Contemporary Mathematics 389

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 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 Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permissioniDams. org. Excluded from these provisions is material in articles for which the author holds copyright. 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Visit the AMS home page at http://www.ams.org/ 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 colleagues 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 activities. 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 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,

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