Combinatorial models for spaces of cubic polynomials

TL;DR

This paper develops a combinatorial model for a significant subset of cubic polynomial parameter space, extending Thurston's model for the Mandelbrot set to more complex, higher-degree cases.

Contribution

It introduces a novel combinatorial model for the space of dendritic cubic polynomials, filling a gap where no models previously existed for higher degrees.

Findings

Model closely resembles Thurston's for quadratic case

Provides a framework for understanding cubic polynomial dynamics

Enhances understanding of parameter spaces with connected Julia sets

Abstract

A model for the Mandelbrot set is due to Thurston and is stated in the language of geodesic laminations. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials, even conjectural models are missing, one possible reason being that the higher degree analog of the MLC conjecture is known to be false. We provide a combinatorial model for an essential part of the parameter space of complex cubic polynomials, namely, for the space of all cubic polynomials with connected Julia sets all of whose cycles are repelling (we call such polynomials \emph{dendritic}). The description of the model turns out to be very similar to that of Thurston.