Rational self-maps of projective surfaces with a regular iterate

Abstract

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