Topological polynomials with a simple core

TL;DR

This paper introduces the concept of a dynamical core for topological polynomials, extending previous notions, and provides explicit descriptions for quadratic and cubic cases, linking them to known structures like the Mandelbrot set.

Contribution

It defines the dynamical core for topological polynomials and characterizes all laminations with a simple core for quadratic and cubic cases.

Findings

Explicit descriptions of the core related to periodic and critical objects

Identification of quadratic laminations with the Main Cardioid of the Mandelbrot set

Extension of core concepts from unimodal maps to topological polynomials

Abstract

We define the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. We describe all laminations associated with quadratic and cubic topological polynomials with a simple core (in the quadratic case, these correspond precisely to points on the Main Cardioid of the Mandelbrot set).