CONTEMPORARY

I 135 '

Symbolic Dynamics and its Applications

Peter Walters Editor http://dx.doi.org/10.1090/conm/135

Symbolic Dynamics and its Applications Recent Titles in This Series

135 Peter Walters, Editor, Symbolic dynamics and its applications, 1992 134 Murray Gerstenhaber and Jim Stashefl', Editors, Deformation theory and quantum groups with applications to mathematical physics, 1992 133 Alan Adolphson, Steven Sperber, and Marvin Tretkofl', Editors, p-Adic methods in number theory and algebraic geometry, 1992 132 Mark Gotay, Jerrold Marsden, and Vincent Moncrief, Mathematical aspects of classical field theory, 1992 131 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, Part l, 2, and 3, 1992 130 L. Fuchs, K. R. Goodearl, J. T. Stafford, and C. Vinsonhaler, Editors, Abelian groups and noncommutative rings, 1992 129 John R. Graef and Jack K. Hale, Oscillation and dynamics in delay equations, 1992 128 Ridgley Lange and Shengwang Wang, New approaches in spectral decomposition, 1992 127 Vladimir Oliker and Andrejs Treibergs, Editors, Geometry and nonlinear partial differential equations, 1992 126 R. Keith Dennis, Claudio Pedrini, and Michael R. Stein, Editors, Algebraic K-theory, commutative algebra, and algebraic geometry, 1992 125 F. Thomas Bruss, Thomas S. Ferguson, and Stephen M. Samuels, Editors, Strategies for sequential search and selection in real time, 1992 124 Darrell Haile and James Osterburg, Editors, Azumaya algebras, actions, and modules, 1992 123 Steven L. Kleiman and Anders Thorup, Editors, Enumerative algebraic geometry, 1991 122 D. H. Sattinger, C. A. Tracy, and S. Venakides, Editors, Inverse scattering and applications, 1991 121 Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries, Spinor construction of vertex operator algebras, triality, and E~ 1 l, 1991 120 RobertS. Doran, Editor, Selfadjoint and nonselfadjoint operator algebras and operator theory, 1991 ll9 Robert A. Metter, Azriel Rosenfeld, and Prabir Bhattacharya, Editors, Vision geometry, 1991 118 Yan Shi-Jian, Wang Jiagang, and Yang Chung-chun, Editors, Probability theory and its applications in China, 1991 117 Morton Brown, Editor, Continuum theory and dynamical systems, 1991 ll6 Brian Harbourne and Robert Speiser, Editors, Algebraic geometry: Sundance 1988, 1991 115 Nancy Flournoy and Robert K. Tsutakawa, Editors, Statistical multiple integration, 1991 114 Jeffrey C. Lagarias and Michael J. Todd, Editors, Mathematical developments arising from linear programming, 1990 113 Eric Grinberg and Eric Todd Quinto, Editors, Integral geometry and tomography, 1990 112 Philip J. Brown and Wayne A. Fuller, Editors, Statistical analysis of measurement error models and applications, 1990 Ill Earl S. Kramer and Spyros S. Magliveras, Editors, Finite geometries and combinatorial designs, 1990 ll 0 and J. Marshall Osborn, Editors, Lie algebras and related topics, 1990 109 Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors, Combinatorial group theory, 1990 l 08 Melvyn S. Berger, Editor, Mathematics of nonlinear science, 1990 107 Mario Milman and Tomas Schonbek, Editors, Harmonic analysis and partial differential equations, 1990 l 06 Wilfried Sieg, Editor, Logic and computation, 1990 (Continued in the back of this publication) CoNTEMPORARY MATHEMATICS

135

Symbolic Dynamics and its Applications

Peter Walters Editor

American Mathematical Society Providence, Rhode Island EDITORIAL BOARD

Richard W. Beals, managing editor Craig Huneke Clark Robinson The AMS Conference on Symbolic Dynamics and its Applications, held in honor of Roy Adler, was held at , New Haven, Connecticut, July 28-August 2, 1991.

1991 Mathematics Subject Classification. Primary 28099, 54H20; Secondary 60J1 0, 58F99.

Library of Congress Cataloging-in-Publication Data Symbolic dynamics and its applications: proceedings of a conference in honor of Roy L. Adler held July 28 to August 2, 1991 /Peter Walters, editor. p. cm.-(Contemporary mathematics; v. 135) Proceedings of a conference held at Yale University. ISBN 0-8218-5146-2 1. Topological dynamics-Congresses. I. Adler, Roy L., 1931-11. Walters, Peter, 1943-. Ill. Series: Contemporary mathematics (American Mathematical Society); v. 135. QA611.5.S96 1992 92-20203 514174-dc20 CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this pub- lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager of Editorial Ser- vices, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance of the code on the first page of an article in this book indicates the copyright owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale.

Copyright @1992 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. § Portions of this volume were printed directly from author-prepared copy. Portions of this volume were typeset by the authors using A~-T(:X, the American Mathematical Society's T(:X macro system. 10 9 8 7 6 5 4 3 2 1 97 96 95 94 93 92 Contents

Introduction IX

Program of the conference XI List of participants xiii The torus and the disk R. 1. ADLER 1 On the work of Roy Adler in ergodic theory and dynamical systems B. WEISS 19 The impact of Roy Adler's work on symbolic dynamics and applications to data storage B. MARCUS 33 LR conjugacies of shifts of finite type are uniquely so J. ASHLEY 57 A polynomial-time for deciding the forcing relation on cyclic permutations C. BERNHARDT AND E. M. COVEN 85 Fully positive topological entropy and topological mixing F. BLANCHARD 95 The stochastic shift equivalence conjecture is false M. BOYLE 107 Predictions with automata A. BROGLIO AND P. LIARDET 111 Common closing extensions and finitary regular isomorphism for synchro- nized systems D. FIEBIG 125 Covers for coded systems D. FIEBIG AND U.-R. FIEBIG 139

v vi CONTENTS

Z-numbers and ;3-transformations 1. FLATTO 181 Quasisymmetric conjugacies for some one-dimensional maps inducing ex- pansion M. JAKOBSON 203 A monotonicity property in one dimensional dynamics J. M. GAMBAUDO AND C. TRESSER 213 Finiteness of conjugacy classes of restricted block upper triangular matrices D. HANDELMAN 223 Polynomials with a positive power D. HANDELMAN 229 Spectral radii of primitive integral companion matrices and log concave polynomials D. HANDELMAN 231 Self-replicating tilings R. KENYON 239

Markov subgroups of (Z/2Z}'l1} B. KITCHENS AND K. SCHMIDT 265 On the dimension of some graphs F. LEDRAPPIER 285 Synchronizing prefix codes and automata and the road coloring problem D. PERRIN AND M.-P. SCHUTZENBERGER 295 A zero entropy, mixing of all orders tiling system S. MOZES 319 A cocycle equation for shifts W.PAA~ ~7

In general a degree 2 map is an automorphism W. PARRY 335 zn versus z actions for systems of finite type C. RADIN 339 Principal vectors of commuting block maps F. RHODES 343 On the recurrence of countable topological Markov chains I. A. SALAMA 349 CONTENTS vii

Substitutions, adic transformations, and beta-expansions B. SoLOMYAK 361 Finitary isomorphism of m-dependent processes M. SMORODINSKY 373 Constant-to-one factor maps and dimension groups P. TROW 377 Faces of Markov chains and matrices of polynomials S. TUNCEL 391 Classification of subshifts of finite type revisited J. B. WAGONER 423 Strong shift equivalence of matrices in G£(2, Z) R. F. WILLIAMS 445 This volume is dedicated to Roy L.Adler on the occasion of his 60th birthday with the affection and respect of his colleagues and friends.

Roy and Audrey Adler Introduction

This volume contains the proceedings of the conference Symbolic Dynamics and Its Applications held at Yale University in New Haven, Connecticut, from July 28 to August 2, 1991. The conference was held in honor of Roy L. Adler. It was organized by Ethan Coven, Shizuo Kakutani (Roy's advisor), Bruce Kitchens, Brian Marcus, Don Ornstein, and Benjamin Weiss. The conference focused on symbolic dynamics and its applications to other fields. These include ergodic theory, smooth dynamical systems, information the- ory, automata theory, and statistical mechanics. There were eighteen one-hour talks, including two on Roy Adler's work, and eighteen thirty-minute talks. There were one hundred and thirty-nine participants from thirteen countries, representing mathematics, applied mathematics, electrical engineering, and physics departments at universities as well as industry. The conference was supported by grants from the NSF and IBM. Roy is honored for both his mathematical and personal contributions to the scientific community. Among his scientific achievements are the invention of topo- logical entropy, the use of Markov partitions for invertible maps, and the application of symbolic dynamics to practical coding problems in information theory. His ideas were crucial in the formation of the subject we call symbolic dynamics. The simple but elegant way he approached difficult problems led many of us into dynamics. Roy had a wonderful partner. He was married to Audrey Wanner Adler from March, 1953 until she died in June, 1990. Together they were a central part of the mathematical community. They made all of us feel at home. As hosts, they cooked hotdogs in Stanford, hamburgers in Warwick, turkeys in Chapel Hill, and lobsters at home. In each location, they provided the central place to gather: a place to sit, talk, and eat. They helped many of us appreciate beautiful mathematics, good food, and really good friends. Thanks.

ix Program One Hour Talks

1. Jonathan Ashley, Uniqueness of a certain decomposition of conjugacies of shifts of finite type 2. Mike Boyle, Quotients of subshifts 3. Elise Cawley, Gibbs measures, symbolic dynamics, and toral diffeomor- phisms 4. Leo Flatto, On a theorem of Mahler, or much ado about nothing 5. David Handelman, Shift equivalent matrices that aren't shift equivalent 6. Mike Keane, isomorphisms 7. , Pleating coordinates for the Riley slice of Schottky space 8. Rick Kenyon, Automatic number systems 9. Wolfgang Krieger, Towards a theory of algebraic subshifts 10. Doug Lind, The entropies of renewal systems 11. Brian Marcus, The Roy Adler story, Part 2 12. Klaus Schmidt, Relative entropies for higher dimensional Markov shifts 13. Caroline Series, From coding curves to deforming Kleinian groups: An ex- ample 14. Paul Trow, Degrees of factor maps between sofic systems 15. Selim Tuncel, Finite equivalence and positivity of polynomials 16. Jack Wagoner, Methods of algebraic topology in symbolic dynamics 17. Benjamin Weiss, On Roy Adler's work in ergodic theory 18. Susan Williams, Core dimension group constraints for factors of sofic shifts Thirty Minute Talks

1. Alexander Blokh, Abstract sets of periodic points for graph maps 2. Annie Broglio, Automaton and prediction 3. Valerio DeAngelis, Polynomial beta functions 4. Doris Fiebig, Common closing extensions for synchronized systems 5. Ulf Fiebig, Covers for coded systems 6. Roza Galeeva, Growth rates for the number of periodic points in countable Markov chains 7. Charlie Jacobson, Totally amalgamatible one sided sofic systems and the information rate function 8. , Slicing sofic sets for toral automorphisms 9. Raj Prasad, Mixing homeomorphisms of prescribed mean rotation vectors

xi xii PROGRAM

10. Jim Propp, The fundamental group of a '!} shift 11. Charles Radin, 2-dimensional uniquely ergodic symbolic dynamics of finite type 12. Mitsuhiro Shishikura, The boundary of the Mandelbrot set 13. Meir Smorodinsky, Finitary isomorphism of 1-dependent processes 14. Boris Somolyak, P expansions and spectral theory of substitutions and adic transformations 15. Jeff Steif, An exact critical value for a cellular automaton 16. Serge Troubetzkoy, Recurrence properties of Lorentz lattice gas cellular au- tomata 17. Tom Ward, Distribution of periodic points for N2 actions on compact groups 18. Zhou Zuo-Ling, The topological and the nonnegative matrix PARTICIPANTS

Roy Adler, IBM Watson Research Center Jawad Yusuf Al-Khal, University of Maryland Paul Algoet, Venkat Anantharam, Cornell University Jonathan Ashley, IBM Almaden Research Center Joseph Auslander, University of Maryland Kirby Baker, University of California, Los Angeles Marie-Pierre Beal, Universite Paris 7 Alexandra Bellow, Northwestern University Ken Berg, University of Maryland Daniel Berend, Ben-Gurion University Vitaly Bergelson, Ohio State University Chris Bernhardt, Fairfield University S. I. Bezuglyi, Universitat Heidelberg Lou Block, University of Florida Alexander Blokh, Wesleyan University Mike Boyle, University of Maryland Rob Bradley, Northwestern University Jonathan Brezin, IBM Watson Research Center Annie Broglio, Universite de Provence Karen Brucks, University of Wisconsin-Milwaukee Bob Burton, Oregon State University Catherine Carroll, Northwestern University Elise Cawley, M.S.R.I Hsin-Ta Frank Cheng, City University of New York J. Choksi, McGill University Ethan Coven, Wesleyan University Larry J. Cummings, University of Waterloo Karma Dajani, University of South Alabama Andres del Junco, University of Toronto Valerio DeAngelis, David Delchamps, Cornell University Beverly Diamond, College of Charleston Stanley Eigen, Northeastern University Jacob Feldman, University of California Sebastien Ferenczi, CNRS - URA 225

xiii xiv PARTICIPANTS

Doris Fiebig, Universitat Heidelberg Ulf Fiebig, Universitat Heidelberg Adam Fieldsteel, Wesleyan University Albert Fisher, Yale University Leo Flatto, Bell Laboratories, Inc. Christiane Frougny, LITP, Institut Blaise Pascal Roza Galeeva, LCTA JINR Will Geller, University of Maryland Bob Gilman, Stevens Institute of Technology Richard Goldberg, IBM Watson Research Center Geoff Goodson, Towson State University Matthew Grayson, IBM Watson Research Center Piotr Grzegorczyk, Stanford University Kamel Haddad, University of Maryland Arshag B. Hajian, Northeastern University Michael Handel, CUNY, Graduate Center David Handelman, University of Ottawa Georges Hansel, Universite de Rouen Jane Hawkins, University of North Carolina Nicolai Haydn, University of Southern California Chris Heegard, Cornell University G. A. Hedlund, Yale University Melissa Hidalgo, University of Hartford Alan Hoffman, IBM Watson Research Center Jun Hu, City University of New York Dan-run Huang, University of Maryland Kristin Hubner, Wesleyan University Charlie Jacobson, Elmira College Aimee Johnson, Tufts University Natasha Jonoska, SUNY at Binghamton Shizuo Kakutani, Yale University Steve Kalikow, Ithaca, NY Anatole Katok, Pennsylvania State University Svetlana Katok, Pennsylvania State University Mike Keane, Delft University of Technology Linda Keen, Lehman College, CUNY Rick Kenyon, Institut des Hautes Etudes Scientifiques Jun Kigami, Osaka University Bruce Kitchens, IBM Watson Research Center Wolfgang Krieger, Universitat Heidelberg Cor Kraaikamp, University of Washington Takashi Kumagai, Osaka University Jeff Lagarias, AT&T MichelL. Lapidus, University of California Yuri Latushkin, San Francisco, CA Fran<;ois Ledrappier, Universite Paris 6 Shihai Li, University of Florida Pierre Liardet, Universite de Provence PARTICIPANTS XV

Douglas Lind, University of Washington Zhou Zuo-Ling, Lingnan College Brian Marcus, IBM Almaden Research Center Christian Mauduit, Universite Claude Bernard - Lyon 1 Dan Mauldin, University of North Texas John Milnor, SUNY at Stony Brook Donna Molinek, University of North Carolina Brigitte Mosse, Universite de Provence Shahar Mozes, M.S.R.I. Irene Mulvey, Fairfield University Masakazu Nasu, Mie University Kyewon Koh Park, Ajou University Alan Parks, Lawrence University Yuval Peres, Yale University Dominique Perrin, Universite Paris 7 Karl Petersen, University of North Carolina Mark Pollicott, Universidade do Porto Brenda Praggastis, University of Washington V. S. Prasad, University of Lowell Jim Propp, Massachusetts Institute of Technology Charles Radin, University of Texas Frank Rhodes, University of Southampton Ted Rivlin, IBM Watson Research Center Ronny Roth, IBM Almaden Research Center Dan Rudolph, University of Maryland Ibrahim Salama, North Carolina Central University Klaus Schmidt, University of Warwick Talal Shamoon, Cornell University Caroline Series, University of Warwick Paul Shields, University of Toledo Sandra Shields, College of William and Mary Mitsuhiro Shishikura, SUNY at Stony Brook Michael Shub, IBM Watson Research Center Fred Shultz, Wellesley College Cesar Silva, Williams College N. T. Sindhushayana, Cornell University Meir Smorodinsky, Tel-Aviv University Boris Solomyak, University of Washington Matthew Stafford, University of Texas Jeff Steif, Chalmers University of Technology Jean-Paul Thouvenot, Universite Paris 6 Charles Tresser, IBM Watson Research Center Serge Troubetzkoy, Universitaet Bielefeld Paul Trow, Memphis State University Selim Tuncel, University of Washington Jack Wagoner, University of California Peter Walters, University of Warwick Tom Ward, Ohio State University xvi PARTICIPANTS

Benjamin Weiss, Hebrew University Robert F. Williams, University of Texas Susan Williams, University of South Alabama Carol Wood, Wesleyan University Michael Yakobson, University of Maryland Michiko Yuri, University of Maryland George Zettler, Columbia University Recent Titles in This Series (Continued from the front of this publication)

105 Jerome Kaminker, Editor, Geometric and topological invariants of elliptic operators, 1990 104 Michael Makkai and Robert Pare, Accessible categories: The foundations of categorical , 1989 103 Steve Fisk, Coloring theories, 1989 102 Stephen McAdam, Primes associated to an ideal, 1989 101 S.-Y. Cheng, H. Choi, and Robert E. Greene, Editors, Recent developments in geometry, 1989 100 W. Brent Lindquist, Editor, Current progress in hyperbolic systems: Riemann problems and computations, 1989 99 Basil Nicolaenko, Ciprian Foias, and Roger Temam, Editors, The connection between infinite dimensional and finite dimensional dynamical systems, 1989 98 Kenneth Appel and Wolfgang Haken, Every planar map is four colorable, 1989 97 J. E. Marsden, P. S. Krishnaprasad, and J. C. Simo, Editors, Dynamics and control of multibody systems, 1989 96 Mark Mahowald and Stewart Priddy, Editors, Algebraic topology, 1989 95 Joel V. Brawley and George E. Schnibben, Infinite algebraic extensions of finite fields, 1989 94 R. Daniel Mauldin, R. M. Shortt, and Cesar E. Silva, Editors, Measure and measurable dynamics, 1989 93 M. Isaacs, A. Lichtman, D. Passman, S. Sehgal, N.J. A. Sloane, and H. Zassenhaus, Editors, Representation theory, group rings, and coding theory, 1989 92 John W. Gray and Andre Scedrov, Editors, Categories in computer science and logic, 1989 91 David Colella, Editor, Commutative harmonic analysis, 1989 90 Richard Randell, Editor, Singularities, 1989 89 R. Bruce Richter, Editor, Graphs and , 1989 88 R. Fossum, W. Haboush, M. Hochster, and V. Lakshmibai, Editors, Invariant theory, 1989 87 Laszlo Fuchs, Riidiger GObel, and Phillip Schultz, Editors, Abelian group theory, 1989 86 J. Ritter, Editor, Representation theory and number theory in connection with the local Langlands conjecture, 1989 85 Bor-Luh Lin, Editor, Banach space theory, 1989 84 Stevo Todorcevic, Partition problems in topology, 1989 83 Michael R. Stein and R. Keith Dennis, Editors, Algebraic K-theory and algebraic number theory, 1989 82 Alexander J. Hahn, Donald G. James, and Zhe-xian Wan, Editors, Classical groups and related topics, 1989 81 Kenneth R. Meyer and Donald G. Saari, Editors, Hamiltonian dynamical systems, 1988 80 N. U. Prabhu, Editor, Statistical inference from stochastic processes, 1988 79 Ernst Kunz and Rolf Waldi, Regular differential forms, 1988 78 Joan S. Birman and Anatoly Libgober, Editors, Braids, 1988 77 Wang Yuan, Yang Chung-chun, and Pan Chengbiao, Editors, Number theory and its applications in China, 1988 76 David C. Hobby and Ralph McKenzie, The structure of finite algebras, 1988

(See the AMS catalog for earlier titles) Symbolic Dynamics and its Applications Peter Walters, Editor This volume contains the proceedings of the conference, Symbolic Dynam- ics and its Applications, held at Yale University in the summer of 1991 in honor of Roy L. Adler on his sixtieth birthday. The conference focused on symbolic dynamics and its applications to other fields, including ergodic theory, smooth dynamical systems, information theory, automata theory, and statisti- cal mechanics. One hundred thirty-nine participants attended from thirteen countries, representing mathematics, applied mathematics, electrical engineer- ing, and physics departments in universities and in industry. Featuring a range of contributions from some of the leaders in the field, this volume presents an excellent overview of the subject.

ISBN 0-8218-5146-2

9780821851463