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

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.