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Symbolic Dynamics and Its Applications http://dx.doi.org/10.1090/psapm/060 AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures published as a subseries of Proceedings of Symposia in Applied Mathematics 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 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. 2000 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, Jan• uary 4-5, 2002, San Diego, California / Susan G. Williams, editor. p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634 ; v. 60) Includes bibliographical references and index. ISBN 0-8218-3157-7 (alk. paper) 1. Symbolic dynamics—Congresses. I. Williams, Susan C, 1953- II. Title. III. Series. QA614.85.A44 2002 514/.74—dc22 2003062891 Copying and reprinting. Material in this book may be reproduced by any means for edu• cational 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 [email protected]. 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 America. @ 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 http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 09 08 07 06 05 04 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 Rd 81 E. ARTHUR ROBINSON, JR. Strong shift equivalence theory 121 J.B. WAGONER Index 155 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 1898. 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, 2002. I would like to take this opportunity to thank Jim Maxwell, Wayne Drady and the other AMS staff members who coordi• nated the short course and worked behind the scenes to make it run smoothly. Susan G. Williams Index (n, M) block code, 21 Douady-Hubbard theory of external rays, \n,M,d) codeC, 21 42 K^R), 141 edge shift, 6, 123 K2(R), 141 #3, 147 edge-to-edge, 83 ^-Construction, 132 eigenfunction, 104 A-strategy, 140 eigenvalue, 104 c-charge-constrained shift, 17 Perron-Frobenius, 103 d-dimensional shift of finite type, 63 eigenvector q-th power graph, 21 Perron-Frobenius, 103 2-dimensional golden mean shift, 64 einstein, 89 2-patch closure property, 91 elementary strong shift equivalence, 124 ellipse field, 43 algebraic Zd-action, 72 entropy, 8, 9, 20, 70 Algebraic Shift Equivalence Problem, 126 topological, 109 almost 1:1, 99 Entropy and the Zeta Function, 135 almost periodic, 94 escape locus, 48 Automorphisms of the Shift, 38 escape rate, 42 automorphisms of the two-sided shift, 58 even shift, 4 existence problem for higher-dimensional beta expansions, 3 shifts of finite type, 68 Bowen-Franks group, 9 factor map, 4 Cantor set, 37 finite-state (5, n)-encoder, 18 cellular automata, 66 finite-state code, 18 code, 4 forbidden patches, 86 complex structure, 43 full shift, 3 complexity, 108 Furstenberg's conjecture, 77 conjugacy, 5, 20 generator matrix, 22 topological, 5, 98, 123 golden mean shift, 3 control point, 96 Gottschalk's Theorem, 94 Cubic, 51 Henon map, 58 decomposition, 89 higher block codes, 5 Dehn twist, 44 Higher Degree Polynomials, 47 Delone set, 108 holographic data storage, 71 diffraction, 105 inert automorphisms, 146 dilatation, 44 dimension, 22 Julia set, 37 Dimension Group, 135 dimension group, 125 Ledrappier's example, 64 dimension group homomorphism, 129 Lehmer's Conjecture, 76 discrete spectrum, 105 local complexity 155 156 INDEX finite, 83 standard structure, 43 local matching rule, 86 Strong Shift Equivalence Problem, 126 locally derivable, 98 strong shift equivalence spaces, 127 mutually, 98 strong shift equivalent over Z+, 124 locally isomorphic, 96 structure matrix, 92 subshift, 3 Mahler measure, 75 subshift of finite type, 122 Mandelbrot set, 39 support, 83 mar-ker automorphism, 38 symbolic trajectory, 1, 2 marker sets, 57 Markov chain, 6 Theorem topological, 6 Perron-Frobenius, 103 maximum distance separable, 23 tile, 82 Measurable Riemann Mapping Theorem, 43 tiles minimal, 94 Penrose, 86 minimal markers, 57 Wang, 87 mixing tiling, 82 strong, 105 aperiodic, 88 weak, 105 periodic, 88 monodromy map, 42, 47 self-affine, 96 self-similar, 96 no Z-cycles condition, 130 tiling dynamical system, 85 Nonnegative Row and Column Operations, tiling metric, 84 137 tiling space, 85 finite type, 86 one-sided d-shift, 38 full, 82 one-sided shift, 3 tiling substitution, 89 parity insertion scheme, 25 invert ible, 92 parity-check matrix, 22 primitive, 92 patch, 83 tiling topology, 84 Path Space Construction, 131 tilings Pisot number, 106 Ammann-Beenker, 112 polynomial strong shift equations, 143 binary, 91, 92 prototiles, 82 chair, 89, 93 pushing deformation, 55 folding table, 90 generalized Penrose, 113 Quadratic Polynomials, 39 Kari & Culik, 88 quasiconformal mapping, 44 octagonal, 112 Penrose, 86 rate p: q finite-state encoder, 18 pinwheel, 91, 108 rate of escape function, 42 quasiperiodic, 112 repetitive, 94 substitution, 92 reversed concatenation, 29 table, 89 Riemann map, 42 triangular Penrose, 90 Robinson tiles, 68 unmarked Penrose, 99 runlength-limited (RLL)(d, k) shift, 16 topological quantum field theory, 148 translation, 83 shift equivalence, 9 Triangle Identities, 127, 147 Shift Equivalence Problem, 125 two-sided d-shift map, 58 shift equivalent, 124 shift of finite type, 4, 17 uniquely ergodic, 102 Singleton bound, 23 sliding block code, 5, 69 vertex shift, 5 sliding-block decodable, 19 Wang tiles, 65 sliding-block decoder, 19 Wang's Conjecture, 88 sofic shift, 6, 16 Spinning, 45 zeta function, 8 spinning construction, 48 standard concatenation, 25 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
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