Quadratic-like dynamics of cubic polynomials

TL;DR

This paper investigates the dynamics of cubic polynomials with non-repelling fixed points, revealing complex behaviors and characterizing those that do not perturb into polynomials with Jordan curve Julia sets, using quadratic-like restrictions.

Contribution

It provides a new description of certain cubic polynomials with non-repelling fixed points via quadratic-like restrictions, especially those resistant to perturbation into Jordan curve Julia sets.

Findings

Identification of cubic polynomials with non-repelling fixed points that do not perturb into Jordan curve Julia sets.

Characterization of these polynomials through their quadratic-like restrictions.

Insights into the structure of the closure of the Cubic Principal Hyperbolic Domain.

Abstract

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.