Typical dynamics of plane rational maps with equal degrees

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Let f:CP²² be a rational map with algebraic and topological degrees both equal to d≥ 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:CP²², the perturbed map T∘ f admits exactly two ergodic measures of maximal entropy d, one of saddle and one of repelling type. Neither measure is supported in an algebraic curve, and T∘ 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.