Rational self-maps of projective surfaces with a regular iterate

Abstract: We show that if $Φ: X \dashrightarrow X$ is a dominant rational self-map of a projective surface $X$ over $\mathbb{C}$ with a regular and non-invertible iterate $Φ^n$, then we can take $n \leq 12$. This bound is sharp and realized on $X = \mathbb{P}^2$. In the case where $Φ$ is a birational self-map of $\mathbb{P}^2$ we prove that as long as $Φ$ does not preserve a non-constant fibration, if some iterate $Φ^n$ is regular then $Φ$ itself must be regular.