  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.
