Countable State Markov Shifts

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


This chapter is concerned with countable state Markov shifts. The first problem is to extend the Perron-Frobenius theory to nonnegative, countably infinite matrices. There are several difficulties. Countable matrices are classified by recurrence properties. There are three classes: positive recurrent, null recurrent and transient. The corresponding version of the Perron-Frobenius Theorem is successively weakened for each class. The necessary matrix theory is treated in the first section. The treatment is complete and there are a number of examples.


Finite Type Topological Entropy Nonnegative Matrix Maximal Measure Borel Probability Measure 
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. [AKM]
    R. Adler, A. Konheim and M.H. MacAndrew, Topological Entropy, Transactions of the American Mathematical Society no. 114 (1965).Google Scholar
  2. [Bo2]
    R. Bowen, Entropy for Group Endomorphisms and Homogeneous Spaces, Transactions of the American Mathematical Society 153 (1971), 401–414.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [EFP]
    P. Erdős, W. Feller, H. Pollard, A Property of Power Series with Positive Coef-ficients, Bulletin of the American Mathematical Society 55 (1949), 201–204.MathSciNetCrossRefGoogle Scholar
  4. [Fe]
    W. Feller, An Introduction to Probability Theory and Its Applications, 3rd. edition, Wiley, 1968.Google Scholar
  5. [FF1]
    D. Fiebig and U. Fiebig, Topological Boundaries for Countable State Markov Shifts, Proceedings of the London Mathematical Society, III 70 (1995), 625–643.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [FF2]
    D. Fiebig and U. Fiebig, Entropy and Finite Generators for Locally Compact Subshifts, Ergodic Theory and Dynamical Systems 70 (1997), 349–368.MathSciNetCrossRefGoogle Scholar
  7. [Hb]
    F. Hofbauer, On the Intrinsic Ergodicity of Piecewise Monotonic transformations with positive entropy, Israel Journal of Mathematics 34 (1979), 213–237.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [Pe2]
    K. Petersen, Chains, Entropy, Codings, Ergodic Theory and Dynamical Systems 6 (1986), 415–448.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Rd]
    H.L. Royden, Real Analysis, MacMillan, 1968.Google Scholar
  10. [Sl1]
    I. Salama, Topological entropy and the Classification of Countable Chains, Ph.D Thesis, University of North Carolina, Chapel Hill, 1984.Google Scholar
  11. [Sl2]
    I. Salama, Topological Entropy and Recurrence of Countable Chains, Pacifac Journal of Mathematics 134 (1988), 325–341; errata Pac. J. of Math. 140 (1889) 397.MathSciNetzbMATHGoogle Scholar
  12. [Sl3]
    I. Salama, On the Recurrence of Countable Topological Markov Chains, American Mathematica Society Contemporary Mathematics 35 (1992), 349–360, Symbolic Dynamics and its Applications.MathSciNetGoogle Scholar
  13. [Se]
    E. Seneta, Non-negative Matrices and Markov Chains, Springer-Verlag, 1981.Google Scholar
  14. [Tk]
    Y. Takahashi, Isomorphisms of β-Automorphisms to Markov Automorphisms, Osaka Journal of Mathematics 10 (1973), 175–184.MathSciNetzbMATHGoogle Scholar
  15. [V-J1]
    D. Vere-Jones, Geometric Ergodicity in Denumerable Markov Chains, Quarterly journal of Mathematics 13 (1962), 7–28.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [V-J2]
    D. Vere-Jones, Ergodic Properties of Nonnegative Matrices, Pacific Journal of Mathematics 22 (1967), 361–386.MathSciNetzbMATHGoogle 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

Personalised recommendations