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
DOI:
https://doi.org/10.1090/S0002-9947-2012-05637-6
Published electronically:
August 9, 2012
MathSciNet review:
3003267
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
