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

Academic Article


  • We interpret the combinatorial Mandelbrot set in terms of quadratic laminations (equivalence relations ∼ on the unit circle invariant under σ2). To each lamination we associate a particular geolamination (the collection λ∼ of points of the circle and edges of convex hulls of ∼-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 minor equivalent if their 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.
  • Published In

    Digital Object Identifier (doi)

    Author List

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

  • 37
  • End Page

  • 62
  • Volume

  • 669