Combinatorial models for spaces of cubic polynomials

Academic Article


  • W. Thurston constructed a combinatorial model of the Mandelbrot set M2 such that there is a continuous and monotone projection of M2 to this model. We propose the following related model for the space MD3 of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3, then every point z in the Julia set of the polynomial P defines a unique maximal finite set Az of angles on the circle corresponding to the rays, whose impressions form a continuum containing z. Let G(z) denote the convex hull of Az. The convex sets G(z) partition the closed unit disk. For (P,c1,c2)∈MD3 let c1⁎ be the co-critical point of c1. We tag the marked dendritic polynomial (P,c1,c2) with the set G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by MD3comb their collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3 to MD3comb so that MD3comb serves as a model for MD3.
  • Digital Object Identifier (doi)

    Author List

  • Blokh A; Oversteegen L; Ptacek R; Timorin V
  • Start Page

  • 590
  • End Page

  • 595
  • Volume

  • 355
  • Issue

  • 5