Locally connected models for Julia sets

Academic Article

Abstract

  • Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map φ onto a locally connected continuum J~P, i.e. a monotone map φ:J→J~P such that for any other monotone map Ψ:J→J' there exists a monotone map h with Ψ=hoφ. Then we extend Ψonto the complex plane C (keeping the same notation) and show that Ψ monotonically semiconjugates P|C to a topological polynomial g:C→C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map Ψ not to collapse all of J into a point. © 2010 Elsevier Inc.
  • Published In

    Digital Object Identifier (doi)

    Author List

  • Blokh AM; Curry CP; Oversteegen LG
  • Start Page

  • 1621
  • End Page

  • 1661
  • Volume

  • 226
  • Issue

  • 2