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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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Class degree and relative maximal entropy
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by Mahsa Allahbakhshi and Anthony Quas PDF
Trans. Amer. Math. Soc. 365 (2013), 1347-1368 Request permission

Abstract:

Given a factor code $\pi$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $\pi$ is finite-to-one there is an invariant called the degree of $\pi$ which is defined as the number of preimages of a typical point in $Y$. We generalize the notion of the degree to the class degree which is defined for any factor code on a one-dimensional shift of finite type. Given an ergodic measure $\nu$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $\nu$ and have maximal entropy among all measures in the fibre $\pi ^{-1}\{\nu \}$. We show that this bound and the class degree of the code agree when $\nu$ is ergodic and fully supported. One of the main ingredients of the proof is a uniform distribution property for ergodic measures of relative maximal entropy.
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Additional Information
  • Mahsa Allahbakhshi
  • Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria British Columbia, Canada V8W 3P4
  • Address at time of publication: Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120 Piso 7, Santiago, Chile
  • Anthony Quas
  • Affiliation: Department of Mathematics and Statistics, University of Victoria, PO Box 3045 STN CSC, Victoria British Columbia, Canada V8W 3P4
  • MR Author ID: 317685
  • Received by editor(s): April 28, 2010
  • Received by editor(s) in revised form: April 13, 2011
  • Published electronically: August 9, 2012
  • Additional Notes: The authors thank the referee for detailed and helpful comments.
    This research was supported by NSERC and the University of Victoria.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1347-1368
  • MSC (2010): Primary 37B10; Secondary 37A35
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05637-6
  • MathSciNet review: 3003267