Communications in Mathematical Physics

, Volume 220, Issue 2, pp 333–375

Julia Sets in Parameter Spaces

  • X. Buff
  • C. Henriksen

Abstract:

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {fb(z)=λz+bz2+z3}b contains quasi-conformal copies of the quadratic Julia set Jz+z2). As a corollary, we show that when the Julia set Jz+z2) is not locally connected (for example when z↦λz+z2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense Gδ subset of S1.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • X. Buff
    • 1
  • C. Henriksen
    • 2
  1. 1.Université Paul Sabatier, UFR MIG, Laboratoire E. Picard, 31062 Toulouse Cedex, France.¶E-mail: buff@picard.ups-tlse.frFR
  2. 2.Technical University of Denmark, Department of Mathematics, 2800 Lyngby, Denmark.¶E-mail: chris@mat.dtu.dkDK

Personalised recommendations