Hyperbolic components and cubic polynomials

TL;DR

This paper proves that all bounded hyperbolic components on the curve of cubic polynomials with a periodic critical point are Jordan disks, enhancing understanding of the structure of cubic polynomial parameter spaces.

Contribution

It establishes that for any period p, all bounded hyperbolic components on the curve of cubic polynomials with a critical point of period p are Jordan disks.

Findings

Bounded hyperbolic components are Jordan disks.

The result applies to all integers p ≥ 1.

Provides structural insight into cubic polynomial parameter spaces.

Abstract

In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any bounded hyperbolic component on $\mathcal{S}_p$ is a Jordan disk.