Automorphism Groups of Commuting Polynomial Maps of the Affine Plane

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps $${\mathsf{\text{F}}}_n:{\mathbb{A}}^N\to {\mathbb{A}}^N\text{for}n\ge 1$$ having the property that ${\mathsf{\text{F}}}_n$ has topological degree $n^N$ and $${\mathsf{\text{F}}}_m\circ {\mathsf{\text{F}}}_n={\mathsf{\text{F}}}_n\circ {\mathsf{\text{F}}}_m\text{for all}m,n\ge 1.$$ We derive formulas for the leading terms of the folding maps on ${\mathbb{A}}^2$ associated to 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.