Measures of maximal relative entropy
  Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi : X \to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all measures in the fiber, but there is an explicit bound on the number of such maximal entropy preimages. 
Karl Petersen (University of North Carolina at Chapel Hill), Anthony   Quas (University of Memphis), and Sujin Shin (Korea A...