CONTEMPORARY MATHEMATICS

152

Nielsen Theory and Dynamical Systems

AMS-IMS-SIAM Summer Research Conference on Nielsen Theory and Dynamical Systems

June 20-26, 1992 Mt. Holyoke College

Christopher K. McCord Editor

Recent Titles in This Series Christopher K. McCord, Editor, Nielsen theory and dynamical systems, 1993 Matatyahu Rubin, The reconstruction of trees from their automorphism groups, 1993 Carl-Friedrich Biidigheimer and Richard M. Hain, Editors, Mapping class groups and moduli spaces of Riemann surfaces, 1993 Harry Cohn, Editor, Doeblin and modem probability, 1993 Jeffrey Fox and Peter Haskell, Editors, Index theory and operator algebras, 1993 Neil Robertson and Paul Seymour, Editors, Graph structure theory, 1993 Martin C. Tangora, Editor, Algebraic topology, 1993 Jeffrey Adams, Rebecca Herb, Stephen Kudla, Jian-Shu Li, Ron Lipsman, Jonathan Rosenberg, Editors, Representation theory of groups and algebras, 1993 Bor-Luh Lin and William B. Johnson, Editors, Banach spaces, 1993 Marvin Knopp and Mark Sheingorn, Editors, A tribute to Emil Grosswald: Number theory and related analysis, 1993 Chung-Chun Yang and Sheng Gong, Editors, Several complex variables in China, 1993 A. Y. Cheer and C. P. van Dam, Editors, Fluid dynamics in biology, 1993 Eric L. Grinberg, Editor, Geometric analysis, 1992 Vinay Deodhar, Editor, Kazhdan-Lusztig theory and related topics, 1992 Donald St. P. Richards, Editor, Hypergeometric functions on domains of positivity, Jack polynomials, and applications, 1992 Alexander Nagel and Edgar Lee Stout, Editors, The Madison symposium on complex analysis, 1992 Ron Donagi, Editor, Curves, Jacobians, and Abelian varieties, 1992 Peter Walters, Editor, Symbolic dynamics and its applications, 1992 Murray kerstenhaber and Jim Stasheff, Editors, Deformation theory and quantum groupst with applications to mathematical physics, 1992 la$ Adolphson, Steven Sperber, and Marvin Tretkoff, Editors, p-Adic methods in number theory and algebraic geometry, 1992 Mark Gotay, Jerrold Marsden, and Vincent Moncrief, Editors, Mathematical aspects of classical field theory, 1992 L. A. Bokut', Yu. L. Ershov, and A. I. Kostrikin, Editors, Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Mal'cev, Parts 1, 2, and 3, 1992 L. Fuchs, K. R. Goodearl, J. T. Stafford, and C. Vinsonhaler, Editors, Abelian groups and noncommutative rings, 1992 John R. Graef and Jack K. Hale, Editors, Oscillation and dynamics in delay equations, 1992 Ridgley Lange and Shengwang Wang, New approaches in spectral decomposition, 1992 Vladimir Oliker and Andrejs Treibergs, Editors, Geometry and nonlinear partial differential equations, 1992 R. Keith Dennis, Claudio Pedrini, and Michael R. Stein, Editors, Algebraic K-theory, commutative algebra, and algebraic geometry, 1 992 F. Thomas Bruss, Thomas S. Ferguson, and Stephen M. Samuels, Editors, Strategies for sequential search and selection in real time, 1992 Darrell Haile and James Osterburg, Editors, Azumaya algebras, actions, and modules, 1992 Steven L. Kleiman and Anders Thorup, Editors, Enumerative algebraic geometry, 199 1 D. H. Sattinger, C. A. Tracy, and S. Venakides, Editors, Inverse scattering and applications, 199 1

(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/152

Nielsen Theory and Dynamical Systems

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Nielsen Theory and Dynamical Systems

AMS-IMS-SIAM Summer Research Conference on Nielsen Theory and Dynamical Systems

June 20-26, 1992 supported by the National Science Foundation

Christopher K. McCord Editor

HEM

4 Y: -2

American Mathematical Society Providence, Rhode Island

EDITORIAL BOARD

Craig Huneke, managing editor Clark Robinson J . T . Stafford Linda Preiss Rothschild Peter M. Winkler

The AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Nielsen Theory and Dynamical Systems was held at Mt. Holyoke College, South Hadley, Massachusetts, from June 20-26, 1992, with support from the National Science Foundation, Grant DMS-8918200 02.

1991 Mathematics Subject Classificatio~n. Primary 54H20, 54H25, 55M20; Secondary 34C35, 58C30, 58F20.

Library of Congress Cataloging-in-Publication Data

AMS-IMS-SIAM Joint Summer Research Conference on Nielsen Theory and Dynamical Systems (1992: Mt. Holyoke College)

Nielsen theory and dynamical systems/Christopher K. McCord, editor. p. cm.-(Contemporary mathematics, ISSN 0271-4132; v. 152) "Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Nielsen

Theory and Dynamical Systems, held June 20-26, 1992, with support from the National Science Foundation."

ISBN 0-8218-5181-0 (acid-free) 1. Topological dynamics-Congresses. 2. Fixed point theory-Congresses. I. McCord,

Christopher Keil. 11. Title. 111. Series: Contemporary mathematics (American Mathematical Society); v. 152. QA611.5.A47 1992 93-26685 514--dc20 CIP

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Contents

Preface

Torus maps and Nielsen numbers LL. A L S E D ~ , S. BALDWIN, J . LLIBRE, R. SWANSON, AND W

SZLENK

Wecken properties for manifolds ROBERT F. BROWN

Lefschetz zeta functions and forced set of periods JOSEFINA CASASAYAS, JAUME LLIBRE, AND ANA NUNES

One-parameter fixed point indices for periodic orbits DONEO DIMOVSKI

Dynamical zeta functions, Nielsen theory and Reidemeister torsion ALEXANDER. FEL'SHTYN AND RICHARD HILL

Cycles for disk homeomorphisms and thick trees JOHN FRANKS AND MICHAL MISIUREWICZ

Lefschetz trace formulae, zeta functions and torsion in dynamics Ross GEOGHEGAN AND ANDREW NICAS

Recent developments in Nielsen theory and discrete groups JANE GILMAN

Local Nielsen fixed point theory and the local generalized H-Lefschetz num- ber

EVELYN L. HART

Nielsen theory for periodic orbits and applications to dynamical systems BOJU JIANG

Genericity of homeomorphisms with connected stable and unstable sets JORGE LEWOWICZ AND JUAN TOLOSA

Lefschetz numbers for periodic points JAUME LLIBRE

X CONTENTS

The Bur au representation of the braid group and the Nielsen-Thurston classification

TAKASHI MATSUOKA 229

Computing Nielsen numbers CHRISTOPHER K. McCORD 249

The structure of isolated invariant sets and the Conley index KONSTANTIN MISCHAIKOW 269

A survey of relative Nielsen fixed point theory HELGA SCHIRMER 291

Classification of lifts of automorphisms of surfaces to the unit disk LEV SLUTSKIN 311

Equivariant Nielsen fixed point theory and periodic points PETER WONG 341

Preface

In June, 1992, a one-week conference on Nielsen Theory and Dynamical Sys- tems was held at Mt. Holyoke College in South Hadley, Massachusetts, as part of the AMS-IMS-SIAM Joint Summer Research Conference Series. The idea for this conference grew out of the special session for Nielsen fixed point theory at the Topology '90 conference, held at the University of Hawaii in August, 1990. At that session, I noticed that most speakers indicated a desire to develop dy- namical applications of their work. But many seemed unsure just what form such applications should take, as they were not sufficiently conversant with dy- namical systems to know what its issues and problems were. It was clear that Nielsen fixed point theory had reached a level of development where it had the opportunity to make some real contributions to dynamics; but it was equally clear that some input and guidance from the dynamics community was needed to make those contributions effective.

The Nielsen Theory and Dynamical Systems Conference was the obvious re- sponse to this opportunity/need. The conference organizing committee consisted of Joan Birman, Robert Brown, John Franks, Boju Jiang and Christopher Mc- Cord (chair). It brought together about 50 topologists and dynamicists. During the week, they heard talks on Nielsen theory and its potential for dynamics; on problems in dynamics that might be amenable to Nielsen theory; and on problems in dynamics that have been successfully analyzed via Nielsen theory. More importantly, the week spent together gave the two research communities the opportunity to meet, interact, and form the personal contacts that are so important for interdisciplinary research. These proceedings attest to the value of such interactions. I can cite one personal example: Konstantin Mischaikow's article contains a result whose proof was supplied the night before his talk. Parts of the proof were supplied at the conference by Phil Boyland, David Fried, Ross Geoghegan, Boju Jiang, Konstantin Mischaikow and myself.

The articles in this volume are contributions by speakers at the conference whose papers, after review, were viewed as conveying the conference's theme. The papers include surveys, expository papers, and technical presentations of new results. Collectively, they provide a good impression of the current status of

xii PREFACE

the interface between Nielsen theory and dynamical systems. I hope that they will promote further interaction between the two fields.

I would like to thank the organizing committee, the conference speakers and the conference participants, whose participation was the real success of the con- ference. I would also like to thank the American Mathematical Society and the National Science Foundation for making the conference possible, and the AMS staff, particularly Carole Kohanski and Donna Harmon, for their assistance in the organization of the conference and this proceedings.

Christopher McCord Cincinnati, Ohio

Recent Titles in This Series (Continuedfrom the front of this publication)

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(See the AMS catalog for earlier titles)

Nielsen Theory and Dynamical Systems Christopher K. McCord, Editor

This volume contains the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Nielsen Theory and Dynamical Systems, held in June 1992 at Mount Holyoke College. Focusing on the interface between Nielsen fixed point theory and dynamical systems, this book provides an almost com-plete survey of the state of the art of Nielsen theory. Most of the articles are expository and provide references to more technical works, making them ac-cessible to both graduate students and researchers in algebraic topology, fixed point theory, and dynamical systems.

ISBN 0-8218-5181-0

9 780821 851814