For n ≥ 2, the family of rational maps contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ ≥ 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 κ.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-2-5,
author = {Robert L. Devaney and Kevin M. Pilgrim},
title = {Dynamic classification of escape time Sierpi\'nski curve Julia sets},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {181-198},
zbl = {1160.37362},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-2-5}
}
Robert L. Devaney; Kevin M. Pilgrim. Dynamic classification of escape time Sierpiński curve Julia sets. Fundamenta Mathematicae, Tome 205 (2009) pp. 181-198. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm202-2-5/