# Models for spaces of dendritic polynomials

## TL;DR

This paper develops a new topological model for the space of cubic dendritic polynomials, extending existing models of the Mandelbrot set to better understand the structure of the cubic connectedness locus.

## Contribution

It constructs a continuous map from the space of all cubic dendritic polynomials onto a laminational model, generalizing the pinched disk model of the Mandelbrot set.

## Key findings

- Constructs a continuous map onto a laminational model
- Generalizes the pinched disk model to cubic polynomials
- Provides a step towards modeling the cubic connectedness locus

## Abstract

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connectedness locus.