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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Class degree and relative maximal entropy


Authors: Mahsa Allahbakhshi and Anthony Quas
Journal: Trans. Amer. Math. Soc. 365 (2013), 1347-1368
MSC (2010): Primary 37B10; Secondary 37A35
Published electronically: August 9, 2012
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05637-6
PII: S 0002-9947(2012)05637-6
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.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.