  We prove existence of maximal entropy measures for an open set of
non-uniformly expanding local diffeomorphisms on a compact Riemannian manifold.
In this context the topological entropy coincides with the logarithm of the
degree, and these maximizing measures are eigenmeasures of the transfer
operator. When the map is topologically mixing, the maximizing measure is
unique and positive on every open set.

Dynamical Systems