Rational maps on P^N with regular iterates

Let Phi be a rational self-map of a variety X defined over an algebraically closed field. We say Phi is eventually regular if Phi^n is regular for some n greater than or equal to 1. It is suspected that if X is “rigid” enough and a rational map Phi is eventually regular, then one can make strong conclusions about the map Phi. In particular, it is recently shown by Bell, Ghioca and Reichstein that if Phi is an eventually regular rational self-map of a semiabelian variety G defined over an algebraically closed field of characteristic 0, then either Phi preserves a non-constant fibration or Phi itself must be regular. In this talk, we will focus on the case where X is the projective n-space for some n and will try to prove analogues of Bell, Ghioca and Reichstein’s results in this case.