Abstract
In this chapter we introduce some further topics which are closely tied to subshifts of finite type. In the first section we look at sofic systems. The continuous symbolic image of a subshift of finite type need not be a subshift of finite type. It may have an unbounded memory. Sofic systems are the symbolic systems that arise as continuous images of subshifts of finite type. There are three equivalent characterizations of these systems. The characterizations are explained and then we investigate some of the basic dynamical properties of sofic systems. The second section contains a discussion of Markov measures. These are the measures which have a finite memory. We define the measures, compute their measure-theoretic entropy, characterize the measures using conditional entropy and then prove that for a fixed subshift of finite type there is a unique Markov measure whose entropy is greater than the entropy of any other measure on the subshift of finite type. The third section investigates symbolic systems that have a group structure. These are the Markov subgroups. We show that any symbolic system with a group structure is a subshift of finite type. Then we classify them up to topological conjugacy. The fourth section contains a very brief introduction to cellular automata. The point is to see how they fit into the framework we have developed. The final section discusses channel codes as illustrated in Example 1.2.8. We describe a class of codes and develope an algorithm to construct them.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. Adler, D. Coppersmith and M. Hassner, Algorithms for Sliding Block Codes, IEEE Transactions on Information Theory no. 29 (1983), 5–22.
M. Boyle, B. Kitchens and B. Marcus, A Note on Minimal Covers for Sofic Systems, Proceedings of the American Mathematical Society 95 (1985), 403–411.
E. Coven and M. Paul, Endomorphisms of Irreducible Subshifts of Finite Type, Mathematical Systems Theory 8 (1974), 167–175.
E. Coven and M. Paul, Sofic Systema, Israel Journal of Mathematics 20 (1975) 165–177.
E. Coven and M. Paul, Finite Procedures for Sofic Systems, Monatshefte für Mathematik 83 (1977), 265–278.
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on Compact Spaces Lecture Notes in Mathematics, 527, Springer-Verlag, (1976)
D. Farmer, T. Toffoli and S. Wolfram, editors, Cellular Automata, North-Holland, 1984.
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd. edition, Wiley, 1968.
R. Fischer, Sofic Systems and Graphs, Monatshefte für Mathematik 80 1975, 179–186.
R. Fischer, Graphs and Symbolic Dynamics, Proceedings of the Colloquia Mathematical Society János Bolyai, Information Theory 16 (1975) 229–244.
P. Franaszek, Sequence State Methods for Run-Length Limited Codes, IBM Journal of Research and Development 14 (1970), 376–383.
D. Fried, Finitely Presented Dynamical Systems, Ergodic Theory and Dynamical Systems 7 (1987), 489–507.
H. Gutowitz, editor, Cellular Automata: Theory and Experiment, MIT Press, 1991.
P. Halmos, On Automorphisms of Compact Groups, Bulletin of the American Mathematical Society 49 (1943), 619–624.
J.E Hopcroft and J.D. Ullman, Frmal Languages and Their Relation to Automata, Addison-Wesley Publishing Co., 1969.
A. Khinchin, Mathematical Foundations of Statistical Mechanics, Dover Publications, 1949.
B. Kitchens, Expansive Dynamics on Zer Dimensional Groups, Ergodic Theory and Dynamical Systems 7 (1987), 249–261.
A. Kolmogorov, A New Metric Invariant for Transient Dynamical Systems, Academiia Nauk SSSR, Doklady 119 (1958), 861–864. (Russian)
W. Krieger, On Sofic Systems I, Israel Journal of Mathematics 48 (1984), 305–330.
W. Krieger, On Sofic Systems II, Israel Journal of Mathematics 60 (1987), 167–176.
A. Manning, Axiom A Diffeomorphisms have Rational Zeta Functions, Bulletin of the London Mathematical Society 3 (1971), 215–220.
M. Nasu, An Invariant for Bounded-to-one Factor Maps Between Transitive Sofic Subshifts, Ergodic Theory and Dynamical Systems 5 (1985), 85–105.
D. Ornstein, Bernoulli Shifts with the Same Entropy are Isomorphic, Advances in Mathematics no. 4 (1970), 337–352.
W. Parry, Intrinsic Markov Chains, Transactions of the American Mathematical Society 112 (1964), 55–66.
K. Petersen, Ergodic Theory, Cambridge Press, 1983.
K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser, 1995.
C. Shannon, A Mathematical Theory of Communication, Bell System Technical Journal (1948), 379–473 and 623-656, reprinted in The Mathematical Theory of Communication by C. Shannon and W. Weaver, University of Illinois Press, 1963.
Y. Sinai, On the Concept of Entropy for a Dynamical System, Academiia Nauk SSSR, Doklady 124 (1959), 768–771. (Russian)
J. von Neumann, Theory of Self-reproducing Automata (A.W. Burks, eds.), Univerity of Illinois Press, 1966.
P. Walters, An Introduction to Ergodic Theory GTM, Springer-Verlag, 1981.
B. Weiss, Sub shifts of Finite Type and Sofic Systems, Monatshefte für Mathematik 77 (1973), 462–474.
S. Williams, Lattice Invariants for Sofic Shifts, Ergodic Theory and Dynamical Systems 11 (1991), 787–801.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kitchens, B.P. (1998). Further Topics. In: Symbolic Dynamics. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58822-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-58822-8_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62738-8
Online ISBN: 978-3-642-58822-8
eBook Packages: Springer Book Archive