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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetric Gibbs measures

Authors: Karl Petersen and Klaus Schmidt
Journal: Trans. Amer. Math. Soc. 349 (1997), 2775-2811
MSC (1991): Primary 28D05, 60G09; Secondary 58F03, 60J05, 60K35, 82B05
MathSciNet review: 1422906
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Abstract: We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures—a version of de Finetti’s theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, ‘canonical’ Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

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Additional Information

Karl Petersen
Affiliation: Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, North Carolina 27599
MR Author ID: 201837
ORCID: 0000-0001-5074-7696

Klaus Schmidt
Affiliation: Department of Mathematics, University of Vienna, Vienna, Austria

Keywords: Gibbs measure, subshift of finite type, cocycle, Borel equivalence relation, exchangeability, adic transformation, tail field, interval splitting, Kolmogorov property, ratio limit theorem, Markov chain
Received by editor(s): August 17, 1995
Received by editor(s) in revised form: August 20, 1996
Additional Notes: First author supported in part by NSF Grant DMS-9203489.
Article copyright: © Copyright 1997 American Mathematical Society