Automorphisms of bounded growth

We study birational automorphisms of algebraic varieties of bounded growth, i.e. such that the norms of the inverse images ${(f^n)}^{\ast }\text{\hspace{0.17em}⁣}:\mathrm{N}\mathrm{S}(X)\to \mathrm{N}\mathrm{S}(X)$ of the powers of the automorphism $f\in \mathrm{B}\mathrm{i}\mathrm{r}(X)$ are bounded above for $n⩾0$. We prove that some power of an infinite order automorphism of a variety $X$ with such property factors either through an infinite order translation on the Albanese variety of $X$ or through an infinite order regular automorphism of ${\mathbb{P}}^m$ for $m⩾1$. We deduce from this that if a rationally connected threefold admits an infinite order automorphism whose growth is bounded then the threefold is rational and an iterate of the automorphism is birationally conjugate to a regular automorphism of ${\mathbb{P}}^3$, a generalization of Blanc and Deserti’s result.