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AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures
published as a subseries of Proceedings of Symposia in Applied Mathematics
http://dx.doi.org/10.1090/psapm/060
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
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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
T h e AMS Short Course Series is sponsored by the Society's P rog ram Commit tee for Nat ional Meetings. T h e series is under the direction of the Short Course
Subcommit tee of the P rog ram Commi t t ee for Nat ional Meetings.
2000 Mathematics Subject Classification. P r imary 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 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
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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, J R .
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 coordinated the short course and worked behind the scenes to make it run smoothly.
Susan G. Williams
Index
(n, M) block code, 21 \n,M,d) codeC, 21 K^R), 141 K2(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 Zd-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
156
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
INDEX
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
Titles in This Series
60 Susan G. Wil l iams, 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 1943-1993: 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 D e W i t t 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 t i t les in this series, visit t he AMS Bookstore at w w w . a m s . o r g / b o o k s t o r e / .
ISBN 0-6215-3157-7
9 ' i780821»831571
