Class Degree and Relative Maximal Entropy

Details Class Degree and Relative Maximal Entropy Class Degree and Relative Maximal Entropy

2010-01-29

arxiv.org/abs/1001.5323v2

Mathematics

Dynamical Systems

30 pages, 7 figures

Scientific paper

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 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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