AMS SHORT COURSE LECTURE NOTES

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2 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics

3 Proceedings of Symposia in APPLIED MATHEMATICS Volume 60 Symbolic Dynamics and its Applications American Mathematical Society Short Course January 4-5, 2002 San Diego, California Susan G. Williams Editor jgematf American Mathematical Society $ Providence, Rhode Island

4 Editorial Board Peter S. Constant in (Chair) Eitan Tadmor Marsha J. Berger LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE SYMBOLIC DYNAMICS AND ITS APPLICATIONS HELD IN SAN DIEGO, CALIFORNIA JANUARY 4-5, 2002 The AMS Short Course Series is sponsored by the Society's Program Committee for National Meetings. The series is under the direction of the Short Course Subcommittee of the Program Committee for National Meetings Mathematics Subject Classification. Primary 37B10, 37B50, 37A15, 37F45, 94B05, 19C99. Library of Congress Cataloging-in-Publication Data American Mathematical Society Short Course on Symbolic Dynamics and its Applications : (2002 : San Diego, Calif.) Symbolic dynamics and its applications : American Mathematical Society, Short Course, January 4-5, 2002, San Diego, California / Susan G. Williams, editor. p. cm. (Proceedings of symposia in applied mathematics, ISSN ; v. 60) Includes bibliographical references and index. ISBN (alk. paper) 1. Symbolic dynamics Congresses. I. Williams, Susan C, II. Title. III. Series. QA A /.74 dc 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 acknowledgment 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 Mathematical Society, 201 Charles Street, Providence, Rhode Island , USA. Requests can also be made by to reprint-permission@ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) 2004 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 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at

5 Contents Preface vn Introduction to symbolic dynamics \ SUSAN G. WILLIAMS Combining modulation codes and error correcting codes 13 BRIAN MARCUS Complex dynamics and symbolic dynamics 37 PAUL BLANCHARD, ROBERT L. DEVANEY, and LINDA KEEN Multi-dimensional symbolic dynamics 61 DOUGLAS LIND Symbolic dynamics and tilings of R d 81 E. ARTHUR ROBINSON, JR. Strong shift equivalence theory 121 J.B. WAGONER Index 155

6 Preface The foundation of the field of symbolic dynamics is generally credited to Jacques Hadamard, who used infinite symbol sequences in his analysis of geodesic flow on negatively curved surfaces in Hadamard's symbolic techniques were soon adopted and extended by other authors. However, the field had to wait forty years for its christening by Marston Morse and Gustav Hedlund, who provided the first systematic study of symbolic dynamical systems as objects of interest in their own right. This rather prescient paper at the dawn of the computer age set the stage for the mathematical analysis of codes and finite-alphabet communication systems using the techniques of dynamics and ergodic theory, most notably in the pioneering work of C.E. Shannon on the mathematical theory of communication. Fifty years after Hadamard applied symbolic techniques to dynamics, Shannon and others were applying dynamical techniques to symbols. In the fifty-odd years since then, symbolic dynamics has expanded its reach to apply, and be applied to, many areas. It has broken the confines of one dimension to encompass multi-dimensional arrays. The six chapters of this volume provide an introduction to the field as it is studied today and a sampler of its concerns and applications. They are expanded versions of the lectures given in the American Mathematical Society Short Course on Symbolic Dynamics and its Applications held in San Diego on January 4-5, I would like to take this opportunity to thank Jim Maxwell, Wayne Drady and the other AMS staff members who coordinated the short course and worked behind the scenes to make it run smoothly. Susan G. Williams

7 Index (n, M) block code, 21 \n,m,d) codec, 21 K^R), 141 K 2 (R), 141 #3, 147 ^-Construction, 132 A-strategy, 140 c-charge-constrained shift, 17 d-dimensional shift of finite type, 63 q-th power graph, 21 2-dimensional golden mean shift, 64 2-patch closure property, 91 algebraic Z d -action, 72 Algebraic Shift Equivalence Problem, 126 almost 1:1, 99 almost periodic, 94 Automorphisms of the Shift, 38 automorphisms of the two-sided shift, 58 beta expansions, 3 Bowen-Franks group, 9 Cantor set, 37 cellular automata, 66 code, 4 complex structure, 43 complexity, 108 conjugacy, 5, 20 topological, 5, 98, 123 control point, 96 Cubic, 51 decomposition, 89 Dehn twist, 44 Delone set, 108 diffraction, 105 dilatation, 44 dimension, 22 Dimension Group, 135 dimension group, 125 dimension group homomorphism, 129 discrete spectrum, 105 Douady-Hubbard theory of external rays, 42 edge shift, 6, 123 edge-to-edge, 83 eigenfunction, 104 eigenvalue, 104 Perron-Frobenius, 103 eigenvector Perron-Frobenius, 103 einstein, 89 elementary strong shift equivalence, 124 ellipse field, 43 entropy, 8, 9, 20, 70 topological, 109 Entropy and the Zeta Function, 135 escape locus, 48 escape rate, 42 even shift, 4 existence problem for higher-dimensional shifts of finite type, 68 factor map, 4 finite-state (5, n)-encoder, 18 finite-state code, 18 forbidden patches, 86 full shift, 3 Furstenberg's conjecture, 77 generator matrix, 22 golden mean shift, 3 Gottschalk's Theorem, 94 Henon map, 58 higher block codes, 5 Higher Degree Polynomials, 47 holographic data storage, 71 inert automorphisms, 146 Julia set, 37 Ledrappier's example, 64 Lehmer's Conjecture, 76 local complexity 155

8 156 INDEX finite, 83 local matching rule, 86 locally derivable, 98 mutually, 98 locally isomorphic, 96 Mahler measure, 75 Mandelbrot set, 39 mar-ker automorphism, 38 marker sets, 57 Markov chain, 6 topological, 6 maximum distance separable, 23 Measurable Riemann Mapping Theorem, 43 minimal, 94 minimal markers, 57 mixing strong, 105 weak, 105 monodromy map, 42, 47 no Z-cycles condition, 130 Nonnegative Row and Column Operations, 137 one-sided d-shift, 38 one-sided shift, 3 parity insertion scheme, 25 parity-check matrix, 22 patch, 83 Path Space Construction, 131 Pisot number, 106 polynomial strong shift equations, 143 prototiles, 82 pushing deformation, 55 Quadratic Polynomials, 39 quasiconformal mapping, 44 rate p: q finite-state encoder, 18 rate of escape function, 42 repetitive, 94 reversed concatenation, 29 Riemann map, 42 Robinson tiles, 68 runlength-limited (RLL)(d, k) shift, 16 shift equivalence, 9 Shift Equivalence Problem, 125 shift equivalent, 124 shift of finite type, 4, 17 Singleton bound, 23 sliding block code, 5, 69 sliding-block decodable, 19 sliding-block decoder, 19 sofic shift, 6, 16 Spinning, 45 spinning construction, 48 standard concatenation, 25 standard structure, 43 Strong Shift Equivalence Problem, 126 strong shift equivalence spaces, 127 strong shift equivalent over Z+, 124 structure matrix, 92 subshift, 3 subshift of finite type, 122 support, 83 symbolic trajectory, 1, 2 Theorem Perron-Frobenius, 103 tile, 82 tiles Penrose, 86 Wang, 87 tiling, 82 aperiodic, 88 periodic, 88 self-affine, 96 self-similar, 96 tiling dynamical system, 85 tiling metric, 84 tiling space, 85 finite type, 86 full, 82 tiling substitution, 89 invert ible, 92 primitive, 92 tiling topology, 84 tilings Ammann-Beenker, 112 binary, 91, 92 chair, 89, 93 folding table, 90 generalized Penrose, 113 Kari & Culik, 88 octagonal, 112 Penrose, 86 pinwheel, 91, 108 quasiperiodic, 112 substitution, 92 table, 89 triangular Penrose, 90 unmarked Penrose, 99 topological quantum field theory, 148 translation, 83 Triangle Identities, 127, 147 two-sided d-shift map, 58 uniquely ergodic, 102 vertex shift, 5 Wang tiles, 65 Wang's Conjecture, 88 zeta function, 8

9 Titles in This Series 60 Susan G. Williams, Editor, Symbolic dynamics and its applications (San Diego, California, January 2002) 59 James Sneyd, Editor, An introduction to mathematical modeling in physiology, cell biology, and immunology (New Orleans, Louisiana, January 2001) 58 Samuel J. Lomonaco, Jr., Editor, Quantum computation: A grand mathematical challenge for the twenty-first century and the millennium (Washington, DC, January 2000) 57 David C. Heath and Glen Swindle, Editors, Introduction to mathematical finance (San Diego, California, January 1997) 56 Jane Cronin and Robert E. O'Malley, Jr., Editors, Analyzing multiscale phenomena using singular perturbation methods (Baltimore, Maryland, January 1998) 55 Frederick Hoffman, Editor, Mathematical aspects of artificial intelligence (Orlando, Florida, January 1996) 54 Renato Spigler and Stephanos Venakides, Editors, Recent advances in partial differential equations (Venice, Italy, June 1996) 53 David A. Cox and Bernd Sturmfels, Editors, Applications of computational algebraic geometry (San Diego, California, January 1997) 52 V. Mandrekar and P. R. Masani, Editors, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, Michigan, 1994) 51 Louis H. Kauffman, Editor, The interface of knots and physics (San Francisco, California, January 1995) 50 Robert Calderbank, Editor, Different aspects of coding theory (San Francisco, California, January 1995) 49 Robert L. Devaney, Editor, Complex dynamical systems: The mathematics behind the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994) 48 Walter Gautschi, Editor, Mathematics of Computation : A half century of computational mathematics (Vancouver, British Columbia, August 1993) 47 Ingrid Daubechies, Editor, Different perspectives on wavelets (San Antonio, Texas, January 1993) 46 Stefan A. Burr, Editor, The unreasonable effectiveness of number theory (Orono, Maine, August 1991) 45 De Witt L. Sumners, Editor, New scientific applications of geometry and topology (Baltimore, Maryland, January 1992) 44 Bela Bollobas, Editor, Probabilistic combinatorics and its applications (San Francisco, California, January 1991) 43 Richard K. Guy, Editor, Combinatorial games (Columbus, Ohio, August 1990) 42 C. Pomerance, Editor, Cryptology and computational number theory (Boulder, Colorado, August 1989) 41 R. W. Brockett, Editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, Editor, Matrix theory and applications (Phoenix, Arizona, January 1989) 39 Robert L. Devaney and Linda Keen, Editors, Chaos and fractals: The mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis, Editor, Computational complexity theory (Atlanta, Georgia, January 1988) 37 Henry J. Landau, Editor, Moments in mathematics (San Antonio, Texas, January 1987) For a complete list of titles in this series, visit the AMS Bookstore at

10 ISBN 'i780821»831571

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