Abstract:

We consider the quadratic family of complex maps given by $q$_c(z)=z²+c where $c$ is the center of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding superattracting cycle by adding one pole to each point in the cycle. When $c=-1$ the Julia set of $q$_-1 is the well known basilica and the perturbed map is given by $f$_&lambda(z)=z²-1+ &lambda/(z^d₀(z+1)^d₁) where $d$₀, d₁ > 0 are integers, and is a complex parameter such that is very small. We focus on the topological characteristics of the Julia and Fatou sets of $f$_&lambda that arise when the parameter becomes nonzero.