Class Degree and Relative Maximal Entropy

Mahsa Allahbakhshi; Anthony Quas

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arXiv · 1001.5323

Class Degree and Relative Maximal Entropy

Mahsa Allahbakhshi·Anthony Quas

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Abstract

Given a factor code $π$ from a one-dimensional shift of finite type $X$ onto an irreducible sofic shift $Y$, if $π$ is finite-to-one there is an invariant called the degree of $π$ which is defined 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 $ν$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $ν$ and have maximal entropy among all measures in the fibre $π^{-1}\{ν\}$. We show that this bound and the class degree of the code agree when $ν$ 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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Mahsa Allahbakhshi, Anthony Quas. 2011-04-15. Class Degree and Relative Maximal Entropy. https://arxiv.org/abs/1001.5323

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