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We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.Holomorphic dynamics in one complex variable is largely concerned with the study of rational maps as dynamical systems, along with their parameter spaces. Postcritically finite rational maps have attracted much attention due to their relative simplicity and their structural significance in parameter space. Furthermore, W. Thurston has given a powerful characterization and rigidity theorem for postcritically finite rational maps up to combinatorial equivalence (see Theorem 1.1). With this tool at our disposal, we are free to consider rational maps in the more flexible topological category.

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Thurston equivalence of topological polynomials

Bartholdi

Nekrashevych

2006

Acta Math.

156

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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

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Cited by 13 publications

References 14 publications

Dynamical Belyi Maps

Dynamical Belyi Maps

Quadratic Thurston maps with few postcritical points

Thurston equivalence of topological polynomials

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