Rigidity of the escaping set of certain H\'enon maps

Let $H$ be a H\'enon map of the form $H(x,y)=(y,p(y)-ax)$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the actions of automorphisms of $\mathbb{C}^2$ if the degree of $H=d\le |a|$. Specifically, every automorphism of $\mathbb{C}^2$ that preserves $U^+$, essentially takes the form $C \circ H^s$ where $s \in \mathbb{Z}$, and $C(x,y)=(\eta x, \eta^d y)$ with $\eta$ some $(d^2-1)$-root of unity. Consequently, we show that the automorphisms of the short $\mathbb{C}^2$'s, obtained as the sub-level sets of the (positive) Green's function corresponding to the H\'enon map $H$ for strictly positive values, are essentially linear maps of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism groups of these short $\mathbb{C}^2$'s are the same, finite, and form a subgroup of $\mathbb{Z}_{d^2-1}$.