Rigidity Theorems for H\'{e}non maps-II

The purpose of this note is to explore further the rigidity properties of H'{e}non maps from arXiv:1806.08189. For instance, we show that if $H$ and $F$ are H'{e}non maps with the same Green measure (${\mu }_H={\mu }_F$), or the same filled Julia set ($K_H=K_F$), or the same Green function ($G_H=G_F$), then $H^2$ and $F^2$ have to commute. This, in turn, gives that $H$ and $F$ have the same non-escaping sets. Further we prove that, either of the association of a H'{e}non map $H$ to its Green measure ${\mu }_H$ or to its filled Julia set $K_H$ or to its Green function $G_H$ is locally injective.