Abstract

We consider families of maps of the form $z^n+c/z^n$ where the parameter $c$ is a center of a Sierpinski hole in the parameter plane. So each of these Julia sets is homeomorphic to the Sierpinski carpet and hence to each other. We give a complete dynamic classification of these "escape time Sierpinski curve maps," i.e., those maps for which the free critical orbits escape. In particular, we show that there are exactly $(2n)^{k-3}$different conjugacy classes when $n$ is odd and $(2n)^{k-3}+2^{k-4}$ classes when $n$ is even. Here $k$ is the escape time of the critical orbits. It is known that there are exactly $(n-1) (2n)^{k-3}$ such centers of Sierpinski holes, so very few of these maps have conjugate dynamics. The result holds for all maps in a Sierpinski hole, not just for the centers.