Organizing Committee
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.

Complex.pAdic: Image courtesy of S. Koch

Confirmed Speakers & Participants

Workshop Schedule

Monday, February 13, 2012
TimeEventLocationMaterials
8:30 - 8:55Registration  
8:55 - 9:00Welcome - Jeff Brock, Deputy Director, ICERM11th Floor Lecture Hall 
9:00 - 9:45Geometry of Feigenbaum Julia sets - Mikhail Lyubich, SUNY11th Floor Lecture Hall 
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15Geometric Limits and Renormalization - Jeremy Kahn, Brown University11th Floor Lecture Hall 
11:30 - 12:15Special curves and post‐critically finite maps in the moduli space - Laura DeMarco, University of Illinois11th Floor Lecture Hall 
12:30 - 2:30Break for Lunch Free Time  
2:30 - 3:15Quantitative Equidistribution in Non-Archimedean and Complex Dynamics - Yusuke Okuyama, Kyoto Institute of Technology11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:45Strong bifurcation loci of maximal Hausdorff dimension - Thomas Gauthier, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
5:00 - 6:30Welcome Reception  
Tuesday, February 14, 2012
TimeEventLocationMaterials
9:00 - 9:45The Berkovich Ramification Locus -- Structure and Applications - Xander Faber, University of Hawaii11th Floor Lecture Hall 
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15Wandering Domains in Spherically Complete Fields - Eugenio Trucco, Centro de Estudios Científicos de Santiago11th Floor Lecture Hall 
11:30 - 12:15On the non-archimedean Calabi--Yau Theorem - Xinyi Yuan, Princeton University11th Floor Lecture Hall 
12:30 - 2:30 Break for Lunch Free Time  
2:30 - 3:15Canonical measures on subvarieties of abelian varieties - Walter Gubler, Universität Regensburg11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:45Singular metrics and the Calabi-Yau theorem in non-Archimedan geometry - Mattias Jonsson, University of Michigan11th Floor Lecture Hall 
Wednesday, February 15, 2012
TimeEventLocationMaterials
9:00 - 9:45Critical Orbits and Attracting Cycles in Non-Archimedean Dynamics - Alon Levy, ICERM11th Floor Lecture Hall 
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15Transversality Principles in Holomorphic Dynamics - Adam Epstein, University of Warwick11th Floor Lecture Hall
11:30 - 12:15Critical heights of polynomials - Patrick Ingram, Colorado State University11th Floor Lecture Hall 
12:30 - 12:40Group Photo11th Floor Lecture Hall 
12:40 - 2:30Break for Lunch Free Time  
2:30 - 3:15Fixed-point-free elements of iterated monodromy groups - Rafe Jones, College of the Holy Cross11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00Break for collaborations  
Thursday, February 16, 2012
TimeEventLocationMaterials
9:00 - 9:45Preperiodic points for families of polynomials - Dragos Ghioca, University of British Columbia11th Floor Lecture Hall
9:55 - 10:00Survey Distribution Please return to 11th floor reception desk  
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15Arithmetic differential invariants of dynamical systems - Alexandru Buium, University of New Mexico11th Floor Lecture Hall
11:30 - 7:00Free afternoon for collaborations  
7:00 - 8:00Poster Session & Dessert Reception (Coffee, tea and desserts provided)11th Floor Lecture Hall and 11th Floor Collaborative Space 
Friday, February 17, 2012
TimeEventLocationMaterials
9:00 - 9:45Orbit length statistics and the dynamical Mordell-Lang conjecture - Par Kurlberg, Kungliga Tekniska Hogskolan11th Floor Lecture Hall 
10:00 - 10:30Coffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20Polynomial Matings - Arnaud Cheritat, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
11:30 - 12:30On local linearization in non-Archimedean dynamics - Karl-Olof Lindahl, Linnaeus University11th Floor Lecture Hall
12:30 - 2:30Break for Lunch Free Time  
2:30 - 3:15On approximating non-Archimedean Julia set - Liang-Chung Hsia, National Taiwan Normal University11th Floor Lecture Hall
3:30 - 4:15Algebraic Entropy - Lucien Szpiro, City University of New York (CUNY)11th Floor Lecture Hall

Tutorial Week Schedule

Thursday, February 9, 2012
TimeEventLocationMaterials
10:30 - 11:50Postcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
2:30 - 3:30Denseness of repelling cycles in the Julia set and its parameter analogue - Tomoko Kawahira, Tomoki Kawahira, Nagoya University11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00P-adic parametrization and the cyclic case of the dynamical Mordell-Lang conjecture - Thomas Tucker, University of Rochester11th Floor Lecture Hall 
Friday, February 10, 2012
TimeEventLocationMaterials
11:00 - 12:00Computational Working Group11th Floor Lecture Hall 
2:30 - 3:30Dynamical Manin-Mumford conjectures - Thomas Tucker, University of Rockester11th Floor Lecture Hall 
3:30 - 4:00Coffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00Transversality for postcritically finite rational maps - Xavier Buff, Université de Toulouse III -Paul Sabatier11th 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?

Associated Semester Workshops

Complex and Arithmetic Dynamics
Global Arithmetic Dynamics
Moduli Spaces Associated to Dynamical Systems

Lecture Videos

On approximating non-Archimedean Julia set

Liang-Chung Hsia
National Taiwan Normal University
February 17, 2012

On local linearization in non-Archimedean dynamics

Karl-Olof Lindahl
Linnaeus University
February 17, 2012

Polynomial Matings

Arnaud Cheritat
Universite de Toulouse III (Paul Sabatier)
February 17, 2012

Arithmetic differential invariants of dynamical systems

Alexandru Buium
University of New Mexico
February 16, 2012

Preperiodic points for families of polynomials

Dragos Ghioca
University of British Columbia
February 16, 2012

Fixed-point-free elements of iterated monodromy groups

Rafe Jones
College of the Holy Cross
February 15, 2012

Critical heights of polynomials

Patrick Ingram
Colorado State University
February 15, 2012

Transversality Principles in Holomorphic Dynamics

Adam Epstein
University of Warwick
February 15, 2012

Canonical measures on subvarieties of abelian varieties

Walter Gubler
Eberhard-Karls-Universitat Tubingen
February 14, 2012

On the non-archimedean Calabi-Yau Theorem

Xinyi Yuan
Princeton University
February 14, 2012

Strong bifurcation loci of maximal Hausdorff dimension

Thomas Gauthier
Univesitre Paul Sabatier
February 13, 2012

Quantitative Equidistribution in Non-Archmedean and Complex Dynamics

Yusuke Okuyama
Kyoto Institute of Technology
February 13, 2012

Geometric Limits and Renormalization

Jeremy Kahn
Brown University
February 13, 2012