workshop will be to formulate and prove additional equidistribution results and give further applications.

##### Problem 3: Dynamics of Fatou Components.

In 2000, Rivera-Letelier classified the possible dynamics on periodic components of the Berkovich-space Fatou set of a rational function, a classification that is analogous to the complex results of Fatou and Julia. Many questions remain open, including bounds for the number of periodic cycles of quasiperiodicity domains and the existence of wandering Fatou domains over the p-adic rationals.

##### Problem 4: Transversality in Complex Dynamics.

Thurston's rigidity theorem gives a topological characterization of critically finite rational maps. Recent work of Epstein on an infinitesimal version of Thurston's theorem has an underlying arithmetic flavor. The workshop will provide an opportunity to explore these powerful new analytic/algebraic tools for studying both complex and p-adic dynamics.

##### Problem 5: The Local Connectivity Conjecture.

This longstanding and fundamental conjecture asserts that the Mandelbrot set is locally connected. Its resolution would have major implications, including the density of hyperbolic maps in moduli space. Might there be tools from arithmetic dynamics which could be useful for studying this fundamental conjecture?