CONTEMPORARY MATHEMATICS 389

, APORTACIONES MATEMATICAS SOCIEDAD MATEMATICA MEXICANA Geometry and Dynamics International Conference in Honor of the 60th Anniversary of Alberto Verjovsky January 6-ll, 2003 ,

James Eells Etienne Ghys Mikhail Lyubich Jacob Polis Jose Seade Editors http://dx.doi.org/10.1090/conm/389

Geometry and Dynamics CoNTEMPORARY MATHEMATICS

389

~APORTACIONES MATEMATICAS 'VSOCIEDAD MATEMATICA MEXICANA Geometry and Dynamics International Conference in Honor of the 60th Anniversary of Alberto Verjovsky January 6-11, 2003 Cuernavaca, Mexico

James Eells Etienne Ghys Mikhail Lyubich Jacob Polis Jose Seade Editors

American Mathematical Society Providence, Rhode Island Editorial Board of Contemporary Mathematics Dennis· DeThrck, managing editor George Andrews Carlos Berenstein Andreas Blass ·Abel Klein

Edit.orial Board of Aportaciones Matematicas Luis Gorostiza and Luz De Teresa, Managing Editors Marcelo Aguilar Guillermo Pastor Jose Seade Jose M. Gonzalez-Barrios Raul Quiroga Martha Takane Max Neumann Sergio Rajsbaum Jorge X. Velasco

This volume contains the proceedings of the International Conference held January 6-11, 2003 in Cmirnavaca, Mexico, in honor of the 60th anniversary of Alberto Verjovsky.

2000 Mathematics Subject Classification. Primary 17B37, 20Fxx, 32V10, 34Mxx, 37E05, 37Fxx, 53C10, 53D25, 57M30.

Library of Congress Cataloging-in-Publication Data Geometry and dynamics : international conference in honor of the 60th anniversary of Alberto Verjovsky, Cuernavaca, Mexico, January 6-11, 2003 /James Eells ... [et al.], editors. p. em. - (Contemporary mathematics, ISSN 0271-4132 ; 389) Includes bibliographical references. ISBN 0-8218-3851-2 (alk. paper) 1. Nonassociative algebras-Congresses. 2. Functions of complex variables-Congresses. 3. Differentiable dynamical systems-Congresses. 4. Geometry, Differential-Congresses. 5. Manifolds (Mathematics)-Congresses. I. Verjovsky, A. II. Eells, James, 1926- III. Con- temporary mathematics (American Mathematical Society); v. 389. QA252.G46 2005 512'.48-dc22 2005053624

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Introduction vii List of participants 1x Phase-parameter relation and sharp statistical properties for general families of unimodal maps and CARLOS GUSTAVO MOREIRA 1 Hamiltonian operators in noncommutative world NATIG M. ATAKISHIYEV and ANATOLY U. KLIMYK 43 Is the deformation space of complete affine structures on the 2-torus smooth? OLIVER BAUES and WILLIAM M. GOLDMAN 69 Some remarks on parabolic foliations MARCO BRUNELLA 91 Geometrfa y dimimica en el complemento de un compacto en C ADRIEN DOUADY 103 The H-principle and Pseudoconcave CR manifolds C. DENSON HILL and EGMONT PORTEN 117 Wild knots as limit sets of Kleinian groups GABRIELA HINOJOSA 125 An extension of the Burau representation to a mapping class group associated to Thompson's group T CHRISTOPHE KAPOUDJIAN and VLAD SERGIESCU 141 Sur les Theoremes I et II de Painleve FRANK LORAY 165 On the growth rate of contractible closed geodesics on reducible manifolds GABRIEL P. PATERNAIN and JIMMY PETEAN 191 Program 197

v Introduction

These are the proceedings of the conference held at Cuernavaca, Mexico, on January 6-11, 2003, to celebrate the 60th anniversary of Alberto Verjovsky. It was organized by the the Instituto de Matematicas de la UNAM1 at Cuernavaca, with the support of a number of other institutions and foundations. Alberto Verjovsky is one of the most prominent mathematicians in Latin Amer- ica. His research interests are Dynamical Systems, Geometry and Topology, and he has made very significant contributions to these fields. Among his exceptional qualities are his broad interest and knowledge in Mathe- matics, his intellectual generosity and his great enthusiasm, which combine to create an atmosphere of creativity and joy in doing mathematics within his students and colleagues. Alberto Verjovsky began his mathematical life as a student of Solomon Lef- schetz in Mexico; then he went to Brown University to work on his Ph.D. which was completed at IMPA, , under advice of Mauricio Peixoto and Ja- cob Palis. After returning back to Mexico he assumed a research position at CIN- VESTAV, the Center for Advanced Studies of the National Polytechnic Institute in . Over the years he regularly visited different academic institutions in the USA and Europe, maintaining at the same time very close ties with his col- leagues in Latin America. In 1987 he became the Coordinator of the Mathematics Section at the ICTP, Trieste. While at this position, he played an instrumental role in the dissemination of mathematical knowledge in the Third World, as well as enhancing the scientific collaborations among mathematicians of different coun- tries. Later, he joined the University of Lille in France, where he played a key role in strengthening the research groups in geometry and dynamical systems. During all these years he kept supporting mathematics in Mexico, and a few years ago he finally came back to Mexico to join the Mathematics Institute of UNAM at Cuernavaca. This move has made a real difference in mathematics in Mexico due to Alberto's enthusiastic research, educational and organizational ac- tivities. These activities have strongly affected not only established research centers of the country but also its less developed areas as Alberto has enthusiastically in- volved young people from all over the country in mathematical studies and research. Within this short period of time, Alberto's influence has helped to turn complex geometry and dynamical systems into leading research areas of mathematics in Mexico; but his deep intellectual generosity and love for mathematics as a unity have also influenced other areas, particularly topology and . The program of the conference included three mini-courses, given by Marco Brunella, Etienne Ghys and Yasha Eliashberg, and 17 invited lectures, delivered by

1Universidad Nacional Aut6noma de Mexico, the National University of Mexico.

vii viii INTRODUCTION

Artur Avila, Serge Cantat, Adrien Douady, Jean-Marc Gambaudo, William Gold- man, Xavier G6mez-Mont, Francisco Gonzalez Acuna, Gabriela Hinojosa, Blaine Lawson, Misha Lyubich, Laurent Meersseman, , Yair Minsky, Den- nis Sullivan, Carlos Gustavo Tamm Moreira, , and Jean-Christophe Yoccoz. There were many other participants, some of them being world leading experts whose presence significantly enhanced the lively research atmosphere of the meeting. Our thanks to all the participants, who helped to make the meeting so enjoyable and memorable for all of us. The practical organization of the meeting was carried out by the staff of the Institute of Mathematics at Cuernavaca: Elizabeth Dominguez, Gonzalo Albarran and Marfa Sarabia who dedicated to it many hours, with the support of the Institute of Mathematics at Mexico City, and with a tremendous amount of joyful work by several enthusiastic young mathematicians from Cuernavaca, some of them students of Alberto Verjovsky, particularly Gabriela Hinojosa, Elsa Puente, Margareta Boege and Jose Luis Cisneros. Our thanks to all of them and to many others who helped organizing this Conference. We are also grateful to the institutions and foundations that sponsored the meeting:· Universidad Nacional Aut6noma de Mexico; National Science Foundation, USA; Centre National de la Recherche Scientifique, France; International Mathematical Union; Abdus Salam, I.C.T.P., Trieste, Italia; Consejo Nacional de Ciencia y Tecnologfa, Mexico; Union Matematica de America Latina y el Caribe; Academia Mexicana de Ciencias.

Etienne Ghys Jim Eells Misha Lyubich Jose Seade

November, 2004 List of Participants

BRAZIL Isabelle Liousse, U. de Lille, Jacob Palis, IMPA, Brasil Frank Loray, U. de Rennes 1, Carlos Gustavo Tamm Moreira, IMPA, Pavao Mardesic, U. de Bourgogne, Brasil, Laurent Meersseman, U. de Rennes, Marcelo Viana, IMPA, Brasil FRANCE Gael Meigneiez, Universite de Bretagne Sud, Arthur Avila, College de France e IMPA, Brasil, Jean-Pierre Ramis, U.P.S. Toulouse, Viviane Baladi, IHES, France, Vlad Sergiescu, UJF Grenoble, Thierry Barbot, ENS de Lyon, Hamish Short, CMI Universite de Provence, Jean-Paul Brasselet, I.M.L. Luminy, Mihai Tibar, Universite de Lille 1, Marco Brunella, U. de Bourgogne, Jean-Christophe Yoccoz, College de Serge Cantat, U. de Rennes, France, Bertrand Deroin, ENS Lyon, Anton Zorich, Universite de Rennes 1, Adrien Douady, Universit Paris-Sud GREAT BRITAIN (Orsay), Oscar Bandlow, University of London Romain Dujardin, Universit Paris-Sud, Emmanuel Dufraine, University of Sorin Dumitrescu, Universit de Warwick, Paris-Sud ( Orsay), James Eells, Cambridge University, Marguerite Flexor, Universit Paris-Sud, Gabriel Paternain, Cambridge Damien Gaboriau, ENS de Lyon, University, Jean-Marc Gambaudo, U. de ITALY Bourgogne, Marco Abate, Universidad de Pisa, Etienne Ghys, ENS de Lyon, Enrico Casadio Tarabusi, Universidad Aziz el-Kacimi, Univ. Valenciennes, La Sapienza, Roma, Francois Labourie, Universit de Le Dung Trang, ICTP, Trieste, Paris-Sud, Massimiliano Pontecorvo, Unisidad de Stephane Lamy, Universit de Lyon 1, Roma 3,

ix X PARTICIPANTS

JAPAN Ricardo Capovilla, CINVESTAV lPN, Yoshihiko Matsumatsu, Chua Ramiro Carrillo Catalan, IMUNAM, University, Cuernavaca, Shinegori Matsumoto, Nihon Olga Castaneda Aguirre, IMUNAM, University, Ciudad de Mexico, !sao Nakai, Ochanamizu University, Victor Castellanos, UJAT, ; MEXICO IMUNAM, Cuernavaca, Haydee Aguilar Cabrera, Universidad Abel Castorena, Universidad de , Michoacana, Karin Alcaraz, Facultad de Ciencias, Jose Luis Cisneros, IMUNAM, UAEM, Cuernavaca, Ruben German Almanza IMUNAM, Gonzalo Contreras, CIMAT, Unidad Cuernavaca, , Alvaro Alvarez, Universidad Aut6noma Juan Carlos de la Cruz Raymundo, de Baja , UJAT, Tabasco, Natella Antonyan, ITESM, Campus Victor Dominguez Flores, IMUNAM, Ciudad de Mexico, Cuernavaca, Sergey Antonyan, Facultad de Ciencias, Elizabeth Dominguez Soto, IMUNAM, UNAM, Cuernavaca, Lorena Armas, IMUNAM, Ciudad de Jesus Francisco Espinoza, IMUNAM, Mexico, Cuernavaca, Fuensanta Aroca, IMUNAM, Cuernavaca, Mario Eudave Munoz, IMUNAM, Ciudad de Mexico, Aubin Arroyo Camacho, IMUNAM, Cuernavaca, Maite Fernandez, CIMAT, Guanajuato, Natig Atakishiyev, IMUNAM, Martin Eduardo Frias Armenta, Cuernavaca, Universidad de , Egidio Barrera Yanez, IMUNAM, Fernando Galaz Garcia, CIMAT, Cuernavaca, Guanajuato, Gamaliel Ble, UJAT,Tabasco, Alexis Garcia, CIMAT,Guanajuato, Margareta Boege, IMUNAM, Montserrat Garcia Campos, Cuernavaca, Departamento de Matematicas, UNAM, Leticia Brambila, CIMAT, Guanajuato, Gerardo Gabriel Garcia Castrejon, Shirley Bromberg, UAM-Iztapalapa, Univ. Aut. de , Maria Emilia Caballero, IMUNAM, Johanna Garcia Saldaa, Universidad Ciudad de Mexico Michoacana, Jose Omegar Calvo, CIMAT, Timothy Gendron, IMUNAM, Guanajuato, Cuernavaca, PARTICIPANTS xi

Arturo Giles, Facultad de Ciencias, Jorge Luis Lopez Lopez, Universidad UAEM, Michoacana, Xavier Gomez Mont, CIMAT, Petr Makienko, IMUNAM, Cuernavaca, Guanajuato, Matilde Martinez, CIMAT, Francisco Gonzalez Acuna, IMUNAM, Guanajuato, Ciudad de Mexico, Evangelina Martinez Morales, Rodo Gonzalez Sanchez, IMUNAM, IMUNAM, Cuernavaca, Cuernavaca, Emigdio Martinez Ojeda, IMUNAM, Luis Gorostiza, CINVESTAV, Cuernavaca, Lili Guadarrama Bustos, Facultad de Alvaro Martinez Ramirez, Facultad de Ciencias, UAEM, Ciencias, UNAM, Adolfo Guillot Santiago, IMUNAM, Daniel Massart, CIMAT, Guanajuato, Cuernavaca, Diana Maya, IMUN AM, Ciudad de Araceli Guzman Tristan, IMUNAM, Mexico, Ciudad de Mexico, Efren Morales Amaya, CIMAT, Rosalia Guadalupe Hernandez, Guanajuato, IMUNAM, Ciudad de Mexico, Jacob Mostovoy, IMUNAM, Luis Hernandez Lamoneda, CIMAT, Cuernavaca, Guanajuato, Jesus Mucino Raymundo, IMUNAM, Luis Hernandez Hernandez, IMUNAM, , Cuernavaca, Ricardo Muniz, CIMAT, Guanajuato, Jose Luis Herrera Aguilar, IMUNAM, Juan Pablo Navarrete, IMUNAM, Cuernavaca, Cuernavaca, Gabriela Hinojosa, IMUNAM, Juan Gabriel Ochoa Petatan, Facultad Cuernavaca, de Matematicas, , Renato Iturriaga, CIMAT, Guanajuato, Jorge Olivares, CIMAT, Guanajuato, Edgar Jasso, IMUNAM, Ciudad de Adriana Ortiz, IMUNAM, Ciudad de Mexico, Mexico, Patricia Eugenia Jimenez Gallegos, Laura Ortiz, IMUNAM, Ciudad de CIMAT, Guanajuato, Mexico, Herbert Kanarek, IMUNAM, Ciudad Osvaldo Osuna, CIMAT, Guanajuato, de Mexico, Octavio Paniagua, IMUNAM, Ciudad Elifalet Lopez, CIMAT, Guanajuato, de Mexico, Santiago Lopez de Medrano, IMUN AM, Veronica Perez, IMUNAM, Cuernavaca, Ciudad de Mexico R. Michael Porter, CINVESTAV-IPN, Jesus Lopez Estrada, Departamento de Metematicas, UNAM, Julio Arnoldo Prado Saavedra, UNAM, xii PARTICIPANTS

Elsa Puente Vazquez, IMUNAM, Salvador Valenzuela, IMUNAM, Cuernavaca, Cuernavaca, Enrique Ramirez de Arellano, Alberto Verjovsky Sola IMUNAM, CINVESTAV, Cuernavaca Salomon Rebollo Perdomo, Universidad Ricardo Vila, CIMAT, Guanajuato, Michoacana, Araceli Villa Mireles, Universidad Ana Rechlman, Facultad de Ciencias, Michoacana, UNAM, Sergey Voronin, IMUNAM, Ciudad de Claudia Reynoso, CIMAT, Guanajuato, Mexico, Miriam Esther Rosado Cardenas, Univ. Miguel Alejandro Xicotencatl, Autonoma de Guerrero, CINVESTAV-IPN, Ernesto Rosales, IMUNAM, Ciudad de Luis Jose Yudico Anaya, IMUNAM, Mexico, Ciudad de Mexico, Eduardo Sacristan, IMUNAM, Cruz Zagal, IMUNAM, Ciudad de Cuernavaca, Mexico, Noemi Santana, Facultad de Ciencias, PORTUGAL UAEM, Salvatore Cosentino, U. do Minho, Alfonso Santos Mendoza, IMUNAM, Cuernavaca, URUGUAY Antonio Sarmiento, IMUNAM, Jorge Lewowics, Universidad de la Cuernavaca, Repblica, Jose Antonio Seade, IMUNAM, Ezequiel Maderna, IME (Montevideo), Cuernavaca, Roberto Markarian, Universidad de la Jawad Snoussi, IMUNAM, Cuernavaca, Republica, Ramon Salvador Soler, Univ. Juarez Alejandra Rodriguez, Universidad de la Aut. de Tabasco, Republica, Stephen Sontz, CIMAT, Guanajuato, Martin Sambarino, Universidad de la Repblica, Elhoim Sumano, IMUNAM, Ciudad de Mexico, Raul Ures, Universidad de la Republica, Franco Toledo de la Cruz, IMUNAM, U.S. A. Cuernavaca, Alina Andrei, Michigan University, Meral Tosun, IMUNAM, Cuernavaca, Eric Bedford, Michigan University, Edgardo Ugalde, UASLP, San Luis Carlos Cabrera, SUNY at Stony Brook, Potosi Jack Calcut, University of Maryland, Ferran Valdez, IMUNAM, Ciudad de Mexico, Hellen Colman, University of Illinois at Chicago, Carla Valencia, Facultad de Ciencias, UNAM, Yasha Eliashberg, Stanford University, PARTICIPANTS xiii

David Fisher, Lehman College N.Y., John Erick Fornaess, Michigan University, William Goldman, University of Maryland, C. Denson Hill, SUNY at Stony Brook, Jennifer Hruska, SUNY at Stony Brook, Steven Hurder, University of Illinois at Chicago, Blaine Lawson, SUNY at Stony Brook, Tao Li, SUNY at Stony Brook, Krastio Lilov, University of Michigan, Misha Lyubich, IMS, SUNY at Stony Brook, Eran Makover, Connecticut College, Araceli Medina, SUNY at Stony Brook, John Milnor, IMS, SUNY at Stony Brook, Yair Minsky, SUNY at Stony Brook, Florence Newberger, California State University at Robert Penner, University of South California, Rodrigo Prez, Cornell University, Han Peters, Michigan University, John Smillie, Cornell University, Dennis Sullivan, IMS, SUNY at Stony Brook, Rogelio Valdez, SUNY at Stony Brook, Cynthia Verjovsky Marcotte, St. Edwards University, Elmar Winkelnkemper, University of Maryland, COMPLEX GEOMETRY AND DYNAMICS International Congress in honor of the 60th anniversary of Alberto Verjovsky Cuernavaca, Mexico, January 6th to 11th, 2003 PROGRAM

Mini-Courses: Marco Brunella (U. de Bourgogne, Francia): Uniformization of holomorphic foliation by curves Etienne Ghys (E.N.S. de Lyon, Francia): Alberto, Henri and Bernhard Yasha Eliashberg (Stanford University, E.U.A.): Topology of lagrangian and legendrian embeddings Lectures: Artur Avila de Melo(IMPA, Brasil y College de France): Typical unimodal maps and their invariants Serge Cantat (U. de Rennes, Francia): Holomorphic Anosov diffeomorphims on complex projective manifolds Adrien Douady (E.N.S. de Paris, Francia): The complement of a compact set in C (Dynamics without dynamic} Jean-Marc Gambaudo (U. de Bourgogne, Francia): Tilings and Laminations William Goldman (University of Maryland, E.U.A): Automorphisms of moduli spaces Xavier Gomez Mont (CIMAT, Guanajuato, Mexico): Geometry and dynamics of holomorphic flows Francisco Gonzalez Acuiia (UNAM, Mexico): Weightwatchers: Three equivalent conjectures (Ker- vaire, Z and A) Gabriela Hinojosa (IMATE Cuernavaca, UNAM, Mexico): Wild Knots defined as limit sets of Klenian groups Blaine Lawson (SUNY at Stony Brook, E.U.A): Singularities, flows and characteristic classes Misha Lyubich (IMS, SUNY at Stony Brook, E.U.A): Renormalization in the Hen on family Laurent Meersseman (U. de Rennes, Francia): Real quadrics, complex manifolds and convex poly- topes John Milnor (IMS, SUNY at Stony Brook, E.U.A): Dynamics of evolution

197 198

Yair N. Minsky {SUNY at Stony Brook, E.U.A): Deformation and classification of hyperbolic 3- manifolds Dennis Sullivan (IMS, SUNY at Stony Brook, E.U.A): Moduli of closed string operators and the homol- ogy action Carlos Gustavo Tamm (IMPA, Brasil): Hausdorff dimensions and analytic families of unimodal maps Marcelo Viana (IMPA, Brasil): Equilibrium states Jean-Christophe Yoccoz {College de France, Francia): On the cocycle equation for interval exchange transformations Titles in This Series

389 James Eells, Etienne Ghys, Mikhail Lyubich, Jacob Palis, and Jose Seade, Editors, Geometry and dynamics, 2005 388 Ravi Vakil, Editor, Snowbird lectures in algebraic geometry, 2005 387 Michael Entov, Yehuda Pinchover, and Michah Sageev, Editors, Geometry, spectral theory, groups, and dynamics, 2005 386 Yasuyuki Kachi, S. B. Mulay, and Pavlos Tzermias, Editors, Recent progress in arithmetic and algebraic geometry, 2005 385 Sergiy Kolyada, Yuri Manin, and Thomas Ward, Editors, Algebraic and topological dynamics, 2005 384 B. Diarra, A. Escassut, A. K. Katsaras, and L. Narici, Editors, Ultrametric functional analysis, 2005 383 Z.-C. Shi, Z. Chen, T. Tang, and D. Yu, Editors, Recent advances in adaptive computation, 2005 382 Mark Agranovsky, Lavi Karp, and David Shoikhet, Editors, Complex analysis and dynamical systems II, 2005 381 David Evans, Jeffrey J. Holt, Chris Jones, Karen Klintworth, Brian Parshall, Olivier Pfister, and Harold N. Ward, Editors, Coding theory and quantum computing, 2005 380 Andreas Blass and Yi Zhang, Editors, Logic and its applications, 2005 379 Dominic P. Clemence and Guoqing Tang, Editors, Mathematical studies in nonlinear wave propagation, 2005 378 Alexandre V. Borovik, Editor, Groups, languages, algorithms, 2005 377 G. L. Litvinov and V. P. Maslov, Editors, Idempotent mathematics and mathematical physics, 2005 376 Jose A. de Ia Peiia, Ernesto Vallejo, and Natig Atakishiyev, Editors, Algebraic structures and their representations, 2005 375 Joseph Lipman, Suresh Nayak, and Pramathanath Sastry, Variance and duality for cousin complexes on formal schemes, 2005 374 Alexander Barvinok, Matthias Beck, Christian Haase, Bruce Reznick, and Volkmar Welker, Editors, Integer points in polyhedra-geometry, number theory, algebra, optimization, 2005 373 0. Costin, M.D. Kruskal, and A. Macintyre, Editors, Analyzable functions and applications, 2005 372 Jose Burillo, Sean Cleary, Murray Elder, Jennifer Taback, and Enric Ventura, Editors, Geometric methods in group theory, 2005 371 Gui-Qiang Chen, George Gasper, and Joseph Jerome, Editors, Nonlinear partial differential equations and related analysis, 2005 370 Pietro Poggi-Corradini, Editor, The p-harmonic equation and recent advances in analysis, 2005 369 Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu, Editors, Affine algebraic geometry, 2005 368 Sagun Chanillo, Paulo D. Cordaro, Nicholas Ranges, Jorge Hounie, and Abdelhamid Meziani, Editors, Geometric analysis of PDE and several complex variables, 2005 367 Shu-Cheng Chang, Bennett Chow, Sun-Chin Chu, and Chang-Shou Lin, Editors, Geometric evolution equations, 2005 366 Bernheim BooB-Bavnbek, Gerd Grubb, and Krzysztof P. Wojciechowski, Editors, Spectral geometry of manifolds with boundary and decompositon of manifolds, 2005 TITLES IN THIS SERIES

365 Robert S. Doran and Richard V. Kadison, Editors, Operator algebras, quantization, and non-commutative geometry, 2004 364 Mark Agranovsky, Lavi Karp, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems, 2004 363 Anthony To-Ming Lau and Volker Runde, Editors, Banach algebras and their applications, 2004 362 Carlos Concha, Raul Manasevich, Gunther Uhlmann, and Michael S. Vogelius, Editors, Partial differential equations and inverse problems, 2004 361 Ali Enayat and Roman Kossak, Editors, Nonstandard models of arithmetic and set theory, 2004 360 Alexei G. Myasnikov and Vladimir Shpilrain, Editors, Group theory, statistics, and cryptography, 2004 359 S. Dostoglou and P. Ehrlich, Editors, Advances in differential geometry and general relativity, 2004 358 David Burns, Christian Popescu, Jonathan Sands, and David Solomon, Editors, Stark's Conjectures: Recent work and new directions, 2004 357 John Neuberger, Editor, Variational methods: open problems, recent progress, and numerical algorithms, 2004 356 Idris Assani, Editor, Chapel Hill ergodic theory workshops, 2004 355 William Abikoff and Andrew Haas, Editors, In the tradition of Ahlfors and Bers, III, 2004 354 Terence Gaffney and Maria Aparecida Soares Ruas, Editors, Real and complex singularities, 2004 353 M. C. Carvalho and J. F. Rodrigues, Editors, Recent advances in the theory and applications of mass transport, 2004 352 Marek Kubale, Editor, Graph colorings, 2004 351 George Yin and Qing Zhang, Editors, Mathematics of finance, 2004 350 Abbas Bahri, Sergiu Klainerman, and Michael Vogelius, Editors, Noncompact problems at the intersection of geometry, analysis, and topology, 2004 349 Alexandre V. Borovik and Alexei G. Myasnikov, Editors, Computational and experimental group theory, 2004 348 Hiroshi Isozaki, Editor, Inverse problems and spectral theory, 2004 347 Motoko Kotani, Tomoyuki Shirai, and Toshikazu Sunada, Editors, Discrete geometric analysis, 2004 346 Paul Goerss and Stewart Priddy, Editors, Homotopy theory: Relations with algebraic geometry, group cohomology, and algebraic K-theory, 2004 345 Christopher Heil, Palle E. T. Jorgensen, and David R. Larson, Editors, Wavelets, frames and operator theory, 2004 344 Ricardo Baeza, John S. Hsia, Bill Jacob, and Alexander Prestel, Editors, Algebraic and arithmetic theory of quadratic forms, 2004 343 N. Sthanumoorthy and Kailash C. Misra, Editors, Kac-Moody Lie algebras and related topics, 2004 342 Janos Pach, Editor, Towards a theory of geometric graphs, 2004

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstorej. This volume is based on talks given at the Conference in Honor of the 60th Anniversary of Alberto Verjovsky, a prominent mathematician in Latin America who made significant contributions to dynamical systems, geometry, and topology. Articles in the book present recent work in these areas and are suitable for graduate students and research mathemati- cians.

ISBN 0-8218- 3851-2

9 780821 838518