  Let $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP^2}$ be a rational map with
algebraic and topological degrees both equal to $d\geq 2$. Little is known in
general about the ergodic properties of such maps. We show here, however, that
for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the
perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy
$\log d$, one of saddle and one of repelling type. Neither measure is supported
in an algebraic curve, and $T\circ f$ is `fully two dimensional' in the sense
that it does not preserve any singular holomorphic foliation. Absence of an
invariant foliation extends to all $T$ outside a countable union of algebraic
subsets. Finally, we illustrate all of our results in a more concrete
particular instance connected with a two dimensional version of the well-known
quadratic Chebyshev map.
