Slices of Parameter Space of Cubic Polynomials

TL;DR

This paper investigates specific slices of the parameter space of cubic polynomials, focusing on the main cubioid and its properties related to fixed points and dynamical behavior.

Contribution

It characterizes the location of the main cubioid within these parameter slices, extending understanding of cubic polynomial dynamics.

Findings

Identification of the main cubioid in parameter space

Analysis of the dynamical properties of cubic polynomials

Insights into the structure of polynomial parameter spaces

Abstract

In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the $main\, cubioid$ in this parameter space. The $main\, cubioid$ is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials $z^2+c$ for $c$ in the filled main cardioid.