Automorphism groups of commuting polynomial maps of the affine plane

Abstract

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated with $\mathcal{L}$ is a family of polynomial folding maps

having the property that ${\mathsf{F}}_n$ has topological degree $n^N$ and

We derive formulas for the leading terms of the folding maps on ${\mathbb{A}}^2$ associated with the Lie algebras ${\mathcal{A}}_2$, ${\mathcal{B}}_2$, and ${\mathcal{G}}_2$, and we use these formulas to compute the affine automorphism group of each folding map.