Advertisement

Multi-Dimensional Capacity, Pressure and Hausdorff Dimension

  • Shmuel Friedland
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 134)

Abstract

This paper surveys the major techniques and results for multi-dimensional capacity (entropy), topological pressure and Hausdorff dimension for ℤ d -subshifts of finite type.

Keywords

Statistical Mechanic Spectral Radius HAUSDORFF Dimension Ergodic Theorem Finite Type 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [BeS]
    I. Beichl and F. Sullivan, Approximating the permanent via importance sampling with application to the dimer covering problem, J. Comp. Phys. 149: 128–147 (1999).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [Ber]
    R. Berger, The Undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966).Google Scholar
  3. [BKW]
    H.J. Brascamp, H. Kunz, and F.Y. Wu, Some rigoroxis results for the vertex model in statistical mechanics, J. Math. Phys. 14: 1927–1932 (1973).CrossRefGoogle Scholar
  4. [Bre]
    L. Bregman, Certain properties of nonnegative matrices and their permanents, Dokl Akad. Nauk SSSR 211: 27–30 (1973).MathSciNetGoogle Scholar
  5. [CaW]
    N.J. Calkin and H.S. Wilf, The number of independent sets in a grid graph, SIAM J. Discr. Math. 11: 54–60 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  6. [Ciu]
    M. Ciucu, An improved upper bound for the 3-dimensional dimer problem, Duke Math. J. 94: 1–11 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Ego]
    G.P. Egorichev, Proof of the van der Waerden conjecture for permanents, Siberian Math. J. 22: 854–859 (1981).Google Scholar
  8. [Fal]
    K. Falconer, A subadditive thermodynamics formalism for mixing repellers, J. Phys. A: Math. Gen. 21: L737-L742 (1988).MathSciNetCrossRefGoogle Scholar
  9. [Fan]
    D.I. Falikman, Proof of the van der Waerden conjecture regarding the permanent of doubly stochastic matrix. Math. Notes Acad. Sci. USSR 29: 475–479 (1981).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Fis]
    M.E. Fisher, Statisticsd mechanics of dimers on a plane lattice, Phys. Rev. 124: 1664–1672 (1961).MathSciNetzbMATHCrossRefGoogle Scholar
  11. [FoJ]
    S. Forschhammer and J. Justesen, Entropy bounds for constrained two-dimensional random fileds, IEEE Trans. Info. Theory 46: 118–127 (1999).CrossRefGoogle Scholar
  12. [FoR]
    R.H. Fowler and G.S. Rushbrooke, Statistical theory of perfect solutions. Trans. Faraday Soc. 33: 1272–1294 (1937).CrossRefGoogle Scholar
  13. [Frl]
    S. Friedland, A lower bound for the permanents of a doubly stochastic matrix. Annals Math. 110: 167–176 (1979).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Pr2]
    S. Friedland, On the entropy of Z-d subshifts of finite type, Linear Algebra Appl. 252: 199–220 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  15. [Fr3]
    S. Friedland, Discrete Lyapunov exponents and Hausdorff dimension, Ergod. Th. & Dynam. Sys, 20: 145–172 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [HoJ]
    R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, New York, 1985.zbMATHGoogle Scholar
  17. [Hal]
    J.M. Hammersley, Existence theorems and Monte Carlo methods for the monomer-dimer problem, in 1966 Reseach papers in statistics: Festschrift for J, Neyman, Wiley, London, 1966, 125–146.Google Scholar
  18. [Ha2]
    J.M. Hammersley, An improved lower bound for the multidimensional dimer problem, Proc. Camh. Phil Soc. 64: 455–463 (1966).MathSciNetCrossRefGoogle Scholar
  19. [Gan]
    F.R. Gantmacher, The Theory of Matrices, 2 vols., Chelsea, New York, 1959.zbMATHGoogle Scholar
  20. [Kas]
    P.W. Kasteleyn, The statistics of dimers on a lattice, Physica 27: 1209–1225 (1961).CrossRefGoogle Scholar
  21. [Kel]
    G. Keller, Equilibrium States in Ergodic Theory, Cambridge Univ. Press, 1998.zbMATHGoogle Scholar
  22. [Kin]
    J.F.C. Kingman, The ergodic theory of subadditive stochastic processes, J. Royal Stat Soc. B 30: 499–510 (1968).MathSciNetzbMATHGoogle Scholar
  23. [Kre]
    U. Krengel, Ergodic Theorems, de Gruyter Studies in Math, 1985.zbMATHCrossRefGoogle Scholar
  24. [Lie]
    E.H. Lieb, Residual entropy of square ice, Phys. Review 162: 162–172 (1967).CrossRefGoogle Scholar
  25. [Min]
    H. Ming, Permanents, Encyclopedia of Mathematics and its Applications, Vol. 6, Addison-Wesley, 1978.Google Scholar
  26. [Mis]
    M. Misiurewicz, A short proof of the variational ℤN action on a compact space, Asterisque 40: 147–157 (1976).MathSciNetGoogle Scholar
  27. [Nal]
    J.F. Nagle, Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice, J. Math. Phys. 7: 1484–1491 (1966).MathSciNetCrossRefGoogle Scholar
  28. [Na2]
    J.F. Nagle, New series expansion method for the dimer problem, Phys. Rev. 152: 190–197 (1966).CrossRefGoogle Scholar
  29. [NaZ]
    Z. Nagy and K. Zeger, Capacity bounds for the 3-dimensional (0,1) run length limited channel, IEEE Trans. Info. Theory 46: 1030–1033 (2000).zbMATHCrossRefGoogle Scholar
  30. [Pau]
    L. Pauling, The structure and entropy of ice and of other crystals with some randomness of atomic arrangement, J. Amer. Chem. Soc. 57: 2680–2684 (1935).CrossRefGoogle Scholar
  31. [Pes]
    Y.B. Pesin, Dimension Theory in Dynamical Systems, Univ. of Chicago Press, 1997.Google Scholar
  32. [Rob]
    R. Robinson, Undecidability and nonperiodicity for tiling of the plane. Invent. Math. 53: 139–186 (1989).Google Scholar
  33. [Rue]
    D. Ruelle, Thermodynamics Formalism, Encyclopedia of mathematics and its applications. Vol. 5, Addison-Wesley 1978.Google Scholar
  34. [Sc]
    K. Schmidt, Algebraic Ideas in Ergodic Theory, Amer. Math. Soc., 1990.zbMATHGoogle Scholar
  35. [Sch]
    A. Schrijver, Counting 1-factors in regular bipartite graphs, J. Comb. Theory B 72: 122–135 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  36. [Wal]
    P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.zbMATHCrossRefGoogle Scholar
  37. [WeB]
    W. Weeks and R.E. Blahut, The capacity and coding gain of certain checkerboard codes, IEEE Trans. Info. Theory 44: 1193–1203 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  38. [You]
    L.S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2: 109–124 (1982).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 2003

Authors and Affiliations

  • Shmuel Friedland
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations