  In this paper we establish a dichotomy for the ergodic measures of maximal
entropy for partially hyperbolic diffeomorphisms with one-dimensional compact
center leaves which are virtually skew products over (transitive) Anosov
homeomorphism. We prove that if the whole manifold is the unique minimal
invariant set saturated by unstable foliation, then either there exists a
unique measure of maximal entropy which is non-hyperbolic or there are exactly
two hyperbolic ergodic measures of maximal entropy.
