Typical Dynamics of Plane Rational Maps with Equal Degrees

Let f : CP 99K CP 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 : CP → CP, the perturbed map T ◦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 fT is ‘fully two dimensional’ in the sense that it does not preserve any singular holomorphic foliation of CP2. In fact, absence of an invariant foliation extends to all T outside a countable union of algebraic subsets of Aut(P2). 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.