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.\; We\; focus\; on\; the\; topological\; characteristics\; of\; the\; Julia\; and\; Fatou\; sets\; of$ f$$$_&lambda that arise when the parameter becomes nonzero. In particular, provided the orders of the poles satisfy a certain arithmetic condition and is chosen sufficiently small, the Julia set consists of a homeomorphic copy of the unperturbed Julia set, countably many Cantor sets of concentric closed curves, and Cantor sets of point components that accumulate on the Cantor sets of curves.