From Topology to Computation: Proceedings of the Smalefest M.W. Hirsch J.E. Marsden M.Shub Editors From Topology to Computation: Proceedings of the Smalefest With 76 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Morris W. Hirsch Michael Shub Department of Mathematics IBM Research University of California T.J. Watson Research Center Berkeley, CA 94720 USA Yorktown Heights, NY 10598-0218 USA Jerrold E. Marsden Department of Mathematics University of California Berkeley, CA 94720 USA Library of Congress Cataloging-in-Publication Data Smalefest (1990: Berkeley, Calif.) From topology to computation: proceedings ofthe Smalefest / M. W. Hirsch, J.E. Marsden & M. Shub, editors. p. em. Papers presented at an international research conference in honor of Stephen Smale's 60th birthday, held Aug. 5-9, 1990 in Berkeley. Includes bibliographical references. ISBN-13: 978-1-4612-7648-7 e-ISBN-13: 978-1-4612-2740-3 DOl: 10.1007/978-1-4612-2740-3 1. Differential topology-Congresses. 2. Numerical calculations­ Congresses. 3. Differentiable dynamical systems-Congresses. I. Hirsch, Morris W., 1933- . II. Marsden, Jerrold E. III. Shub, Michael, 1943- . IV. Smale, Stephen, 1930- V. Title. QA613.6.S63 1990 514'.72-dc20 92-31991 Printed on acid-free paper. © 1993 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1993 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue. New York, NY 10010, USA), except for brief excepts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. 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Paul Bishop, Jr., Berkeley, CA, photographer Preface An extraordinary mathematical conference was held 5-9 August 1990 at the University of California at Berkeley: From Topology to Computation: Unity and Diversity in the Mathematical Sciences An International Research Conference in Honor of Stephen Smale's 60th Birthday The topics of the conference were some of the fields in which Smale has worked: • Differential Topology • Mathematical Economics • Dynamical Systems • Theory of Computation • Nonlinear Functional Analysis • Physical and Biological Applications This book comprises the proceedings of that conference. The goal of the conference was to gather in a single meeting mathemati­ cians working in the many fields to which Smale has made lasting con­ tributions. The theme "Unity and Diversity" is enlarged upon in the section entitled "Research Themes and Conference Schedule." The organizers hoped that illuminating connections between seemingly separate mathematical sub­ jects would emerge from the conference. Since such connections are not easily made in formal mathematical papers, the conference included discussions after each of the historical reviews of Smale's work in different fields. In addition, there was a final panel discussion at the end of the conference. These discussions, with many contributions from the audience, turned out to be extremely useful in bringing out insights and patterns in Smale's work, and in bringing to light informative incidents and facts-some from decades ago-that are too often lost to the history of mathematics. We considered these discussions to be so valuable that we persuaded Springer-Verlag to viii Preface have them transcribed from the rather crude videotapes of the conference. The transcription was done with great patience and precision by Esther Zack. Some of the speakers had the opportunity to edit their transcriptions or submit manuscripts of their remarks; for mistakes in the other transcrip­ tions, the editors are responsible. Whereas these discussions are not all of the utmost clarity, they present remarkable commentaries on important parts of contemporary mathematics by some of the people who developed it. The main part of this volume consists of research papers contributed by speakers, colleagues, and students of Smale. All of these were refereed under the supervision of the editors. Also included are informal talks about Smale's life and career which were given at social events, and autobio­ graphical writings by Smale, some of which are published here for the first time. No discussion of Smale's work would be complete without reference to his political activities against oppression, some of which are discussed in his autobiographical articles. An interesting evening panel on Smale's politics was chaired by Serge Lang, but unfortunately we were not able to make a transcript. The choice of topics and speakers inevitably reflects the interests of the organizing committee. Whereas we had intended to cover all aspects of Smale's mathematics, as it turned out some very interesting parts of his work were neglected: There were unfortunately no talks or contributed papers about mathematical biology, game theory, linear programming, or smooth solutions to partial differential equations. But to do justice to the full breadth of Smale's research would have required more than five days. The organizing committee consisted of Gerard Debreu, Morris Hirsch, Nancy Kopell, Jerrold Marsden, Jacob Palis, Michael Shub, Anthony Tromba, and Alan Weinstein-all colleagues or former students of Smale. The conference was generously supported by IBM Research Mathematical Sciences Research Institute National Science Foundation Springer -Verlag University of California at Berkeley Contents Preface vii Contributors xv Research Themes XIX Conference Schedule, Smalefest, August 5-9, 1990 XXVll Part 1: Autobiographical Material Some Autobiographical Notes ............................. 3 2 On How I Got Started in Dynamical Systems, 1959-1962 ...... 22 3 The Story of the Higher Dimensional Poincare Conjecture (What Actually Happened on the Beaches of Rio) ............. 27 4 On the Steps of Moscow University ........................ 41 5 Professional Biography, Bibliography, and Graduate Students .. 53 Part 2: Informal Talks 6 Luncheon Talk and Nomination for Stephen Smale 67 R. Bott 7 Some Recollections of the Early Work of Steve Smale ............................................. 73 M.M. Peixoto 8 Luncheon Talk 76 R. Thorn x Contents 9 Banquet Address at the Smalefest 79 E.c. Zeeman Part 3: Differential Topology 10 The Work of Stephen Smale in Differential Topology 83 Morris W. Hirsch DISCUSSIONS 11 Discussion.............................................. 107 J. Stallings, A. Haejliger, and Audience Members 12 Note on the History of Immersion Theory 114 David Spring CONTRIBUTED PAPER 13 The Smale-Hirsch Principle in Catastrophe Theory ........... 117 V.A. Vassil'ev Part 4: Economics 14 Stephen Smale and the Economic Theory of General Equilibrium ............................................. 131 Gerard Debreu DISCUSSION 15 Topology and Economics: The Contribution of Stephen Smale .. 147 Graciella Chichilnisky 16 Comments 162 Y.-H. Wan Part 5: Dynamical Systems 17 On the Contribution of Smale to Dynamical Systems .......... 165 J. Palis DISCUSSION 18 Discussion .............................................. 179 S. Newhouse, R.F. Williams, and Audience Members Contents xi CONTRIBUTED PAPERS 19 Recurrent Sets for Planar Homeomorphisms ................. 186 Marcy Barge and John Franks 20 Convergence of Finite-Element Solutions for Nonlinear PDEs .. 196 X iaohua X uan 21 Ergodic Theory of Chaotic Dynamical Systems ............... 201 L.-S. Young 22 Beyond Hyperbolicity: Expansion Properties of One-Dimensional Mappings .............................................. 227 John Guckenheimer and Stewart Johnson 23 Induced Hyperbolicity, Invariant Measures, and Rigidity ....... 237 M.Jakobson 24 On the Enumerative Geometry of Geodesics ................. 243 Ivan A.K. Kupka and M.M. Peixoto 25 A Relation Between Newton's Method and Successive Approximations for Quadratic Irrationals 254 Julian Palmore 26 On Dynamical Systems and the Minimal Surface Equation ..... 259 Per Tomter 27 A New Zeta Function, Natural for Links .................... 270 R.F. Williams Part 6: Theory of Computation 28 On the Work of Steve Smale on the Theory of Computation .... 281 Michael Shub DISCUSSIONS 29 Smale's Work on the Dynamics of Numerical Analysis ......... 302 Steve Batterson 30 Steve Smale and the Geometry of Ill-Conditioning ............ 305 James Demmel Xli Contents 31 On Smale's Work in the Theory of Computation: From Polynomial Zeros to Continuous Complexity ................ 317 J.F. Traub CONTRIBUTED PAPERS 32 The G6del Incompleteness Theorem and Decidability over a Ring ................................................. 321 Lenore Blum and Steve Smale 33 Ill-Posed Problem Instances ............................... 340 J ames Renegar 34 Cohomology of Braid Groups and Complexity ............... 359 V.A. Vassil'ev 35 The Dynamics of Eigenvalue Computation .................. 368 Steve Batterson 36 On the Role of Computable Error Estimates in the Analysis of Numerical