Class degree and relative maximal entropy
Mahsa Allahbakhshi, Anthony Quas · Transactions of the American Mathematical Society · 2012
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 $ u$ on $Y$, we find an invariant upper bound on the number of ergodic measures on $X$ which project to $ u$ and have maximal entropy among all measures in the fibre $\pi ^{-1}\{ u \}$. We show that this bound and the class degree of the code agree when $ u$ 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.