Ergodic Theory and Dynamical Systems



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
a1 Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)

Article author query
blanchard p   [Google Scholar] 
devaney rl   [Google Scholar] 
look dm   [Google Scholar] 
seal p   [Google Scholar] 
shapiro y   [Google Scholar] 
 

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.

(Received January 14 2004)
(Revised May 20 2004)