Dynamic classification of escape time Sierpiński curve Julia sets
Volume 202 / 2009
Fundamenta Mathematicae 202 (2009), 181-198
MSC: Primary 37F10; Secondary 37F45.
DOI: 10.4064/fm202-2-5
Abstract
For $n \geq 2$, the family of rational maps $F_\lambda(z) = z^n + \lambda/z^n$ contains a countably infinite set of parameter values for which all critical orbits eventually land after some number $\kappa$ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if $\kappa \geq 3$. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of $n$ and $\kappa$.
