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October  2011, 29(4): 1497-1516. doi: 10.3934/dcds.2011.29.1497

Subshifts of finite type which have completely positive entropy

Christopher Hoffman 1,

1. 

Department of Mathematics, University of Washington, Seattle, WA 98195-4350, United States

Received  November 2009 Revised  July 2010 Published  December 2010

Domino tilings have been studied extensively for both their statistical properties [5], [12], [15] and their dynamical properties [3]. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results about domino tilings to show that our subshift of finite type has a measure of maximal entropy with which the subshift has completely positive entropy but is not isomorphic to a Bernoulli shift.
Keywords: domino tilings, completely positive entropy., Subshift of finite type.
Mathematics Subject Classification: 37B10, 28D20, 60D0.
Citation: Christopher Hoffman. Subshifts of finite type which have completely positive entropy. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1497-1516. doi: 10.3934/dcds.2011.29.1497
References:
[1]

R. Berger, The undecidability of the domino problem, Memoirs Amer. Math. Soc., 66 (1966).  Google Scholar

[2]

H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A, 15 (1982), L631. doi: 10.1088/0305-4470/15/11/011.  Google Scholar

[3]

R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab., 21 (1993), 1329-1371. doi: 10.1214/aop/1176989121.  Google Scholar

[4]

R. Burton and J. E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of finite type, Ergodic Theory Dynam. Systems, 14 (1994), 213-235. doi: 10.1017/S0143385700007859.  Google Scholar

[5]

H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, Journal Of The AMS, 14 (2001), 297-346.  Google Scholar

[6]

J. P. Conze, Entropie d'un groupe abélian de transformations1. Z Wasrscheinlichkeitstheorie Verw. Geiete, 25 (1972), 11-30.  Google Scholar

[7]

J. Feldman, New $K$-automorphisms and a problem of Kakutani, Israel J. Math., 24 (1976), 16-38. doi: 10.1007/BF02761426.  Google Scholar

[8]

N. A. Friedman and D. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1.  Google Scholar

[9]

C. Hoffman, A Markov random field which is $K$ but not Bernoulli, Israel J. Math., 112 (1999), 249-269. doi: 10.1007/BF02773484.  Google Scholar

[10]

F. den Hollander and J. Steif, On K-automorphisms, Bernoulli shifts and Markov random fields,, Ergodic Theory and Dynamical Systems, ().   Google Scholar

[11]

R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math., 150 (2002), 409-439. doi: 10.1007/s00222-002-0249-4.  Google Scholar

[12]

R. Kenyon, Conformal invariance of domino tiling, Ann. Probab., 28 (2000), 759-795. doi: 10.1214/aop/1019160260.  Google Scholar

[13]

R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré, Probabilités, 33 (1997), 591-618.  Google Scholar

[14]

R. Kenyon, Dominos and the Gaussian free field, Ann. Probab., 29 (2001), 1128-1137. doi: 10.1214/aop/1015345599.  Google Scholar

[15]

R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, Ann. of Math. (2), 163 (2006), 1019-1056. doi: 10.4007/annals.2006.163.1019.  Google Scholar

[16]

F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.  Google Scholar

[17]

D. Lind, B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

I. Meilijson, Mixing properties of a class of skew-products, Israel J. Math., 19 (1974), 266-270. doi: 10.1007/BF02757724.  Google Scholar

[19]

S. Sheffield, Uniqueness of maximal entropy measure on essential spanning forests, Ann. Probab., 34 (2006), 857-864. doi: 10.1214/009117905000000765.  Google Scholar

[20]

W. P. Thurston, Conway's tiling groups, Amer. Math. Monthly, 97 (1990), 757-773. doi: 10.2307/2324578.  Google Scholar

show all references

References:
[1]

R. Berger, The undecidability of the domino problem, Memoirs Amer. Math. Soc., 66 (1966).  Google Scholar

[2]

H. W. J. Blöte and H. J. Hilhorst, Roughening transitions and the zero-temperature triangular Ising antiferromagnet, J. Phys. A, 15 (1982), L631. doi: 10.1088/0305-4470/15/11/011.  Google Scholar

[3]

R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Probab., 21 (1993), 1329-1371. doi: 10.1214/aop/1176989121.  Google Scholar

[4]

R. Burton and J. E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of finite type, Ergodic Theory Dynam. Systems, 14 (1994), 213-235. doi: 10.1017/S0143385700007859.  Google Scholar

[5]

H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings, Journal Of The AMS, 14 (2001), 297-346.  Google Scholar

[6]

J. P. Conze, Entropie d'un groupe abélian de transformations1. Z Wasrscheinlichkeitstheorie Verw. Geiete, 25 (1972), 11-30.  Google Scholar

[7]

J. Feldman, New $K$-automorphisms and a problem of Kakutani, Israel J. Math., 24 (1976), 16-38. doi: 10.1007/BF02761426.  Google Scholar

[8]

N. A. Friedman and D. Ornstein, On isomorphism of weak Bernoulli transformations, Advances in Math., 5 (1970), 365-394. doi: 10.1016/0001-8708(70)90010-1.  Google Scholar

[9]

C. Hoffman, A Markov random field which is $K$ but not Bernoulli, Israel J. Math., 112 (1999), 249-269. doi: 10.1007/BF02773484.  Google Scholar

[10]

F. den Hollander and J. Steif, On K-automorphisms, Bernoulli shifts and Markov random fields,, Ergodic Theory and Dynamical Systems, ().   Google Scholar

[11]

R. Kenyon, The Laplacian and Dirac operators on critical planar graphs, Invent. Math., 150 (2002), 409-439. doi: 10.1007/s00222-002-0249-4.  Google Scholar

[12]

R. Kenyon, Conformal invariance of domino tiling, Ann. Probab., 28 (2000), 759-795. doi: 10.1214/aop/1019160260.  Google Scholar

[13]

R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré, Probabilités, 33 (1997), 591-618.  Google Scholar

[14]

R. Kenyon, Dominos and the Gaussian free field, Ann. Probab., 29 (2001), 1128-1137. doi: 10.1214/aop/1015345599.  Google Scholar

[15]

R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae, Ann. of Math. (2), 163 (2006), 1019-1056. doi: 10.4007/annals.2006.163.1019.  Google Scholar

[16]

F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.  Google Scholar

[17]

D. Lind, B. Marcus, "An Introduction to Symbolic Dynamics and Coding," Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

I. Meilijson, Mixing properties of a class of skew-products, Israel J. Math., 19 (1974), 266-270. doi: 10.1007/BF02757724.  Google Scholar

[19]

S. Sheffield, Uniqueness of maximal entropy measure on essential spanning forests, Ann. Probab., 34 (2006), 857-864. doi: 10.1214/009117905000000765.  Google Scholar

[20]

W. P. Thurston, Conway's tiling groups, Amer. Math. Monthly, 97 (1990), 757-773. doi: 10.2307/2324578.  Google Scholar

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