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

TL;DR

This paper proves the rigidity of the escaping set for polynomial automorphisms of ^2 with positive entropy, showing automorphisms preserving this set are essentially iterates of the map combined with finite cyclic affine maps.

Contribution

It establishes the rigidity of the escaping set under holomorphic automorphisms and characterizes the automorphism groups of associated Short ^2 spaces.

Findings

Automorphisms preserving the escaping set are of the form L  H^s.

Automorphism groups of Short ^2 spaces are finite cyclic groups.

The automorphism group is the same for all sub-level sets ^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.