Rigidity of the Julia set for H\'enon-Sibony maps

TL;DR

This paper proves that two Hénon-Sibony maps with identical forward Julia sets must share a common iterate, extending previous results from two dimensions to higher dimensions.

Contribution

It establishes a rigidity result for Hénon-Sibony maps, showing the uniqueness of their dynamics given the Julia set in higher dimensions.

Findings

Maps with the same Julia set share a common iterate

Extends Lamy's results from dimension 2 to higher dimensions

Provides new insights into the structure of Hénon-Sibony maps

Abstract

Let $f$ and $g$ be two H\'enon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.