Skip to main content
SpringerLink
Log in

Symbolic Dynamics

  • Reference work entry
Encyclopedia of Complexity and Systems Science

  • 540 Accesses

Definition of the Subject

Symbolic dynamics is the study of shift spaces, which consist of infinite or bi‐infinite sequences defined by a shift‐invariantconstraint on the finite‐length sub-words. Mappings between two such spaces can be regarded as codes or encodings. Shift spaces are classified, upto various kinds of invertible encodings, by combinatorial, algebraic, topological and measure‐theoretic invariants.

The subject is intimately related to many other areas of research, including dynamical systems, ergodic theory, automata theory and informationtheory. Shift spaces and their associated shift mappings are used to model a rich and important class of smooth dynamical systems and ergodicmeasure‐preserving transformations. These models have provided a valuable tool for classifying and understanding fundamental properties ofdynamical systems. In addition, techniques from symbolic dynamics have had profound applications for data recording applications, such as algorithms andanalysis...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Abbreviations

Glossary:

In this glossary, we give only brief descriptions of key terms. We refer to specific sections in the text for more precise definitions.

Almost conjugacy:

(Sect. “Other Coding Problems”) A common extension of two shift spaces given by factor codes that are one-to-one almost everywhere.

Automorphism:

(Sect. “The Conjugacy Problem”) An invertible sliding block code from a shift space to itself; equivalently, a shift‐commuting homeomorphism from a shift space to itself; equivalently, a topological conjugacy from a shift space to itself.

Dimension group:

(Sect. “The Conjugacy Problem”) A particular group associated to a shift of finite type. This group, together with a distinguished sub‐semigroup and an automorphism, captures many invariants of topological conjugacy for shifts of finite type.

Embedding:

(Sect. “Shift Spaces and Sliding Block Codes”) A one-to-one sliding block code from one shift space to another; equivalently, a one-to-one continuous shift‐commuting mapping from one shift space to another.

Factor map:

(Sect. “Shift Spaces and Sliding Block Codes”) An onto sliding block code from one shift space to another; equivalently, an onto continuous shift‐commuting mapping from one shift space to another. Sometimes called Factor Code.

Finite equivalence:

(Sect. “Other Coding Problems”) A common extension of two shift spaces given by finite-to-one factor codes.

Full shift:

(Sect. “Shift Spaces and Sliding Block Codes”) The set of all bi‐infinite sequences over an alphabet (together with the shift mapping). Typically, the alphabet is finite.

Higher dimensional shift space:

(Sect. “Higher Dimensional Shift Spaces”) A set of bi‐infinite arrays of a given dimension, determined by a collection of finite forbidden arrays. Typically, the alphabet is finite.

Markov partition:

(Sect. “Origins of Symbolic Dynamics: Modeling of Dynamical Systems”) A finite cover of the underlying phase space of a dynamical system, which allows the system to be modeled by a shift of finite type. The elements of the cover are closed sets, which are allowed to intersect only on their boundaries.

Measure of maximal entropy:

(Sect. “Connections with Information Theory and Ergodic Theory”) A shift‐invariant measure of maximal measure‐theoretic entropy on a shift space. Its measure‐theoretic entropy coincides with the topological entropy of the shift space.

Road problem:

(Sect. “Other Coding Problems”) A recently‐solved classical problem in symbolic dynamics, graph theory and automata theory.

Run‐length limited shift:

(Sect. “Coding for Data Recording Channels”) The set of all bi‐infinite binary sequences whose runs of zeros, between two successive ones, are bounded below and above by specific numbers.

Shift equivalence:

(Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. This relation characterizes the corresponding shifts of finite type, up to an eventual notion of topological conjugacy.

Shift space:

(Sect. “Shift Spaces and Sliding Block Codes”) A set of bi‐infinite sequences determined by a collection of finite forbidden words; equivalently, a closed shift‐invariant subset of a full shift.

Shift of finite type:

(Sect. “Shifts of Finite Type and Sofic Shifts”) A set of bi‐infinite sequences determined by a finite collection of finite forbidden words.

Sliding block code:

(Sect. “Shift Spaces and Sliding Block Codes”) A mapping from one shift space to another determined by a finite sliding block window; equivalently, a continuous shift‐commuting mapping from one shift space to another.

Sofic shift:

(Sect. “Shifts of Finite Type and Sofic Shifts”) A shift space which is a factor of a shift of finite type; equivalently, a set of bi‐infinite sequences determined by a finite directed labeled graph.

State splitting:

(Sect. “The Conjugacy Problem”) A splitting of states in a finite directed graph that creates a new graph, whose vertices are the split states. The operation that creates the new graph from the original graph is a basic building block for all topological conjugacies between shifts of finite type.

Strong shift equivalence:

(Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. In principle, this relation characterizes the corresponding shifts of finite type, up to topological conjugacy.

Topological conjugacy:

(Sect. “Shift Spaces and Sliding Block Codes”) A bijective sliding block code from one shift space to another; equivalently, a shift‐commuting homeomorphism from one shift space to another. Sometimes called conjugacy.

Topological entropy:

(Sect. “Entropy and Periodic Points”) The asymptotic growth rate of the number of finite sequences of given length in a shift space (as the length goes to infinity).

Zeta function:

(Sect. “Entropy and Periodic Points”) An expression for the number of periodic points of each given period in a shift space.

Bibliography

  1. Adler RL, Coppersmith D, Hassner M (1983) Algorithms for sliding blockcodes – an application, of symbolic dynamics to information theory. Trans IEEE Inf Theory 29:5–22

    MathSciNet  Google Scholar 

  2. Adler RL, Goodwyn LW, Weiss B (1977) Equivalence of topological Markovshifts. Isr J Math 27:48–63

    MathSciNet  Google Scholar 

  3. Adler R, Konheim A, McAndrew M (1965) Topological entropy. Trans Amer Math Soc114:309–319

    MathSciNet  Google Scholar 

  4. Adler R, Marcus B (1979) Topological entropy and equivalence of dynamicalsystems. Mem Amer Math Soc 219. AMS, Providence

    Google Scholar 

  5. Adler R, Weiss B (1970) Similarity of automorphisms of the torus. Mem Amer MathSoc 98. AMS, Providence

    Google Scholar 

  6. Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computeralgorithms. Addison‐Wesley, Reading

    Google Scholar 

  7. Ashley J (1988) A linear bound for sliding block decoder window size. TransIEEE Inf Theory 34:389–399

    MathSciNet  Google Scholar 

  8. Ashley J (1996) A linear bound for sliding block decoder window size,II. Trans IEEE Inf Theory 42:1913–1924

    MathSciNet  Google Scholar 

  9. Ashley J (1991) Resolving factor maps for shifts of finite type with equalentropy. Ergod Theory Dynam Syst 11:219–240

    MathSciNet  Google Scholar 

  10. Béal M-P (1990) The method of poles: a coding method for constrainedchannels. Trans IEEE Inf Theory 36:763–772

    Google Scholar 

  11. Béal M-P (1993) Codage symbolique. Masson, Paris

    Google Scholar 

  12. Béal M-P (2003) Extensions of the method of poles for code construction. TransIEEE Inf Theory 49:1516–1523

    Google Scholar 

  13. Bedford T (1986) Generating special Markov partitions for hyperbolic toralautomorphisms using fractals. Ergod Theory Dynam Syst 6:325–333

    MathSciNet  Google Scholar 

  14. Berger R (1966) The undecidability of the Domino problem. Mem Amer MathSoc. AMS, Providence

    Google Scholar 

  15. BerstelJ, Perrin D (1985) Theory of codes. Academic Press, New York

    Google Scholar 

  16. Blanchard P, Devaney R, Keen L (2004) Complex dynamics and symbolicdynamics. In: Williams S (ed) Symbolic dynamics and its applications. Proc Symp Appl Math. AMS, Providence,pp 37–59

    Google Scholar 

  17. Blanchard F, Maass A, Nogueira A (2000) Topics in symbolic dynamics andapplications. In: Blanchard F, Maass A, Nogueira A (eds) LMS Lecture Notes, vol 279. Cambridge University Press,Cambridge

    Google Scholar 

  18. Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Amer JMath 92:725–747

    MathSciNet  ADS  Google Scholar 

  19. Bowen R (1973) Symbolic dynamics for hyperbolic flows. Amer J Math95:429–460

    MathSciNet  Google Scholar 

  20. Bowen R, Franks J (1977) Homology for zero‐dimensional basic sets. AnnMath 106:73–92

    MathSciNet  Google Scholar 

  21. Bowen R, Lanford OE (1970) Zeta functions of restrictions of the shifttransformation. Proc Symp Pure Math AMS 14:43–50

    MathSciNet  Google Scholar 

  22. Boyle M (1983) Lower entropy factors of sofic systems. Ergod Theory Dynam Syst3:541–557

    MathSciNet  Google Scholar 

  23. Boyle M (1993) Symbolic dynamics and matrices. In: Brualdi R et al (eds)Combinatorial and graph theoretic problems in linear algebra. IMA Vol Math Appl 50:1–38

    MathSciNet  Google Scholar 

  24. Boyle M (2007) Open problems in symbolic dynamics. Contemponary Math (toappear)

    Google Scholar 

  25. Boyle M, Handelman D (1991) The spectra of nonnegative matrices via symbolicdynamics. Ann Math 133:249–316

    MathSciNet  Google Scholar 

  26. Boyle M, Krieger W (1986) Almost Markov and shift equivalent soficsystems. In: Aleixander J (ed) Proceedings of Maryland Special Year in Dynamics 1986–87. Lecture Notes in Math, vol 1342. Springer,Berlin, pp 33–93

    Google Scholar 

  27. Boyle M, Marcus BH, Trow P (1987) Resolving maps and the dimension group forshifts of finite type. Mem Amer Math Soc 377. AMS, Providence

    Google Scholar 

  28. Boyle M, Pavlov R, Schraudner M (2008) preprint

    Google Scholar 

  29. Burton R, Steif J (1994) Nonuniqueness of measures of maximal entropy forsubshifts of finite type. Ergod Theory Dynam Syst 14(2):213–235

    MathSciNet  Google Scholar 

  30. Calkin N, Wilf H (1998) The number of independent sets in a gridgraph. SIAM J Discret Math 11:54–60

    MathSciNet  Google Scholar 

  31. Cidecyian R, Evangelos E, Marcus B, Modha M (2001) Maximum transition runcodes for generalized partial response channels. IEEE J Sel Area Commun 19:619–634

    Google Scholar 

  32. Coven E, Paul M (1974) Endomorphisms of irreducible shifts of finitetype. Math Syst Theory 8:167–175

    MathSciNet  Google Scholar 

  33. Coven E, Paul M (1975) Sofic systems. Isr J Math20:165–177

    MathSciNet  Google Scholar 

  34. Coven E, Paul M (1977) Finite procedures for sofic systems. Monats Math83:265–278

    MathSciNet  Google Scholar 

  35. Cover T, Thomas J (1991) Elements of information theory. Wiley, NewYork

    Google Scholar 

  36. Desai A (2006) Subsystem entropy for Z d sofic shifts. Indag Math17:353–360

    MathSciNet  Google Scholar 

  37. Desai A (2008) A class of Z d-subshifts which factor onto lower entropy full shifts. Proc AmerMath Soc, to appear

    Google Scholar 

  38. Devaney R (1987) An introduction to chaotic dynamicalsystems. Addison‐Wesley, Reading

    Google Scholar 

  39. Fischer R (1975) Sofic systems and graphs. Monats Math80:179–186

    Google Scholar 

  40. Fischer R (1975) Graphs and symbolic dynamics. Colloq Math Soc János Bólyai:Top Inf Theory 16:229–243

    Google Scholar 

  41. Franaszek PA (1968) Sequence‐state coding for digital transmission. BellSyst Tech J 47:143–155

    Google Scholar 

  42. Franaszek PA (1982) Construction of bounded delay codes for discrete noiselesschannels. J IBM Res Dev 26:506–514

    MathSciNet  Google Scholar 

  43. Franaszek PA (1989) Coding for constrained channels: a comparison of twoapproaches. J IBM Res Dev 33:602–607

    Google Scholar 

  44. Friedman J (1990) On the road coloring problem. Proc Amer Math Soc110:1133–1135

    MathSciNet  Google Scholar 

  45. Gomez R (2003) Positive K‑theory for finitary isomoprhisms of Markovchains. Ergod Theory Dynam Syst 23:1485–1504

    Google Scholar 

  46. Hadamard J (1898) Les surfaces a courbures opposées et leurs lignesgeodesiques. J Math Pure Appl 4:27–73

    Google Scholar 

  47. Hassellblatt B, Katok A (1995) Introduction to the modern theory of dynamicalsystems. Cambridge University Press, Cambridge

    Google Scholar 

  48. Hedlund GA (1939) The dynamics of geodesic flows. Bull Amer Math Soc45:241–260

    MathSciNet  Google Scholar 

  49. Hedlund GA (1944) Sturmian minimal sets. Amer J Math66:605–620

    MathSciNet  Google Scholar 

  50. Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamicalsystem. Math Syst Theory 3:320–375

    MathSciNet  Google Scholar 

  51. Hochman M, Meyerovitch T (2007) A characterization of the entropies ofmultidimensional shifts of finite type. Ann Math, to appear

    Google Scholar 

  52. Hochman M (2007) On the dynamics and recursive properties of multidimensionalsymbolic systems. Preprint

    Google Scholar 

  53. Hollmann HDL (1995) On the construction of bounded‐delay encodable codesfor constrained systems. Trans IEEE Inf Theory 41:1354–1378

    MathSciNet  ADS  Google Scholar 

  54. Immink KAS (2004) Codes for mass data storage, 2nd edn. Shannon FoundationPress, Eindhoven

    Google Scholar 

  55. Johnson A, Madden K (2005) Factoring higher‐dimensional shifts of finitetype onto full shifts. Ergod Theory Dynam Syst 25:811–822

    MathSciNet  Google Scholar 

  56. Karabed R, Siegel P, Soljanin E (1999) Constrained coding for binary channelswith high intersymbol intereference. Trans IEEE Inf Theory 45:1777–1797

    MathSciNet  Google Scholar 

  57. Kari J (2001) Synchronizing finite automata on Eulerian digraphs. SpringerLect Notes Comput Sci 2136:432–438

    Google Scholar 

  58. Kastelyn PW (1961) The statistics of dimers on a lattice. Physica A27:1209–1225

    Google Scholar 

  59. Kenyon R (2008) Lectures ondimers. http://www.math.brown.edu/%7Erkenyon/papers/dimerlecturenotes.pdf

  60. Kim KH, Roush FW (1979) Some results on decidability of shiftequivalence. J Comb Inf Syst Sci 4:123–146

    MathSciNet  Google Scholar 

  61. Kim KH, Roush FW (1988) Decidability of shift equivalence. In: Alexander J(ed) Proceedings of Maryland special year in dynamics 1986–87. Lecture Notes in Math, vol 1342. Springer, Berlin, pp 374–424

    Google Scholar 

  62. Kim KH, Roush FW (1990) An algorithm for sofic shift equivalence. Ergod TheoryDynam Syst 10:381–393

    MathSciNet  Google Scholar 

  63. Kim KH, Roush FW (1999) Williams conjecture is false for irreduciblesubshifts. Ann Math 149:545–558

    MathSciNet  Google Scholar 

  64. Kim KH, Ormes N, Roush F (2000) The spectra of nonnegative integer matricesvia formal power series. Amer J Math Soc 13:773–806

    MathSciNet  Google Scholar 

  65. Kitchens B (1998) Symbolic dynamics: One-sided, two-sided and countable stateMarkov chains. Springer, Berlin

    Google Scholar 

  66. Kitchens B, Marcus B, Trow P (1991) Eventual factor maps and compositions ofclosing maps. Ergod Theory Dynam Syst 11:85–113

    MathSciNet  Google Scholar 

  67. Kitchens B, Schmidt K (1988) Periodic points, decidability and Markovsubgroups, dynamical systems. In: Alexander JC (ed) Proceedings of the special year. Springer Lect Notes Math1342:440–454

    MathSciNet  Google Scholar 

  68. Krieger W (1980) On a dimension for a class of homeomorphism groups. Math Ann 252:87–95

    MathSciNet  Google Scholar 

  69. Krieger W (1980) On dimension functions and topological Markov chains. InventMath 56:239–250

    MathSciNet  ADS  Google Scholar 

  70. Krieger W (1982) On the subsystems of topological Markov chains. Ergod TheoryDynam Syst 2:195–202

    MathSciNet  Google Scholar 

  71. Krieger W (1983) On the finitary isomorphisms of Markov shifts that havefinite expected coding time. Wahrscheinlichkeitstheorie Z 65:323–328

    MathSciNet  Google Scholar 

  72. Krieger W (1984) On sofic systems I. Isr J Math48:305–330

    MathSciNet  Google Scholar 

  73. Lightwood S (2003/04) Morphisms form non‐periodic Z 2 subshifts I and II. Ergod Theory Dynam Syst 23:587–609, 24:1227–1260

    Google Scholar 

  74. Lind D (1984) The entropies of topological Markov shifts and a relatedclass of algebraic integers. Ergod Theory Dynam Syst 4:283–300

    MathSciNet  Google Scholar 

  75. Lind D (1989) Perturbations of shifts of finite type. SIAM J Discret Math2:350–365

    MathSciNet  Google Scholar 

  76. Lind D (2004) Multi‐dimensional symbolic dynamics. In: Williams S (ed)Symbolic dynamics and its applications. Proc Symp Appl Math 60:81–120

    MathSciNet  Google Scholar 

  77. Lind D (1996) A zeta function for Z d-actions. In: Pollicott M, Schmidt K (eds) Proceedings of WarwickSymposium on Z d-actions. LMS Lecture Notes,vol 228. Cambridge University Press, Cambridge, pp 433–450

    Google Scholar 

  78. Lind D, Marcus B (1995) An introduction to symbolic dynamics andcoding. Cambridge University Press, Cambridge

    Google Scholar 

  79. Lind D, Schmidt K (2002) Symbolic and algebraic dynamical systems. In:Hasselblatt B, Katok A (eds) Handbook of Dynamics Systems. Elsevier, Amsterdam, pp 765–812

    Google Scholar 

  80. Manning A (1971) Axiom A diffeomorphisms have rational zetafunctions. Bull Lond Math Soc 3:215–220

    MathSciNet  Google Scholar 

  81. Marcus B (1979) Factors and extensions of full shifts. Monats Math88:239–247

    Google Scholar 

  82. Marcus BH, Roth RM (1991) Bounds on the number of states in encoder graphs forinput‐constrained channels. Trans IEEE Inf Theory 37:742–758

    MathSciNet  Google Scholar 

  83. Marcus BH, Roth RM, Siegel PH (1998) Constrained systems and coding for recording chapter. In: Brualdi R, Huffman C, Pless V (eds) Handbook on coding theory. Elsevier, New York; updated version at http://www.math.ubc.ca/%7Emarcus/Handbook/

  84. Marcus B, Tuncel S (1990) Entropy at a weight‐per‐symbol andembeddings of Markov chains. Invent Math 102:235–266

    MathSciNet  ADS  Google Scholar 

  85. Marcus B, Tuncel S (1991) The weight‐per‐symbol polytope andscaffolds of invariants associated with Markov chains. Ergod Theory Dynam Syst 11:129–180

    MathSciNet  Google Scholar 

  86. Marcus B, Tuncel S (1993) Matrices of polynomials, positivity, and finiteequivalence of Markov chains. J Amer Math Soc 6:131–147

    MathSciNet  Google Scholar 

  87. Markley N, Paul M (1981) Matrix subshifts for \( { \mathbf{Z}^\nu } \) symbolic dynamics. Proc Lond Math Soc43:251–272

    MathSciNet  Google Scholar 

  88. Markley N, Paul M (1981) Maximal measures and entropyfor \( { \mathbf{Z}^\nu } \) subshifts offinite type. In: Devaney R, Nitecki Z (eds) Classical mechanics and dynamical systems. Dekker Notes 70:135–157

    MathSciNet  Google Scholar 

  89. Meester R, Steif J (2001) Higher‐dimensional subshifts of finite type,factor maps and measures of maximal entropy. Pac Math J 200:497–510

    MathSciNet  Google Scholar 

  90. Mouat R, Tuncel S (2002) Constructing finitary isomorphisms with finiteexpected coding time. Isr J Math 132:359–372

    MathSciNet  Google Scholar 

  91. Morse M (1921) Recurrent geodesics on a surface of negativecurvature. Trans Amer Math Soc 22:84–100

    MathSciNet  Google Scholar 

  92. Morse M, Hedlund GA (1938) Symbolic dynamics. Amer J Math60:815–866

    MathSciNet  Google Scholar 

  93. Morse M, Hedlund GA (1940) Symbolic dynamics II, Sturmiantrajectories. Amer J Math 62:1–42

    MathSciNet  Google Scholar 

  94. Mozes S (1989) Tilings, substitutions and the dynamical systems generated bythem. J Anal Math 53:139–186

    MathSciNet  Google Scholar 

  95. Mozes S (1992) A zero entropy, mixing of all orders tiling system. In:Walters P (ed) Symbolic dynamics and its applications. Contemp Math 135:319–326

    MathSciNet  Google Scholar 

  96. Nagy Z, Zeger K (2000) Capacity bounds for the three‐dimensional(0, 1) run length limited channel. Trans IEEE Inf Theory 46:1030–1033

    Google Scholar 

  97. Nasu M (1986) Topological conjugacy for sofic systems. Ergod Theory Dynam Syst6:265–280

    MathSciNet  Google Scholar 

  98. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. AdvMath 4:337–352

    MathSciNet  Google Scholar 

  99. Parry W (1964) Intrinsic Markov chains. Trans Amer Math Soc112:55–66

    MathSciNet  Google Scholar 

  100. Parry W (1977) A finitary classification of topological Markov chainsand sofic systems. Bull Lond Math Soc 9:86–92

    MathSciNet  Google Scholar 

  101. Parry W (1979) Finitary isomorphisms with finite expectedcode‐lengths. Bull Lond Math Soc 11:170–176

    MathSciNet  Google Scholar 

  102. Parry W (1991) Notes on coding problems for finite state processes. BullLond Math Soc 23:1–33

    MathSciNet  Google Scholar 

  103. Parry W, Schmidt K (1984) Natural coefficients and invariants for Markovshifts. Invent Math 76:15–32

    MathSciNet  ADS  Google Scholar 

  104. Parry W, Tuncel S (1981) On the classification of Markov chains by finiteequivalence. Ergod Theory Dynam Syst 1:303–335

    MathSciNet  Google Scholar 

  105. Parry W, Tuncel S (1982) Classification problems in ergodic theory. In:LMS Lecture Notes, vol 67. Cambridge University Press, Cambridge

    Google Scholar 

  106. Pavlov R (2007) Perturbations of multi‐dimensional shifts of finitetype. Preprint

    Google Scholar 

  107. Petersen K (1989) Ergodic theory. Cambridge University Press,Cambridge

    Google Scholar 

  108. Quas A, Sahin A (2003) Entropy gaps and locally maximal entropy in Z d-subshifts. Ergod Theory Dynam Syst 23:1227–1245

    MathSciNet  Google Scholar 

  109. Quas A, Trow P (2000) Subshifts of multidimensional shifts of finitetype. Ergod Theory Dynam Syst 20:859–874

    MathSciNet  Google Scholar 

  110. Radin C (1996) Miles of tiles. In: Pollicott M, Schmidt K (eds) ErgodicTheory of Z d-actions. LMS Lecture Notes,vol 228. Cambridge University Press, Cambridge, pp 237–258

    Google Scholar 

  111. Robinson RM (1971) Undecidability and nonperiodicity for tilings of theplane. Invent Math 12:177–209

    MathSciNet  ADS  Google Scholar 

  112. Robinson EA (2004) Symbolic dynamics and tilings of \( { \mathbf{R}^d } \). In: Williams S (ed) Symbolic dynamicsand its applications. Proc Symp Appl Math 60:81–120

    Google Scholar 

  113. Rudolph D (1990) Fundamentals of measurable dynamics. Oxford UniversityPress, Oxford

    Google Scholar 

  114. Schmidt K (1984) Invariants for finitary isomorphisms with finite expectedcode lengths. Invent Math 76:33–40

    MathSciNet  ADS  Google Scholar 

  115. Schmidt K (1990) Algebraic ideas in ergodic theory. AMS-CBMS Reg Conf76

    Google Scholar 

  116. Schmidt K (1995) Dynamical systems of algebraic origin. Birkhauser, Basel

    Google Scholar 

  117. Schwartz M, Bruck S (2008) Constrained codeds as networks of relations. IEEETrans Inf Theory 54:2179–2195

    MathSciNet  Google Scholar 

  118. Seneta E (1980) Non‐negative matrices and Markov chains, 2ndedn. Springer, Berlin

    Google Scholar 

  119. Shannon C (1948) A mathematical theory of communication. Bell SystTech J 27:379–423,623–656

    MathSciNet  Google Scholar 

  120. Sinai YG (1968) Markov partitions and C‑diffeomorphisms. Funct AnalAppl 2:64–89

    MathSciNet  Google Scholar 

  121. Smale S (1967) Differentiable dynamical systems. Bull Amer Math Soc73:747–817

    MathSciNet  Google Scholar 

  122. Trachtman A (2007) The road coloring problem. Israel J Math, toappear

    Google Scholar 

  123. Tuncel S (1981) Conditional pressure and coding. Isr J Math39:101–112

    MathSciNet  Google Scholar 

  124. Tuncel S (1983) A dimension, dimension modules and Markov chains. ProcLond Math Soc 46:100–116

    MathSciNet  Google Scholar 

  125. Wagoner J (1992) Classification of subshifts of finite type revisited. In:Walters P (ed) Symbolic dynamics and its applications. Contemp Math 135:423–444

    MathSciNet  Google Scholar 

  126. Wagoner J (2004) Strong shift equivalence theory. In: Walters P (ed)Symbolic dynamics and its applications. Proc Symp Appl Math 60:121–154

    MathSciNet  Google Scholar 

  127. Walters P (1982) An introduction to ergodic theory. Springer Grad Text Math79. Springer, Berlin

    Google Scholar 

  128. Walters P (1992) Symbolic dynamics and its applications. In: Walter P (ed)Contemp Math 135. AMS, Providence

    Google Scholar 

  129. Ward T (1994) Automorphisms of Z d-subshifts of finite type. Indag Math5:495–504

    MathSciNet  Google Scholar 

  130. Weiss B (1973) Subshifts of finite type and sofic systems. Monats Math77:462–474

    Google Scholar 

  131. Williams RF (1973/74) Classification of subshifts of finite type. Ann Math98:120–153; Erratum: Ann Math 99:380–381

    Google Scholar 

  132. Williams S (2004) Introduction to symbolic dynamics. In: Williams S (ed)Symbolic dynamics and its applications. Proc Symp Appl Math 60:1–12

    Google Scholar 

  133. Williams S (2004) Symbolic dynamics and its applications. In: Williams S (ed) Proc Symp ApplMath 60. AMS, Providence

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, Canada

    Brian Marcus

Authors
  1. Brian Marcus
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

Marcus, B. (2009). Symbolic Dynamics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_531

Download citation

Publish with us

Policies and ethics

Search

Navigation