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Julia sets converging to filled quadratic Julia sets

Published online by Cambridge University Press:  21 August 2012

ROBERT T. KOZMA  and
ROBERT L. DEVANEY
ROBERT T. KOZMA
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (email: bob@bu.edu)
ROBERT L. DEVANEY
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (email: bob@bu.edu)
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Abstract

In this paper we consider singular perturbations of the quadratic polynomial $F(z) = z^2 + c$ where $c$ is the center of a hyperbolic component of the Mandelbrot set, i.e., rational maps of the form $z^2 + c + \lambda /z^2$. We show that, as $\lambda \rightarrow 0$, the Julia sets of these maps converge in the Hausdorff topology to the filled Julia set of the quadratic map $z^2 + c$. When $c$ lies in a hyperbolic component of the Mandelbrot set but not at its center, the situation is much simpler and the Julia sets do not converge to the filled Julia set of $z^2 + c$.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 171 - 184
Copyright
Copyright © 2012 Cambridge University Press 

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