The combinatorial Mandelbrot set as the quotient of the space of geolaminations

TL;DR

This paper provides a topological interpretation of the combinatorial Mandelbrot set using quadratic laminations and geolaminations, establishing a homeomorphism with the boundary of the set.

Contribution

It introduces a novel topological model of the Mandelbrot set via geolaminations and their minors, connecting quadratic laminations to the set's boundary.

Findings

The quotient space of geolaminations is homeomorphic to the Mandelbrot boundary.

Minor equivalence of geolaminations corresponds to the Mandelbrot boundary points.

Each geolamination class corresponds to a unique quadratic lamination and polynomial.

Abstract

We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations $\sim$ on the unit circle invariant under $\sigma_2$). To each lamination we associate a particular {\em geolamination} (the collection $\mathcal{L}_\sim$ of points of the circle and edges of convex hulls of $\sim$-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology.