# Rigidity of Julia sets of families of biholomorphic mappings in higher dimension

## TL;DR

This paper investigates the rigidity properties of Julia sets for certain automorphisms in higher-dimensional complex spaces, focusing on polynomial shift-like maps and skew products of Hénon maps in , revealing conditions under which Julia sets determine the mappings.

## Contribution

It establishes new rigidity results linking Julia sets to the automorphisms in , especially for polynomial shift-like maps and Hénon map skew products.

## Key findings

- Shared Julia sets imply map equivalence in certain classes.
- Characterization of Julia set invariance under specific automorphisms.
- Rigidity conditions for polynomial shift-like maps and Hénon map skew products.

## Abstract

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that share the same backward and forward Julia sets (or non-escaping sets). Secondly, we study the relationship between any pair of skew products of H\'{e}non maps in $\mathbb{C}^3$ having the same forward and backward Julia sets.