Topological Conjugacy

  • Bruce P. Kitchens
Part of the Universitext book series (UTX)


In this chapter we investigate the problem of determining when two subshifts of finite type are the “same”. Two subshifts of finite type are dynamically the same if there is a homeomorphism between them which commutes with the shifts. We examine this problem for both one and two-sided subshifts of finite type. In the one-sided setting we develop a simple algorithm which allows us to determine when two one-sided subshifts of finite type are the same. In the two-sided setting we will see that it is not known whether such an algorithm can exist.


Transition Matrix Finite Type Transition Matrice Inverse Limit Dimension Group 
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  1. [Ba]
    K. Baker, Strong Shift Equivalence of 2 × 2 matrices of Non-negative Integers, Ergodic Theory and Dynamical Systems 3 (1983), 501–508.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [BF]
    R. Bowen and J. Franks, Homology for Zero-dimensional Nonwandering Sets, Annals of Mathematics 106 (1977), 73–92.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [CK1]
    J. Cuntz and W Krieger, A Class of C*-Algebras and Topological Markov Chains, Inventiones Mathematicae 56 (1980), 251–268.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [CK2]
    J. Cuntz and W. Krieger, Topological Markov Chains with Dicyclic Dimension Groups, Journal für die reine und angewandte Mathematik 320 (1980), 44–50.MathSciNetzbMATHGoogle Scholar
  5. [Fk]
    J. Franks, Flow Equivalence of Sub shifts of Finite Type, Ergodic Theory and Dynamical Systems 4 (1984), 53–66.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [HR]
    E. Hewitt and K. Ross, Abstract Harmonic Analysis, Academic Press and Springer-Verlag, 1963.Google Scholar
  7. [KR1]
    K.H. Kim and F. Roush, Decidability of Shift Equivalence, Dynamical Systems: Proceedings, University of Maryland 1886–87 (J.C. Alexander, ed.), Springer-Verlag, 1988, pp. 374–424.Google Scholar
  8. [KR2]
    K.H. Kim and F. Roush, William’s Conjecture is False for Reducible Subshifts, Journal of the American Mathematical Society 5 (1992), 213–215.MathSciNetzbMATHGoogle Scholar
  9. [KRW]
    K.H. Kim, F. Roush and J. Wagoner, Automorphisms of the Dimension Group and Gyration Numbers, Journal of the American Mathematical Society 5 (1992), 191–212.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Kg1]
    W. Krieger, On Topological Markov Chains, Astérisque 50 (1977), 193–196.MathSciNetGoogle Scholar
  11. [Kg2]
    W. Krieger, On Dimension for a Class of Homeomorphism Groups, Mathematische Annalen 252 (1980), 87–95.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [Kg3]
    W. Krieger, On Dimension Functions and Topological Markov Chains, Inventiones Mathematicae 56 (1980), 239–250.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [PS]
    W. Parry and D. Sullivan, A Topological Invariant for Flows on One-dimensional Spaces, Topology 14 (1975), 297–299.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [PT]
    W. Parry and S. Tuncel, Classification Problems in Ergodic Theory London Mathematical Society Lecture Series, 67, Cambridge University Press, 1982.Google Scholar
  15. [Wa1]
    J. Wagoner, Markov Partitions and K 2, Publications Mathematétiques IHES no. 65 (1987), 91–129.MathSciNetzbMATHGoogle Scholar
  16. [Wi1]
    R.F. Williams, Classification of One-dimensional Attractors, 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. 341–361.Google Scholar
  17. [Wi2]
    R.F. Williams, Classification of Subshifts of Finite Type, Annals of Mathematics 98 (1973), 120–153; Errata 99 (1974), 380-381.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Bruce P. Kitchens
    • 1
  1. 1.Mathematical Sciences DepartmentIBM T.J. Watson Research CenterYorktown HeightsUSA

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