Rigidity of the escaping set of certain Hénon maps

Let $H$ be a Hénon 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 $H=d\le \mathrm{\mid }a\mathrm{\mid }$. 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 Hénon 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 ${\mathbb{Z}}_{d^2-1}$.