Rigidity of Julia Sets of Families of Biholomorphic Mappings in Higher Dimensions

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 forward and backward Julia sets. Secondly, we study the relation between any pair of skew products of Henon maps in $$\mathbb C^3$$ having the same (forward and backward) Julia sets. Also in the same spirit, we further establish a similar relation between skew products of Henon maps fibered over a compact metric space, sharing the same Julia sets.