Degree growth of polynomial automorphisms and birational maps: some examples

Abstract

We provide the existence of new degree growths in the context of polynomial automorphisms of ${\mathbb{C}}^k$: if $k$ is an integer $\ge 3$, then for any $\ell \le \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of ${\mathbb{C}}^k$ such that $\mathrm{deg}{f}^n\sim n^{\ell }$. We also give counter-examples in dimension $k\ge 3$ to some classical properties satisfied by polynomial automorphisms of ${\mathbb{C}}^2$. We provide the existence of new degree growths in the context of birational maps of ${\mathbb{P}}_{\mathbb{C}}^k$: assume $k\ge 3$; forall $0\le \ell \le k$ there exist birational maps $\varphi$ of ${\mathbb{P}}_{\mathbb{C}}^k$ such that $\mathrm{deg}{\varphi }^n\sim n^{\ell }$.