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.

Image for "Complex and p-adic Dynamics"
Complex.pAdic: Image courtesy of S. Koch

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, February 13, 2012
TimeEventLocationMaterials
8:30 - 8:55am ESTRegistration  
8:55 - 9:00am ESTWelcome - Jeff Brock, Deputy Director, ICERM11th Floor Lecture Hall 
9:00 - 9:45am ESTGeometry of Feigenbaum Julia sets - Mikhail Lyubich, SUNY11th Floor Lecture Hall 
10:00 - 10:30am ESTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15am ESTGeometric Limits and Renormalization - Jeremy Kahn, Brown University11th Floor Lecture Hall 
11:30 - 12:15pm ESTSpecial curves and post‐critically finite maps in the moduli space - Laura DeMarco, University of Illinois11th Floor Lecture Hall 
12:30 - 2:30pm ESTBreak for Lunch Free Time  
2:30 - 3:15pm ESTQuantitative Equidistribution in Non-Archimedean and Complex Dynamics - Yusuke Okuyama, Kyoto Institute of Technology11th Floor Lecture Hall 
3:30 - 4:00pm ESTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:45pm ESTStrong bifurcation loci of maximal Hausdorff dimension - Thomas Gauthier, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
5:00 - 6:30pm ESTWelcome Reception  
Tuesday, February 14, 2012
TimeEventLocationMaterials
9:00 - 9:45am ESTThe Berkovich Ramification Locus -- Structure and Applications - Xander Faber, University of Hawaii11th Floor Lecture Hall 
10:00 - 10:30am ESTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15am ESTWandering Domains in Spherically Complete Fields - Eugenio Trucco, Centro de Estudios Científicos de Santiago11th Floor Lecture Hall 
11:30 - 12:15pm ESTOn the non-archimedean Calabi--Yau Theorem - Xinyi Yuan, Princeton University11th Floor Lecture Hall 
12:30 - 2:30pm EST Break for Lunch Free Time  
2:30 - 3:15pm ESTCanonical measures on subvarieties of abelian varieties - Walter Gubler, Universität Regensburg11th Floor Lecture Hall 
3:30 - 4:00pm ESTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 4:45pm ESTSingular 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 - 9:45am ESTCritical Orbits and Attracting Cycles in Non-Archimedean Dynamics - Alon Levy, ICERM11th Floor Lecture Hall 
10:00 - 10:30am ESTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15am ESTTransversality Principles in Holomorphic Dynamics - Adam Epstein, University of Warwick11th Floor Lecture Hall
11:30 - 12:15pm ESTCritical heights of polynomials - Patrick Ingram, Colorado State University11th Floor Lecture Hall 
12:30 - 12:40pm ESTGroup Photo11th Floor Lecture Hall 
12:40 - 2:30pm ESTBreak for Lunch Free Time  
2:30 - 3:15pm ESTFixed-point-free elements of iterated monodromy groups - Rafe Jones, College of the Holy Cross11th Floor Lecture Hall 
3:30 - 4:00pm ESTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm ESTBreak for collaborations  
Thursday, February 16, 2012
TimeEventLocationMaterials
9:00 - 9:45am ESTPreperiodic points for families of polynomials - Dragos Ghioca, University of British Columbia11th Floor Lecture Hall
9:55 - 10:00am ESTSurvey Distribution Please return to 11th floor reception desk  
10:00 - 10:30am ESTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:15am ESTArithmetic differential invariants of dynamical systems - Alexandru Buium, University of New Mexico11th Floor Lecture Hall
11:30 - 7:00pm ESTFree afternoon for collaborations  
7:00 - 8:00pm ESTPoster Session & Dessert Reception (Coffee, tea and desserts provided)11th Floor Lecture Hall and 11th Floor Collaborative Space 
Friday, February 17, 2012
TimeEventLocationMaterials
9:00 - 9:45am ESTOrbit length statistics and the dynamical Mordell-Lang conjecture - Par Kurlberg, Kungliga Tekniska Hogskolan11th Floor Lecture Hall 
10:00 - 10:30am ESTCoffee/Tea Break11th Floor Collaborative Space 
10:30 - 11:20am ESTPolynomial Matings - Arnaud Cheritat, Université de Toulouse III (Paul Sabatier)11th Floor Lecture Hall 
11:30 - 12:30pm ESTOn local linearization in non-Archimedean dynamics - Karl-Olof Lindahl, Linnaeus University11th Floor Lecture Hall
12:30 - 2:30pm ESTBreak for Lunch Free Time  
2:30 - 3:15pm ESTOn approximating non-Archimedean Julia set - Liang-Chung Hsia, National Taiwan Normal University11th Floor Lecture Hall
3:30 - 4:15pm ESTAlgebraic Entropy - Lucien Szpiro, City University of New York (CUNY)11th Floor Lecture Hall

Tutorial Week Schedule

Thursday, February 9, 2012
TimeEventLocationMaterials
10:30 - 11:50am ESTPostcritically Finite Rational Maps and their Deformations - Jeremy Kahn, Brown University11th Floor Lecture Hall 
2:30 - 3:30pm ESTDenseness of repelling cycles in the Julia set and its parameter analogue - Tomoko Kawahira, Tomoki Kawahira, Nagoya University11th Floor Lecture Hall 
3:30 - 4:00pm ESTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm ESTP-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 - 12:00pm ESTComputational Working Group11th Floor Lecture Hall 
2:30 - 3:30pm ESTDynamical Manin-Mumford conjectures - Thomas Tucker, University of Rockester11th Floor Lecture Hall 
3:30 - 4:00pm ESTCoffee/Tea Break11th Floor Collaborative Space 
4:00 - 5:00pm ESTTransversality 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
Image for "Complex and Arithmetic Dynamics"
Jan 30 - May 4, 2012
Semester Program
Global Arithmetic Dynamics
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Mar 19 - 23, 2012
Semester Workshop
Moduli Spaces Associated to Dynamical Systems
Image for "Moduli Spaces Associated to Dynamical Systems"
Apr 16 - 20, 2012
Semester Workshop

Lecture Videos

On approximating non-Archimedean Julia set

Liang-Chung Hsia
National Taiwan Normal University
February 17, 2012
Complex and p-adic Dynamics
Play     Slides

On local linearization in non-Archimedean dynamics

Karl-Olof Lindahl
Linnaeus University
February 17, 2012
Complex and p-adic Dynamics
Play     Slides

Polynomial Matings

Shmuel Weinberger
University of Chicago
February 17, 2012
Complex and p-adic Dynamics
Play     Slides

Arithmetic differential invariants of dynamical systems

Alexandru Buium
University of New Mexico
February 16, 2012
Complex and p-adic Dynamics
Play     Slides

Preperiodic points for families of polynomials

Dragos Ghioca
University of British Columbia
February 16, 2012
Complex and p-adic Dynamics
Play     Slides

Fixed-point-free elements of iterated monodromy groups

Rafe Jones
College of the Holy Cross
February 15, 2012
Complex and p-adic Dynamics
Play

Critical heights of polynomials

Patrick Ingram
York University
February 15, 2012
Complex and p-adic Dynamics
Play

Transversality Principles in Holomorphic Dynamics

Adam Epstein
University of Warwick
February 15, 2012
Complex and p-adic Dynamics
Play     Slides

Critical Orbits and Attracting Cycles in Non-Archimedean Dynamics

Alon Levy
Institute for Computational and Experimental Research in Mathematics (ICERM)
February 15, 2012
Complex and p-adic Dynamics
Play

Singular metrics and the Calabi-Yau theorem in non-archimedean geometry

Mattias Jonsson
University of Michigan
February 14, 2012
Complex and p-adic Dynamics
Play

Canonical measures on subvarieties of abelian varieties

Walter Gubler
Universität Regensburg
February 14, 2012
Complex and p-adic Dynamics
Play

On the non-archimedean Calabi-Yau Theorem

Xinyi Yuan
Princeton University
February 14, 2012
Complex and p-adic Dynamics
Play

Strong bifurcation loci of maximal Hausdorff dimension

Thomas Gauthier
Universite de Toulouse III (Paul Sabatier)
February 13, 2012
Complex and p-adic Dynamics
Play

Quantitative Equidistribution in Non-Archmedean and Complex Dynamics

Yûsuke Okuyama
Kyoto Institute of Technology
February 13, 2012
Complex and p-adic Dynamics
Play     Slides

Special curves and post-critically finite maps in the moduli space

Laura DeMarco
University of Illinois
February 13, 2012
Complex and p-adic Dynamics
Play

Geometric Limits and Renormalization

Jeremy Kahn
Brown University
February 13, 2012
Complex and p-adic Dynamics
Play