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Sierpinski-curve Julia sets and singular perturbations of complex polynomials

Published online by Cambridge University Press:  19 April 2005

PAUL BLANCHARD,
ROBERT L. DEVANEY,
DANIEL M. LOOK,
PRADIPTA SEAL and
YAKOV SHAPIRO
PAUL BLANCHARD
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)
ROBERT L. DEVANEY
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)
DANIEL M. LOOK
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)
PRADIPTA SEAL
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)
YAKOV SHAPIRO
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: bob@bu.edu)
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Abstract

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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.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 25 , Issue 4 , August 2005, pp. 1047 - 1055
Copyright
2005 Cambridge University Press