# Further remarks on rigidity of H\'{e}non maps

## TL;DR

This paper characterizes polynomial automorphisms that preserve Green function level sets of Hénon maps in complex two-space, showing the uniqueness of automorphisms acting on associated Short c2^2 spaces and exploring the commutation of He9non maps with coinciding Green level sets.

## Contribution

It provides a complete characterization of automorphisms preserving Green function level sets and establishes new properties of Short c2^2 spaces related to He9non maps.

## Key findings

- Interior of non-zero sublevel sets of Green functions are Short c2^2 spaces.
- No polynomial automorphism other than affine automorphisms acts on these spaces.
- Coinciding Green level sets imply the He9non maps almost commute.

## Abstract

For a H\'{e}non map $H$ in $\mathbb{C}^2$, we characterize the polynomial automorphisms of $\mathbb{C}^2$ which keep any fixed level set of the Green function of $H$ completely invariant. The interior of any non-zero sublevel set of the Green function of a H\'{e}non map turns out to be a Short $\mathbb{C}^2$ and as a consequence of our characterization, it follows that there exists no polynomial automorphism apart from possibly the affine automorphisms which acts as an automorphism on any of these Short $\mathbb{C}^2$'s. Further, we prove that if any two level sets of the Green functions of a pair of H\'{e}non maps coincide, then they almost commute.