## 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.## References

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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