Singular perturbations in the quadratic family with multiple poles

Abstract

We consider the quadratic family of complex maps given by , where c is the centre of a hyperbolic component in the Mandelbrot set. Then, we introduce a singular perturbation on the corresponding bounded super-attracting 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 , where are integers, and λ is a complex parameter such that |λ| is very small. We focus on the topological characteristics of the Julia and Fatou sets of f λ that arise when the parameter λ becomes non-zero. We give sufficient conditions on the order of the poles so that for small λ, the Julia sets consist of the union of homeomorphic copies of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on them.