Complex and p-adic Dynamics (February 13-17, 2012)


Organizing Committee
Complex.pAdic


[Complex.pAdic: Image courtesy of S. Koch]

Description

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.




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?

  • Robert Rumely
    (University of Georgia)
  • Zachary Louis Scherr
    (University of Michigan)
  • Dierk Schleicher
    (Jacobs University)
  • Nikita Selinger
    (Jacobs University)
  • Thomas Sharland
    (University of Michigan)
  • Mitsuhiro Shishikura
    (Kyoto University)
  • Joseph Silverman
    (Brown University)
  • Lucien Szpiro*
    (City University of New York)
  • Bianca Thompson
    (University of Hawaii)
  • Eugenio Trucco*
    (Universidad Austral de Chile)
  • Thomas Tucker
    (University of Rochester)
  • Eva Uhre
    (SUNY at Stony Brook)
  • Bianca Viray
    (Brown University)
  • Paul Vojta
    (University of California, Berkeley)
  • Chi-Hao Wang
    (National Central University)
  • Xiaoguang Wang
    (Chinese Academy of Sciences)
  • Kazuhiko Yamaki
    (Kyoto University)
  • Xinyi Yuan*
    (Columbia University)
  • Ilies Zidane
    (Université de Toulouse III, Paul Sabatier)
  • Michael Zieve
    (University of Michigan)
Thursday February 9, 2012
Time Description Speaker Location Abstracts Slides
10:30 - 11:50 Postcritically Finite Rational Maps and their Deformations Jeremy Kahn, Brown University 11th Floor Lecture Hall
PDF
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 Rockester 11th Floor Lecture Hall
Friday February 10, 2012
Time Description Speaker Location Abstracts Slides
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

Download all Abstracts: PDF

Monday February 13, 2012
Time Description Speaker Location Abstracts Slides
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
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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
PDF
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
PDF
PDF
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
PDF
5:00 - 6:30 Welcome Reception  
Tuesday February 14, 2012
Time Description Speaker Location Abstracts Slides
9:00 - 9:45 The Berkovich Ramification Locus -- Structure and Applications Xander Faber, University of Hawaii 11th Floor Lecture Hall
PDF
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
PDF
PDF
11:30 - 12:15 On the non-archimedean Calabi--Yau Theorem Xinyi Yuan, Princeton University 11th Floor Lecture Hall
PDF
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
PDF
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
PDF
PDF
Wednesday February 15, 2012
Time Description Speaker Location Abstracts Slides
9:00 - 9:45 Critical Orbits and Attracting Cycles in Non-Archimedean Dynamics Alon Levy, ICERM 11th Floor Lecture Hall
PDF
10:00 - 10:30 Coffee/Tea Break   11th Floor Collaborative Space
10:30 - 11:15 Transversality Principles in Holomorphic Dynamics Adam Epstein, University of Warwick 11th Floor Lecture Hall
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11:30 - 12:15 Critical heights of polynomials Patrick Ingram, Colorado State University 11th Floor Lecture Hall
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12:30 - 12:40 Group Photo   11th Floor Lecture Hall
12:40 - 2:30

Break for Lunch

Free Time

 
2:30 - 3:15 Fixed-point-free elements of iterated monodromy groups Rafe Jones, College of the Holy Cross 11th Floor Lecture Hall
PDF
3:30 - 4:00 Coffee/Tea Break   11th Floor Collaborative Space
4:00 - 5:00 Break for collaborations  
Thursday February 16, 2012
Time Description Speaker Location Abstracts Slides
9:00 - 9:45 Preperiodic points for families of polynomials Dragos Ghioca, University of British Columbia 11th Floor Lecture Hall
PDF
PDF
9:55 - 10:00

Survey Distribution

Please return to 11th floor reception desk

 
10:00 - 10:30 Coffee/Tea Break   11th Floor Collaborative Space
10:30 - 11:15 Arithmetic differential invariants of dynamical systems Alexandru Buium, University of New Mexico 11th Floor Lecture Hall
PDF
PDF
11:30 - 7:00 Free afternoon for collaborations  
7:00 Poster Session & Dessert Reception
(Coffee, tea and desserts provided)
  11th Floor Lecture Hall and 11th Floor Collaborative Space
Friday February 17, 2012
Time Description Speaker Location Abstracts Slides
9:00 - 9:45 Orbit length statistics and the dynamical Mordell-Lang conjecture Par Kurlberg, Kungliga Tekniska Hogskolan 11th Floor Lecture Hall
PDF
10:00 - 10:30 Coffee/Tea Break   11th Floor Collaborative Space
10:30 - 11:20 Polynomial Matings Arnaud Cheritat, Université de Toulouse III (Paul Sabatier) 11th Floor Lecture Hall
11:30 - 12:30 On local linearization in non-Archimedean dynamics Karl-Olof Lindahl, Linnaeus University 11th Floor Lecture Hall
PDF
PDF
12:30 - 2:30 Break for Lunch
Free Time
 
2:30 - 3:15 On approximating non-Archimedean Julia set Liang-Chung Hsia, National Taiwan Normal University 11th Floor Lecture Hall
PDF
PDF
3:30 - 4:15 Algebraic Entropy Lucien Szpiro, City University of New York (CUNY) 11th Floor Lecture Hall
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PDF