A Myriad of Sierpinski Curve Julia Sets
by
Robert L. Devaney
Boston University

In this talk we describe a number of dynamically different ways that Sierpinski curve Julia sets occur in families of rational maps of the form zⁿ + C / zⁿ. In the parameter plane for these families, we show that there are countably many parameters for which the critical orbits eventually land on ∞ and so the corresponding Julia sets are Sierpinski curves. We also show that there is a Cantor set of simple closed curves in the parameter plane for which the corresponding maps also have Sierpinski curve Julia sets. And finally, we show that there are infinitely many buried copies of the Mandelbrot set in the parameter plane, and each parameter drawn from the center of the main cardioid of this set has a Sierpinski curve as its Julia set. As a consequence, all of these Julia sets are homeomorphic, but we show that any pair of them are dynamically distinct, i.e., the maps are not topologically conjugate on their Julia sets.

Date received: February 18, 2006