An escape time Sierpiński map is a rational map drawn from the McMullen family z ↦ zⁿ + λ/zⁿ with escaping critical orbits and Julia set homeomorphic to the Sierpiński curve continuum. We address the problem of characterizing postcritically finite escape time Sierpiński maps in a combinatorial way. To accomplish this, we define a combinatorial model given by a planar tree whose vertices come with a pair of combinatorial data that encodes the dynamics of critical orbits. We show that each escape time Sierpiński map realizes a subgraph of the combinatorial tree and the combinatorial information is a complete conjugacy invariant.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-2-1,
author = {M\'onica Moreno Rocha},
title = {A combinatorial invariant for escape time Sierpi\'nski rational maps},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {99-130},
zbl = {1338.37058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-2-1}
}
Mónica Moreno Rocha. A combinatorial invariant for escape time Sierpiński rational maps. Fundamenta Mathematicae, Tome 220 (2013) pp. 99-130. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-2-1/