Julia sets converging to filled quadratic Julia sets

Abstract

AbstractIn 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 $z^2 + c$.