Higher-dimensional Subshifts of Finite Type, Factor Maps and Measures of Maximal Entropy

We investigate factor maps of higher-dimensional subshifts of finite type. In particular, we are interested in how the number of ergodic measures of maximal entropy behaves under such factor maps. We show that this number is preserved under almost invertible maps, but not in general under finite to one factor maps. One of our tools, which is of independent interest, is a higher-dimensional characterization of entropy-preserving factor maps that extends the well-known one-dimensional characterization result.