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Abstract

The purpose of this note is to explore further the rigidity properties of Hénon maps from [S. Bera et al., Eur. J. Math. 6 (2020)]. For instance, we show that if $H$ and $F$ are Hénon maps with either 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 and they share the same non-escaping sets. Further, we prove that assigning to an Hénon map $H$ its Green measure $\mu _H$, or its filled Julia set $K_H$, or its Green function $G_H$ is locally injective in the space of Hénon maps (with the topology of uniform convergence on compact sets).