Degree growth of polynomial automorphisms and birational maps: some examples

TL;DR

This paper constructs examples of polynomial automorphisms and birational maps in higher dimensions with prescribed polynomial degree growth rates, expanding understanding of their dynamical complexity and countering classical properties in dimension three and above.

Contribution

It introduces new polynomial automorphisms and birational maps with specific degree growth behaviors in dimensions three and higher, and provides counter-examples to classical properties in these contexts.

Findings

Existence of polynomial automorphisms with degree growth $ ext{deg}^n o n^ ext{ell}$ for $ ext{ell} o [(k-1)/2]$ in $ ext{C}^k$.

Existence of birational maps with degree growth $ ext{deg}^n o n^ ext{ell}$ for $ ext{ell} o k$ in $ ext{P}^k_ ext{C}$.

Counter-examples to classical properties of polynomial automorphisms in dimension $k o 3$ and above.

Abstract

We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ 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 $k\geq 3$ 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 $\mathbb{P}^k_\mathbb{C}$: assume $k\geq 3$; forall $0\leq\ell\leq k$ there exist birational maps $\phi$ of $\mathbb{P}^k_\mathbb{C}$ such that $\mathrm{deg}\, \phi^n\sim n^\ell$.