Baby Julia sets in parameter slices of generalized McMullen maps

Let a one-dimensional subfamily of the bifurcation locus of the generalized McMullen family be given, with two critical orbits having independent behavior. Suppose one critical value generates a baby Mandelbrot set, and the necklace structure of the bifurcation locus of the other critical point intersects the interior of that baby Mandelbrot set. Location of baby Julia sets in parameter slices of generalized McMullen maps. In that situation, rather than baby Mandelbrot sets being dense in the necklace structure, there are “baby” topological Julia sets or partial baby Julia sets. These sets are expected to be homeomorphic, but not necessarily quasiconformal, copies of Julia sets. This conjecture is based on observed parameter-space images and describes a phenomenon arising from the interaction of two independent bounded critical orbits; its general validity remains open.