Rigidity of the escaping set of polynomial automorphisms of $\mathbb{C}^2$

Abstract

Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that preserves $U^+$ essentially takes the form $L \circ H^s$ where $s \in \mathbb{Z}$ and $L$ belongs to a finite cyclic group of affine maps that preserve the escaping set. Second, note that the sub-level sets $\{G^+ < c\}$, $c > 0$, of the Greens function $G^+$ associated with the map $H$ are canonical examples of Short $\mathbb{C}^2$s. As a consequence of the above theorem, we show that the holomorphic automorphisms of these Short $\mathbb{C}^2$s are affine automorphisms of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism group of these Short $\mathbb{C}^2$s are the same for every $c>0$ and is a finite cyclic group.