On degrees of birational mappings

Abstract

We prove that the degrees of the iterates $\mathrm{deg}(f^n)$ of a birational map satisfy $\mathrm{liminf}(\mathrm{deg}(f^n))<+\infty$ if and only if the sequence $\mathrm{deg}(f^n)$ is bounded, and that the growth of $\mathrm{deg}(f^n)$ can not be arbitrarily slow, unless $\mathrm{deg}(f^n)$ is bounded.