Embedding Unicritical Connectedness Loci

TL;DR

This paper introduces an embedding of the connectedness locus of degree d+1 polynomials into the locus of degree 2d+1 bicritical odd polynomials, revealing new structural relationships in complex dynamics.

Contribution

It constructs a novel embedding $\\Phi_d$ linking degree d+1 and degree 2d+1 polynomial connectedness loci, expanding understanding of polynomial parameter spaces.

Findings

Established an explicit embedding between connectedness loci

Revealed structural relationships in polynomial dynamics

Enhanced understanding of bicritical polynomial parameter spaces

Abstract

In this article, for degree $d\geq 1$, we construct an embedding $\Phi_d $ of the connectedness locus $\mathcal{M}_{d+1}$ of the polynomials $z^{d+1}+c$ into the connectedness locus of degree $2d+1$ bicritical odd polynomials.