Dynamics of Birational Maps of P

Inspired by work done for polynomial automorphisms, we apply pluripotential theory to study iteration of birational maps of P2. A major theme is that success of pluripotential theoretic constructions depends on separation between orbits of the forward and backward indeterminacy sets. In particular, we show that a very mild separation hypothesis guarantees the existence of a plurisubharmonic escape function G̃+ and the induced current μ+. We show that under normalized pullback by the birational map, a large class of currents are attracted to μ+. Under stronger separation hypotheses, we establish relationships between the set of normality, stable manifolds of saddle periodic points, and the support of μ+. We illustrate this work in the more concrete setting of quadratic polynomial maps of C2 with merely rational inverses.