Symbolic Dynamics, Group Automorphisms and Markov Partitions

Part of the NATO ASI Series book series (ASIC, volume 464)


These lectures are about symbolic dynamics and the relationship of subshifts of finite type to toral automorphisms and solenoids through topological entropy and Markov partitions. The first part is about subshifts of finite type. Subshifts of finite type have been studied in dynamics and in information theory from a number of different points of view. C. Shannon in 1948 [S] studied them in an information theory context because they model information channels with a finite memory. W. Parry in 1964 [P] investigated them in a topological and ergodic theoretic setting. R. Adler and B. Weiss in 1970 [AW] used them as a combinatorial model for two dimensional toral automorphisms. This allowed a measure theoretic classification of these automorphisms and lead to the use of subshifts of finite type as models for all hyperbolic dynamical systems. In recent years the study of subshifts of finite type has become a subject in its own right. Here, the basic definitions and dynamical properties of these systems are presented. There is also a discussion of topological conjugacy and continuous factor maps. Most of this was known by about ten years ago. Since then there have been new developments but many of the central problems remain unresolved. Some of these problems are discussed here. The next part describes toral automorphisms and solenoids. Toral automorphisms were important in the development of hyperbolic dynamics. An expansive toral automorphism exhibits in a simple, concrete setting the important features of any hyperbolic dynamical system. An expansive toral automorphism has local product structure, its periodic points are dense, its zeta function can be computed, it is structurally stable and it has a kind of algebraic rigidity. In addition, Markov partitions for invertible maps were first constructed on the two dimensional torus. Solenoids are generalizations of toral automorphisms and have all of these properties. The final part of these lectures discusses topological entropy and Markov partitions. Markov partitions tie together the dynamics of subshifts of finite type, toral automorphisms and solenoids. A Markov partition is constructed for a particular two dimensional toral automorphism. It is very easily drawn and exhibits clearly the relationship between subshifts of finite type and toral automorphisms. Finally there is a short general discussion of Markov partitions, their properties and some of their consequences.


Group Automorphism Periodic Point Finite Type Topological Entropy Symbolic Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Ab]
    L. M. Abramov, The entropy of an automorphism of a solenoidal group, Theory Prob. Appl. 4 (1959), 231–236.MathSciNetCrossRefGoogle Scholar
  2. [AKM]
    R. Adler, A. Konheim and H. MacAndrew, Topological entropy, Trans. AMS 114 (1965), 309–319.zbMATHCrossRefGoogle Scholar
  3. [AM]
    R. Adler and B. Marcus, Topological entropy and the equivalence of dynamical systems, Mem. AMS 219 (1979).Google Scholar
  4. [AW]
    R.L. Adler and B. Weiss, Similarity of Autornorphisms of the torus, Memoir Amer. Math. Soc. 98 (1970).Google Scholar
  5. [Av]
    D. Z. Arov, The computation of the entropy for one class of group endomorphisms, Zap. Mekh.-Matem. Fakulteta Kharkov Matem. 30 (1964), 48–69.MathSciNetGoogle Scholar
  6. [As]
    J. Ashley, Bounded-to-1 factors of an aperiodic shift of finite type are 1-to-1 almost every-where factors also, Ergod. Th. and Dynam. Sys. 10 (1990), 615–625.MathSciNetzbMATHGoogle Scholar
  7. [Be]
    K. Berg, On the Conjugacy Problem for K-systems, Ph.D. Thesis, U. of Minn. (1967).Google Scholar
  8. [Bol]
    R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401–414.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Bo2]
    R. Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 91 (1970), 725–747.MathSciNetCrossRefGoogle Scholar
  10. [BL]
    R. Bowen and O. Lanford, Zeta functions of restrictions of the shift transformation, Proc. Sympos. Pure Math. 14 (1970), Amer. Math. Soc..Google Scholar
  11. [By]
    M. Boyle, Lower entropy factors of sofic systems, Ergod. Th. and Dynam. Sys. 4 (1984), 541–557.MathSciNetCrossRefGoogle Scholar
  12. [BMT]
    M. Boyle, B. Marcus and P. Trow, Resolving maps and the dimension group for shifts of finite type, Memoir Amer. Math. Soc. 377 (1987).Google Scholar
  13. [CP]
    E.M. Coven and M.E. Paul, Endomorphisms of irreducible subshifts of finite type, Math Systems Theory 8 (1974), 167–175.MathSciNetCrossRefGoogle Scholar
  14. [F]
    D. Fried, Finitely presented dynamical systems, Ergod. Th. and Dynam. Sys. 7 (1987), 489–507.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [G]
    W. F. Gantmacher, The Theory of Matrices. Vol. 1, Chelsea Publishing Co., 1959.Google Scholar
  16. [H]
    G.A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math System Theory 3 (1969), 320–375.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [K]
    B. Kitchens, An invariant for continuous factors of Markov shifts, Proc. Amer. Math. Soc. 83 (1981), 825–828.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [KMT]
    B. Kitchens, B. Marcus and P. Trow, Eventual factor maps and compositions of closing maps, Ergod. Th. and Dynam. Sys. 11 (1991), 85–113.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [L]
    W. Lawton, The structure of compact connected groups which admit an expansive automorphism, Recent Advances in Topological. Dynamics, Lecture Notes in Mathematics, Springer, 1973, pp. 182–196.Google Scholar
  20. [P]
    W. Parry, Intrinsic Markov Chains, Trans. AMS 112 (1964), 55–66.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [R]
    W. Reddy, The existence of expansive homeomorphisms on manifolds, Duke J. Math. 32 (1965), 494–509.Google Scholar
  22. [Se]
    E. Seneta, Non-negative Matrices and Markov Chains, Springer-Verlag, 1973 and 1981.Google Scholar
  23. [S]
    C. Shannon, A mathematical theory of communication, Bell Sys. Tech J. (1948), 379–423, 623–656.MathSciNetGoogle Scholar
  24. [Si]
    Ya. G. Sinai, Markov partitions and C-diffeomorphisms, Func. Anal. and its Appl. 2 (1968), 64–89.MathSciNetGoogle Scholar
  25. [W]
    R. F. Williams, Classification of subshifts of finite type, Annals Math. 98 (1973), 120–153;zbMATHCrossRefGoogle Scholar
  26. [W]
    R. F. Williams, Errata 99 (1974), 380–381.Google Scholar
  27. [Y]
    S. A. Yuzvinskii, Computing the entropy of a group of endomorphisms, Siberian Math. J. 8 (1967), 172–178.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Mathematical Sciences DepartmentIBM Watson Research CenterYorktown HeightsUSA

Personalised recommendations