Insanely Twisted Rabbits
نویسندگان
چکیده
Let f : S → S be the “rabbit” polynomial z + c with c ∼= (−0.1+ 0.8i) a root of (c + c) + c = 0. Let T : S → S be the Dehn about c and c + c. We are interested in such maps up to combinatorial equivalence, that is, letting Pf denote the critical values of f and their forward images, we declare two maps f, g : S → S to be equivalent if there exist orientation-preserving homeomorphisms φ0, φ1 : (S, Pf )→ (S, Pg) such that fφ0 = φ1g and φ0, φ1 are isotopic rel Pf . By Thurston’s theorem, there are, up to combinatorial equivalence, only three branched coverings of S whose post-critical points follow the same orbit as f ’s: the “rabbit”, its complex conjugate, and the “airplane” ∼= z − 1.7. Consider the maps T ◦ f . They are all branched coverings with same postcritical orbit as f , and each is therefore equivalent to the rabbit, the corabbit or the airplane. The question is, which? This question was asked by Douady and Hubbard in [2].
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