Laminational models for some spaces of polynomials of any degree

TL;DR

This paper explores laminational models for polynomial parameter spaces of any degree, extending the combinatorial approach used for the Mandelbrot set to higher degrees, and investigates how critical sets influence these models.

Contribution

It introduces new models of polynomial parameter spaces based on geodesic laminations, generalizing the pinched disk model beyond quadratic polynomials.

Findings

Laminations are largely determined by their critical sets.

Different critical sets can produce essentially the same lamination.

The models provide a framework for understanding higher degree polynomial spaces.

Abstract

The so-called "pinched disk" model of the Mandelbrot set is due to A.~Douady, J.~H.~Hubbard and W.~P.~Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, "pinches" the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. 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 no combinatorial model is known. One possible reason may be that the higher degree analog of the MLC conjecture is known to be false. We investigate to which extent a geodesic lamination is determined by the location of its critical sets and when different choices of critical sets lead to essentially the same lamination. This yields models of various parameter spaces of laminations similar to the "pinched disk" model of the Mandelbrot set.