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 Cuernavaca, Mexico

James Eells Etienne Ghys

Mikhail Lyubich Jacob Polis Jose Seade

Editors

Geometry and Dynamics

http://dx.doi.org/10.1090/conm/389

CoNTEMPORARY MATHEMATICS

389

~APORTACIONES MATEMATICAS 'V SOCIEDAD 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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10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05

Contents

Introduction vii

List of participants 1x

Phase-parameter relation and sharp statistical properties for general families of unimodal maps

ARTUR AVILA 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, Rio de Janeiro, 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 Mexico City. 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 mathematical physics.

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, John Milnor, Yair Minsky, Den-nis Sullivan, Carlos Gustavo Tamm Moreira, Marcelo Viana, 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 Jacob Palis Jose Seade

November, 2004

List of Participants

BRAZIL Jacob Palis, IMPA, Brasil

Carlos Gustavo Tamm Moreira, IMPA, Brasil,

Marcelo Viana, IMPA, Brasil

FRANCE

Arthur Avila, College de France e IMPA, Brasil,

Viviane Baladi, IHES, France,

Thierry Barbot, ENS de Lyon,

Jean-Paul Brasselet, I.M.L. Luminy,

Marco Brunella, U. de Bourgogne,

Serge Cantat, U. de Rennes,

Bertrand Deroin, ENS Lyon,

Adrien Douady, Universit Paris-Sud (Orsay),

Romain Dujardin, Universit Paris-Sud,

Sorin Dumitrescu, Universit de Paris-Sud ( Orsay),

Marguerite Flexor, Universit Paris-Sud,

Damien Gaboriau, ENS de Lyon,

Jean-Marc Gambaudo, U. de Bourgogne,

Etienne Ghys, ENS de Lyon,

Aziz el-Kacimi, Univ. Valenciennes,

Francois Labourie, Universit de Paris-Sud,

Stephane Lamy, Universit de Lyon 1,

ix

Isabelle Liousse, U. de Lille,

Frank Loray, U. de Rennes 1,

Pavao Mardesic, U. de Bourgogne,

Laurent Meersseman, U. de Rennes,

Gael Meigneiez, Universite de Bretagne Sud,

Jean-Pierre Ramis, U.P.S. Toulouse,

Vlad Sergiescu, UJF Grenoble,

Hamish Short, CMI Universite de Provence,

Mihai Tibar, Universite de Lille 1,

Jean-Christophe Yoccoz, College de France,

Anton Zorich, Universite de Rennes 1,

GREAT BRITAIN

Oscar Bandlow, University of London

Emmanuel Dufraine, University of Warwick,

James Eells, Cambridge University,

Gabriel Paternain, Cambridge University,

ITALY

Marco Abate, Universidad de Pisa,

Enrico Casadio Tarabusi, Universidad La Sapienza, Roma,

Le Dung Trang, ICTP, Trieste,

Massimiliano Pontecorvo, Unisidad de Roma 3,

X PARTICIPANTS

JAPAN

Yoshihiko Matsumatsu, Chua University,

Shinegori Matsumoto, Nihon University,

!sao Nakai, Ochanamizu University,

MEXICO

Haydee Aguilar Cabrera, Universidad de Veracruz,

Karin Alcaraz, Facultad de Ciencias, UAEM,

Ruben German Almanza IMUNAM, Unidad Cuernavaca,

Alvaro Alvarez, Universidad Aut6noma de Baja California,

Natella Antonyan, ITESM, Campus Ciudad de Mexico,

Sergey Antonyan, Facultad de Ciencias, UNAM,

Lorena Armas, IMUNAM, Ciudad de Mexico,

Fuensanta Aroca, IMUNAM, Cuernavaca,

Aubin Arroyo Camacho, IMUNAM, Cuernavaca,

Natig Atakishiyev, IMUNAM, Cuernavaca,

Egidio Barrera Yanez, IMUNAM, Cuernavaca,

Gamaliel Ble, UJAT,Tabasco,

Margareta Boege, IMUNAM, Cuernavaca,

Leticia Brambila, CIMAT, Guanajuato,

Shirley Bromberg, UAM-Iztapalapa,

Maria Emilia Caballero, IMUNAM, Ciudad de Mexico

Jose Omegar Calvo, CIMAT, Guanajuato,

Ricardo Capovilla, CINVESTAV lPN,

Ramiro Carrillo Catalan, IMUNAM, Cuernavaca,

Olga Castaneda Aguirre, IMUNAM, Ciudad de Mexico,

Victor Castellanos, UJAT, Tabasco; IMUNAM, Cuernavaca,

Abel Castorena, Universidad Michoacana,

Jose Luis Cisneros, IMUNAM, Cuernavaca,

Gonzalo Contreras, CIMAT, Guanajuato,

Juan Carlos de la Cruz Raymundo, UJAT, Tabasco,

Victor Dominguez Flores, IMUNAM, Cuernavaca,

Elizabeth Dominguez Soto, IMUNAM, Cuernavaca,

Jesus Francisco Espinoza, IMUNAM, Cuernavaca,

Mario Eudave Munoz, IMUNAM, Ciudad de Mexico,

Maite Fernandez, CIMAT, Guanajuato,

Martin Eduardo Frias Armenta, Universidad de Sonora,

Fernando Galaz Garcia, CIMAT, Guanajuato,

Alexis Garcia, CIMAT,Guanajuato,

Montserrat Garcia Campos, Departamento de Matematicas, UNAM,

Gerardo Gabriel Garcia Castrejon, Univ. Aut. de Guerrero,

Johanna Garcia Saldaa, Universidad Michoacana,

Timothy Gendron, IMUNAM, Cuernavaca,

PARTICIPANTS xi

Arturo Giles, Facultad de Ciencias, UAEM,

Xavier Gomez Mont, CIMAT, Guanajuato,

Francisco Gonzalez Acuna, IMUNAM, Ciudad de Mexico,

Rodo Gonzalez Sanchez, IMUNAM, Cuernavaca,

Luis Gorostiza, CINVESTAV,

Lili Guadarrama Bustos, Facultad de Ciencias, UAEM,

Adolfo Guillot Santiago, IMUNAM, Cuernavaca,

Araceli Guzman Tristan, IMUNAM, Ciudad de Mexico,

Rosalia Guadalupe Hernandez, IMUNAM, Ciudad de Mexico,

Luis Hernandez Lamoneda, CIMAT, Guanajuato,

Luis Hernandez Hernandez, IMUNAM, Cuernavaca,

Jose Luis Herrera Aguilar, IMUNAM, Cuernavaca,

Gabriela Hinojosa, IMUNAM, Cuernavaca,

Renato Iturriaga, CIMAT, Guanajuato,

Edgar Jasso, IMUNAM, Ciudad de Mexico,

Patricia Eugenia Jimenez Gallegos, CIMAT, Guanajuato,

Herbert Kanarek, IMUNAM, Ciudad de Mexico,

Elifalet Lopez, CIMAT, Guanajuato,

Santiago Lopez de Medrano, IMUN AM, Ciudad de Mexico

Jesus Lopez Estrada, Departamento de Metematicas, UNAM,

Jorge Luis Lopez Lopez, Universidad Michoacana,

Petr Makienko, IMUNAM, Cuernavaca,

Matilde Martinez, CIMAT, Guanajuato,

Evangelina Martinez Morales, IMUNAM, Cuernavaca,

Emigdio Martinez Ojeda, IMUNAM, Cuernavaca,

Alvaro Martinez Ramirez, Facultad de Ciencias, UNAM,

Daniel Massart, CIMAT, Guanajuato,

Diana Maya, IMUN AM, Ciudad de Mexico,

Efren Morales Amaya, CIMAT, Guanajuato,

Jacob Mostovoy, IMUNAM, Cuernavaca,

Jesus Mucino Raymundo, IMUNAM, Morelia,

Ricardo Muniz, CIMAT, Guanajuato,

Juan Pablo Navarrete, IMUNAM, Cuernavaca,

Juan Gabriel Ochoa Petatan, Facultad de Matematicas, Acapulco,

Jorge Olivares, CIMAT, Guanajuato,

Adriana Ortiz, IMUNAM, Ciudad de Mexico,

Laura Ortiz, IMUNAM, Ciudad de Mexico,

Osvaldo Osuna, CIMAT, Guanajuato,

Octavio Paniagua, IMUNAM, Ciudad de Mexico,

Veronica Perez, IMUNAM, Cuernavaca,

R. Michael Porter, CINVESTAV-IPN,

Julio Arnoldo Prado Saavedra, UNAM,

xii PARTICIPANTS

Elsa Puente Vazquez, IMUNAM, Cuernavaca,

Enrique Ramirez de Arellano, CINVESTAV,

Salomon Rebollo Perdomo, Universidad Michoacana,

Ana Rechlman, Facultad de Ciencias, UNAM,

Claudia Reynoso, CIMAT, Guanajuato,

Miriam Esther Rosado Cardenas, Univ. Autonoma de Guerrero,

Ernesto Rosales, IMUNAM, Ciudad de Mexico,

Eduardo Sacristan, IMUNAM, Cuernavaca,

Noemi Santana, Facultad de Ciencias, UAEM,

Alfonso Santos Mendoza, IMUNAM, Cuernavaca,

Antonio Sarmiento, IMUNAM, Cuernavaca,

Jose Antonio Seade, IMUNAM, Cuernavaca,

Jawad Snoussi, IMUNAM, Cuernavaca,

Ramon Salvador Soler, Univ. Juarez Aut. de Tabasco,

Stephen Sontz, CIMAT, Guanajuato,

Elhoim Sumano, IMUNAM, Ciudad de Mexico,

Franco Toledo de la Cruz, IMUNAM, Cuernavaca,

Meral Tosun, IMUNAM, Cuernavaca,

Edgardo Ugalde, UASLP, San Luis Potosi

Ferran Valdez, IMUNAM, Ciudad de Mexico,

Carla Valencia, Facultad de Ciencias, UNAM,

Salvador Valenzuela, IMUNAM, Cuernavaca,

Alberto Verjovsky Sola IMUNAM, Cuernavaca

Ricardo Vila, CIMAT, Guanajuato,

Araceli Villa Mireles, Universidad Michoacana,

Sergey Voronin, IMUNAM, Ciudad de Mexico,

Miguel Alejandro Xicotencatl, CINVESTAV-IPN,

Luis Jose Yudico Anaya, IMUNAM, Ciudad de Mexico,

Cruz Zagal, IMUNAM, Ciudad de Mexico,

PORTUGAL

Salvatore Cosentino, U. do Minho,

URUGUAY

Jorge Lewowics, Universidad de la Repblica,

Ezequiel Maderna, IME (Montevideo),

Roberto Markarian, Universidad de la Republica,

Alejandra Rodriguez, Universidad de la Republica,

Martin Sambarino, Universidad de la Repblica,

Raul Ures, Universidad de la Republica,

U.S. A.

Alina Andrei, Michigan University,

Eric Bedford, Michigan University,

Carlos Cabrera, SUNY at Stony Brook,

Jack Calcut, University of Maryland,

Hellen Colman, University of Illinois at Chicago,

Yasha Eliashberg, Stanford University,

PARTICIPANTS

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,

xiii

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 (UN AM, 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