The Mandelbrot set is the shadow of a Julia set

TL;DR

This paper presents a novel perspective on bifurcations in quadratic polynomials by linking them to projections of Julia sets in higher-dimensional complex dynamical systems, potentially applicable to broader families.

Contribution

It introduces a new approach that connects bifurcations to Julia sets in three dimensions, offering a fresh geometric insight into complex dynamical systems.

Findings

Bifurcations can be viewed as projections of higher-dimensional Julia sets.

The approach provides a new geometric framework for understanding bifurcations.

Potential extension of the method to other holomorphic families.

Abstract

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems.