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Non-uniqueness of measures of maximal entropy for subshifts of finite type

  • Robert Burton (a1) and Jeffrey E. Steif (a2)
  • DOI: http://dx.doi.org/10.1017/S0143385700007859
  • Published online: 01 September 2008
Abstract
Abstract

It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

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[2]R. Burton . An asymptotic definition of K-groups of automorphisms and a non-Bernoullian counter example. Z. Wahr. 47 (1979), 205212.

[3]R. Burton & R. Pemantle . Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Prob. 21(3) (1993), 13291371.

[5]R.L. Dobrushin , J. Kolafa & S. B. Shlosman . Phase diagram of the two-dimensional Ising antiferromagnet (computer-assisted proof). Commun. Math. Phys. 102 (1985), 89103.

[7]R. S. Ellis . Entropy, Large Deviations, and Statistical Mechanics. Springer: New York, 1985.

[8]H. Georgii . Gibbs Measures and Phase Transitions. de Gruyter: New York, 1988.

[9]H. Kesten . Percolation Theory for Mathematicians. Birkhauser: New York, 1982.

[11]T.M. Liggett . Interacting Particle Systems. Springer: Berlin, 1985.

[16]W. Parry . Intrinsic markov chains. Trans. Amer. Math. Soc. 112 (1964), 5565.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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