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 $$\textsf{F}_{n}:\mathbb{A}^N\to\mathbb{A}^N\quad\text{for}\quad n\ge1$$ having the property that $\textsf{F}_{n}$ has topological degree $n^N$ and $$\textsf{F}_{m}\circ\textsf{F}_{n}=\textsf{F}_{n}\circ\textsf{F}_{m}\quad\text{for all}\quad m,n\ge1.$$ 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.