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  • Ergodic Theory and Dynamical Systems, Volume 25, Issue 4
  • August 2005, pp. 1047-1055

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

  • PAUL BLANCHARD (a1), ROBERT L. DEVANEY (a1), DANIEL M. LOOK (a1), PRADIPTA SEAL (a1) and YAKOV SHAPIRO (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385704000380
  • Published online: 01 April 2005
Abstract

In this paper we consider the family of rational maps of the complex plane given by \[z^2+\frac{\lambda}{z^2}\] where $\lambda$ is a complex parameter. We regard this family as a singular perturbation of the simple function $z^2$. We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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