Buried Sierpinski Curve Julia Sets
by
Daniel M. Look
Boston University
Coauthors: Robert L. Devaney

For the family of functions F_l(z)=zⁿ+l/z^d it has been shown that for n=d=2 or n=2, d=1, in every neighborhood of the parameter value l = 0 there are infinitely many parameter values for which the Julia set is a Sierpinski curve on which the dynamics are distinct. In each of the above cases where the Julia set is a Sierpinski curve, the complementary domains (or the Fatou components) are always preimages of the immediate basin of attraction of ¥, which is a superattracting fixed point for these maps (provided n ³ 2). In this talk, we exhibit a similar infinite collection of dynamically distinct Julia sets, but now the Fatou components are quite different. Instead of being preimages of a single superattracting basin at ¥, we give examples where the complementary domains consist of a collection of a number of different attracting basins together with the basin at ¥ and all of the preimages of these basins.

We prove that the dynamics on these Julia sets are all distinct from one another as well as from those mentioned above, but again, all of these Julia sets are homeomorphic.

Date received: January 28, 2005