Thurston equivalence of topological polynomials

Abstract

We answer Hubbard's question on determining the Thurston equivalence class of ``twisted rabbits'', i.e. images of the ``rabbit'' polynomial under n-th powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.Comment: 40 pages, lots of figure

Twisted matings and equipotential gluings

“…When P is postcritically finite, K(P ) and J(P ) are connected. 2 In this situation, the complement of K(P ) is isomorphic to C − D, and there is an isomorphism C−D → C−K(P ) conjugating z → z d to P . Since P is monic, there is a unique such Böttcher coordinate böt : C − D → C − K(P ) tangent to the identity at infinity.…”

Book Review: Early days in complex dynamics: a history of complex dynamics in one variable during 1906–1942

“…To illustrate the first direction, consider Thurston's topological characterization of branched covers that are equivalent to rational maps (see [7]). The language of self-similar groups is ideally suited to describe Thurston obstructions, and gave Nekrashevych the means for his solution with Laurent Bartholdi of the famous "Twisted Rabbit Problem" in [3].…”

Mapping schemes realizable by obstructed topological polynomials

“…In Figure 1, we give diagrams in the style of [BN06] of two different topological polynomials. For these diagrams (here and in the rest of this paper), we choose a basepoint t near infinity and draw a loop around ∞ in the negative direction.…”

“…Theorem 2.5 (see [DH93] Note that a Levy cycle need not be stable (as in the definition in [BN06]). However, in this paper we will only consider stable Levy cycles, since they are easier to identify with our method.…”