Abstract
Given a factor code π from a shift of finite type X onto an irreducible sofic shift Y , and a fully supported ergodic measure ν on Y , we give an explicit upper bound on the number of ergodic measures on X which project to ν and have maximal entropy among all measures in the fiber π−1{ν}. This bound is invariant under conjugacy. We relate this to an important construction for finite-to-one symbolic factor maps.
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