For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family F lambda = zn + lambda/z d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial zn. This work presents several results related to these goals for the family Flambda , particularly regarding a structure of "necklaces" in the lambda parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of "subnecklaces," wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces.
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