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Gibbs and equilibrium measures for some families of subshifts

Published online by Cambridge University Press:  16 May 2012

TOM MEYEROVITCH
TOM MEYEROVITCH*
Affiliation:
University of British Columbia, 1933 West Mall, Vancouver, British Columbia, Canada V6T1Z2 (email: [email protected])
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Abstract

For subshifts of finite type (SFTs), any equilibrium measure is Gibbs, as long as $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly irreducible subshifts, shift-invariant Gibbs measures are equilibrium measures. Here we prove a generalization of the Lanford–Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is ‘topologically Gibbs’. This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta $-shifts, Dyck shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford–Ruelle-type theorem holds. For each of these families, we provide a specific proof of the result.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 3 , June 2013 , pp. 934 - 953
Copyright
Copyright © 2012 Cambridge University Press

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References

[1]Aaronson, J. and Nakada, H.. Exchangeable, Gibbs and equilibrium measures for Markov subshifts. Ergod. Th. & Dynam. Sys. 27(2) (2007), 321–339.Google Scholar
[2]Bowen, R.. Some systems with unique equilibrium states. Math. Syst. Theory 8(3) (1974/75), 193–202.Google Scholar
[3]Burton, R. and Steif, J. E.. Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergod. Th. & Dynam. Sys. 14(2) (1994), 213–235.Google Scholar
[4]Dobrušin, R. L.. Gibbsian random fields for lattice systems with pairwise interactions. Funkcional. Anal. i Priložen. 2(4) (1968), 31–43.Google Scholar
[5]Feldman, J. and Moore, C. C.. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324.Google Scholar
[6]Hamachi, T. and Inoue, K.. Embedding of shifts of finite type into the Dyck shift. Monatsh. Math. 145(2) (2005), 107–129.CrossRefGoogle Scholar
[7]Kalikow, S. A.. $T, T^{-1}$ transformation is not loosely Bernoulli. Ann. of Math. (2) 115(2) (1982), 393–409.Google Scholar
[8]Krieger, W.. On the uniqueness of the equilibrium state. Math. Syst. Theory 8(2) (1974/75), 97–104.Google Scholar
[9]Krieger, W.. On dimension functions and topological Markov chains. Invent. Math. 56(3) (1980), 239–250.Google Scholar
[10]Krieger, W. and Matsumoto, K.. A lambda-graph system for the Dyck shift and its $K$-groups. Doc. Math. 8 (2003), 79–96 (electronic).Google Scholar
[11]Lanford, O. E. and Robinson, D. W.. Statistical mechanics of quantum spin systems. III. Comm. Math. Phys. 9 (1968), 327–338.Google Scholar
[12]Lanford, O. E. and Ruelle, D.. Observables at infinity and states with short range correlations in statistical mechanics. Comm. Math. Phys. 13 (1969), 194–215.Google Scholar
[13]Marcus, B. and Newhouse, S.. Measures of maximal entropy for a class of skew products. Ergodic Theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) (Lecture Notes in Mathematics, 729). Springer, Berlin, 1979, pp. 105–125.Google Scholar
[14]Meyerovitch, T.. Tail invariant measures of the Dyck shift. Israel J. Math. 163 (2008), 61–83.Google Scholar
[15]Parry, W.. On the $\beta $-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416.Google Scholar
[16]Petersen, K. and Schmidt, K.. Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349(7) (1997), 2775–2811.Google Scholar
[17]Ruelle, D.. Thermodynamic Formalism, 2nd edn(Cambridge Mathematical Library). Cambridge University Press, Cambridge, 2004.Google Scholar
[18]Schmeling, J.. Symbolic dynamics for $\beta $-shifts and self-normal numbers. Ergod. Th. & Dynam. Sys. 17(3) (1997), 675–694.Google Scholar
[19]Schmidt, K.. Invariant cocycles, random tilings and the super-$K$ and strong Markov properties. Trans. Amer. Math. Soc. 349(7) (1997), 2813–2825.Google Scholar
[20]Schmidt, K.. Algebraic coding of expansive group automorphisms and two-sided beta-shifts. Monatsh. Math. 129(1) (2000), 37–61.Google Scholar
[21]Walters, P.. Ruelle’s operator theorem and $g$-measures. Trans. Amer. Math. Soc. 214 (1975), 375–387.Google Scholar
[22]Walters, P.. A variational principle for the pressure of continuous transformations. Amer. J. Math. 97(4) (1975), 937–971.Google Scholar
[23]Walters, P.. Equilibrium states for $\beta $-transformations and related transformations. Math. Z. 159(1) (1978), 65–88.Google Scholar