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Abstract

An escape time Sierpiński map is a rational map drawn from the McMullen family $z \mapsto z^n+\lambda /z^n$ 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.