  We characterize two of the most studied non-uniform hyperbolicity conditions
for rational maps, semi-hyperbolicity and the topological Collet-Eckmann
condition, in terms of the maximal entropy measure. Using the same tools in the
proof of these results we give an extension of a result of Carleson, Jones and
Yoccoz, that semi-hyperbolicity characterizes those polynomial maps whose basin
of attraction of infinity is a John domain, to rational maps having a
completely invariant attracting basin.
