Abstract
We will be working with sequence spaces. There will be two types of sequence spaces. One is composed of one-sided infinite sequences and the other is composed of two-sided infinite sequences. These are metric spaces where two one-sided sequences are said to be close together if they agree for a long time at the beginning and two two-sided sequences are said to be close together if they agree for a long time around the center. In the first part of the book the sequences will have entries coming from a finite set and in the second part the entries will come from a countable set. When the entries are from a finite set the sequence space is usually homeomorphic to the standard Cantor set. The transformation we study will be the shift transformation which shifts each sequence once to the left. The richness of the dynamics does not arise from the topology of the space or the definition of the transformation but from the definition of which sequences belong to each space. We study subshifts of finite type. They are the sequence spaces characterized by having a finite rule which determines the sequences belonging to each space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Adler, A. Konheim and M.H. MacAndrew, Topological Entropy, Transactions of the American Mathematical Society no. 114 (1965).
R. Adler and B. Weiss, Entropy, a Complete Metric Invariant for Automorphisms of the Torus, Proceedings of the National Academy of Sciences, USA no. 57 (1967), 1573–1576.
R. Adler and B. Weiss, Similarity of Automorphisms of the Torus, Memoirs of the American Mathematical Society no. 98 (1970).
M. Artin and B. Mazur, On Periodic Points, Annals of Mathematics 81 (1965).
K. Berg, On the Conjugacy Problem for K-systems, Ph. D Thesis, University of Minnesota, 1967.
R. Bowen, Markov Partitions for Axiom A Diffeomorphisms, American Journal of Mathematics 92 (1970) 725–747.
R. Bowen and O. Lanford, Zeta Functions of Restrictions of the Shift Transformation, in Global Analysis, Proceedings of Symposia in Pure and Applied Math (S-S. Chern and S. Smale, eds.), vol. XIV, American Mathematical Society, 1970, pp. 43–49.
P. Fatou, Sur les Equations Fonctionnelles, Bulletin de la Société Mathématique de France 47 (1919), 161–247.
W. Feller, An Introduction to Probability Theory and Its Applications, 3rd. edition, Wiley, 1968.
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.
W. Gantmacher, The Theory of Matrices, vol. 1, Chelsea Publishing Co., 1959.
J. Hadamard, Les surfaces à courbures opposées et leurs lignes géodésiques, Journal de Mathématiques Pures et Appliquées 4 (1898), 27–73.
G.A. Hedlund, Transformations Commuting with the Shift, Topological Dynamics (J. Auslander and W. Gottschalk, eds.), W.A. Benjamin, 1968.
G.A. Hedlund, Endomorphisms and Automorphisms of the Shift Dynamical System, Mathematical Systems Theory 3 no. 4 (1969), 320–375.
G. Julia, Iteration des Applications Fonctionnelles, Journal de Mathématiques Pures et Appliquées 8 (1918), 47–245, reprinted in Oeuvres de Gaston Julia, Gauthier-Villars, volume I, 121-319.
A. Kolmogorov, A New Metric Invariant for Transient Dynamical Systems, Academiia Nauk SSSR, Doklady 119 (1958), 861–864. (Russian)
D. Lind, Entropies and Factorizations of Topological Markov Shifts, Bulletin of the American Mathematical Society 9 (1983), 219–222.
D. Lind, The Entropies of Topological Markov Shifts and a Related Class of Algebraic Integers, Ergodic Theory and Dynamical Systems 4 (1984), 283–300.
M. Morse and G.A. Hedlund, Symbolic Dynamics, American Journal of Mathematics 60 (1938), 815–866.
W. Parry, Intrinsic Markov Chains, Transactions of the American Mathematical Society 112 (1964), 55–66.
W. Parry, Symbolic Dynamics and Transformations of the Unit Interval, Transactions of the American Mathematical Society 122 (1966), 368–378.
E. Seneta, Non-negative Matrices and Markov Chains, Springer-Verlag, 1981.
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, Construction of Markov Partitions, Funkcional’nyi Analiz i Ego Prilozheniya 2 no. 3(1968), 70–80 (Russian); English transl. in Functional Analysis and Its Applications 2 (1968), 245-253.
Y. Sinai, On the Concept of Entropy for a Dynamical System, Academiia Nauk SSSR, Doklady 124 (1959), 768–771. (Russian)
S. Smale, Diffeomorphisms with Many Periodic Points, Differential and Combinatorial Topology, Princeton University Press, 1965, pp. 63–80.
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). Background and Basics. In: Symbolic Dynamics. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58822-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-58822-8_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62738-8
Online ISBN: 978-3-642-58822-8
eBook Packages: Springer Book Archive