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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Symmetric Gibbs measures
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by Karl Petersen and Klaus Schmidt PDF
Trans. Amer. Math. Soc. 349 (1997), 2775-2811 Request permission

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
  • Email: petersen@math.unc.edu
  • Klaus Schmidt
  • Affiliation: Department of Mathematics, University of Vienna, Vienna, Austria
  • Email: klaus.schmidt@univie.ac.at
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 2775-2811
  • MSC (1991): Primary 28D05, 60G09; Secondary 58F03, 60J05, 60K35, 82B05
  • DOI: https://doi.org/10.1090/S0002-9947-97-01934-X
  • MathSciNet review: 1422906