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

## TL;DR

This paper investigates the rigidity properties of Hénon maps, showing that sharing certain dynamical invariants implies commutativity of their squares and identical non-escaping sets, and establishes local injectivity of key associations.

## Contribution

It proves that equal Green measures, filled Julia sets, or Green functions imply the commutativity of squared Hénon maps and the equality of non-escaping sets, extending understanding of their rigidity.

## Key findings

- Hénon maps with the same Green measure have commuting squares.
- Hénon maps with the same filled Julia set have commuting squares.
- Associations of Hénon maps to their Green measure, Julia set, or Green function are locally injective.

## Abstract

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.