Julia sets and bifurcation loci

TL;DR

This paper proves that certain fractals from polynomial dynamical systems in complex spaces are distinct, showing that bifurcation loci and Julia sets of different systems cannot coincide, highlighting their fundamental differences.

Contribution

It establishes that the closure of Misiurewicz PCF cubic polynomials and the Julia sets of Hénon maps are fundamentally different objects, not coinciding with each other or with certain polynomial endomorphisms.

Findings

Closure of Misiurewicz PCF cubic polynomials is not a Julia set of a regular polynomial endomorphism of C^2

Julia set of a Hénon map cannot be the same as that of a polynomial endomorphism

Different polynomial dynamical systems produce inherently distinct fractals

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.