Julia sets and bifurcation loci

Abstract

We prove that several dynamically defined fractals in $\mathbb{C}$ and $\mathbb{C}^2$ which arise from different type of polynomial dynamical systems can not be the same objects. One of our main results is that the closure of Misiurewicz PCF cubic polynomials (the strong bifurcation locus) cannot be the Julia set of a regular polynomial endomorphism of $\mathbb{C}^2$. We also show that the Julia set of a H\'enon map and a polynomial endomorphism cannot coincide.