Class degree and relative maximal entropy

Allahbakhshi, Mahsa; Quas, Anthony

doi:10.1090/S0002-9947-2012-05637-6

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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