Abstract
This paper surveys the major techniques and results for multi-dimensional capacity (entropy), topological pressure and Hausdorff dimension for ℤd-subshifts of finite type.
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
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).
R. Berger, The Undecidability of the Domino Problem, Mem. Amer. Math. Soc. 66 (1966).
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).
L. Bregman, Certain properties of nonnegative matrices and their permanents, Dokl Akad. Nauk SSSR 211: 27–30 (1973).
N.J. Calkin and H.S. Wilf, The number of independent sets in a grid graph, SIAM J. Discr. Math. 11: 54–60 (1998).
M. Ciucu, An improved upper bound for the 3-dimensional dimer problem, Duke Math. J. 94: 1–11 (1998).
G.P. Egorichev, Proof of the van der Waerden conjecture for permanents, Siberian Math. J. 22: 854–859 (1981).
K. Falconer, A subadditive thermodynamics formalism for mixing repellers, J. Phys. A: Math. Gen. 21: L737-L742 (1988).
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).
M.E. Fisher, Statisticsd mechanics of dimers on a plane lattice, Phys. Rev. 124: 1664–1672 (1961).
S. Forschhammer and J. Justesen, Entropy bounds for constrained two-dimensional random fileds, IEEE Trans. Info. Theory 46: 118–127 (1999).
R.H. Fowler and G.S. Rushbrooke, Statistical theory of perfect solutions. Trans. Faraday Soc. 33: 1272–1294 (1937).
S. Friedland, A lower bound for the permanents of a doubly stochastic matrix. Annals Math. 110: 167–176 (1979).
S. Friedland, On the entropy of Z-d subshifts of finite type, Linear Algebra Appl. 252: 199–220 (1997).
S. Friedland, Discrete Lyapunov exponents and Hausdorff dimension, Ergod. Th. & Dynam. Sys, 20: 145–172 (2000).
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, New York, 1985.
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.
J.M. Hammersley, An improved lower bound for the multidimensional dimer problem, Proc. Camh. Phil Soc. 64: 455–463 (1966).
F.R. Gantmacher, The Theory of Matrices, 2 vols., Chelsea, New York, 1959.
P.W. Kasteleyn, The statistics of dimers on a lattice, Physica 27: 1209–1225 (1961).
G. Keller, Equilibrium States in Ergodic Theory, Cambridge Univ. Press, 1998.
J.F.C. Kingman, The ergodic theory of subadditive stochastic processes, J. Royal Stat Soc. B 30: 499–510 (1968).
U. Krengel, Ergodic Theorems, de Gruyter Studies in Math, 1985.
E.H. Lieb, Residual entropy of square ice, Phys. Review 162: 162–172 (1967).
H. Ming, Permanents, Encyclopedia of Mathematics and its Applications, Vol. 6, Addison-Wesley, 1978.
M. Misiurewicz, A short proof of the variational ℤN action on a compact space, Asterisque 40: 147–157 (1976).
J.F. Nagle, Lattice statistics of hydrogen bonded crystals. I. The residual entropy of ice, J. Math. Phys. 7: 1484–1491 (1966).
J.F. Nagle, New series expansion method for the dimer problem, Phys. Rev. 152: 190–197 (1966).
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).
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).
Y.B. Pesin, Dimension Theory in Dynamical Systems, Univ. of Chicago Press, 1997.
R. Robinson, Undecidability and nonperiodicity for tiling of the plane. Invent. Math. 53: 139–186 (1989).
D. Ruelle, Thermodynamics Formalism, Encyclopedia of mathematics and its applications. Vol. 5, Addison-Wesley 1978.
K. Schmidt, Algebraic Ideas in Ergodic Theory, Amer. Math. Soc., 1990.
A. Schrijver, Counting 1-factors in regular bipartite graphs, J. Comb. Theory B 72: 122–135 (1998).
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.
W. Weeks and R.E. Blahut, The capacity and coding gain of certain checkerboard codes, IEEE Trans. Info. Theory 44: 1193–1203 (1998).
L.S. Young, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dynam. Sys. 2: 109–124 (1982).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Friedland, S. (2003). Multi-Dimensional Capacity, Pressure and Hausdorff Dimension. In: Rosenthal, J., Gilliam, D.S. (eds) Mathematical Systems Theory in Biology, Communications, Computation, and Finance. The IMA Volumes in Mathematics and its Applications, vol 134. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21696-6_6
Download citation
DOI: https://doi.org/10.1007/978-0-387-21696-6_6
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2326-4
Online ISBN: 978-0-387-21696-6
eBook Packages: Springer Book Archive