Relations between Escape Regions in the Parameter Space of Cubic Polynomials

TL;DR

This paper explores the topological relationships between escape regions in the parameter space of cubic polynomials, focusing on the boundaries of connectedness loci in specific parameter slices.

Contribution

It establishes a topological connection between the boundaries of connectedness loci in the parameter slices  and  of cubic maps, extending previous work by Milnor.

Findings

Describes the relationship between boundaries of connectedness loci in  and  slices.

Provides a topological framework linking escape regions in cubic polynomial parameter space.

Abstract

We describe a topological relationship between slices of the parameter space of cubic maps. In the paper \cite{CP1}, Milnor defined the curves $\mathcal{S}_n$ as the set of all cubic polynomials with a marked critical point of period $p$. In this paper, we will describe a relationship between the boundaries of the connectedness loci in the curves $\mathcal{S}_1$ and $\mathcal{S}_2$.