A proof of Thurston’s topological characterization of rational functions.

*(English)*Zbl 0806.30027From authors’ introduction: “The criterion proved in this paper was stated by Thurston in November 1982. Thurston lectured on its proof on several occasions, notably at the NSF summer conference in Duluth, 1983, where one of the authors (JHH) was present. Using the notes of various attendants at these lectures, we have reconstructed a proof that we have made as precise as we could. Since this required a certain amount of work on our part, we thought it might be of some use to present this proof to the reader.”

Now, let us consider an orientation-preserving branched covering map \(f: S^ 2\to S^ 2\). The critical set of \(f\), post-critical set of \(f\) and the notion of the orbifold, multi-curve, Thurston linear transformation are defined in section 1, which are needed to formulate Thurston’s criterion, i.e., \(f\) is equivalent to a rational function iff a largest eigenvalue of Thurston’s linear transformation is less than 1.

To prove the theorem, the basic construction is a mapping \(\sigma_ f\) of an appropriate Teichmüller space \({\mathfrak T}_ f\) to itself. The space \({\mathfrak T}_ f\) is the space of smooth almost-complex structures on \(S^ 2\). The paper is well written and organized.

Now, let us consider an orientation-preserving branched covering map \(f: S^ 2\to S^ 2\). The critical set of \(f\), post-critical set of \(f\) and the notion of the orbifold, multi-curve, Thurston linear transformation are defined in section 1, which are needed to formulate Thurston’s criterion, i.e., \(f\) is equivalent to a rational function iff a largest eigenvalue of Thurston’s linear transformation is less than 1.

To prove the theorem, the basic construction is a mapping \(\sigma_ f\) of an appropriate Teichmüller space \({\mathfrak T}_ f\) to itself. The space \({\mathfrak T}_ f\) is the space of smooth almost-complex structures on \(S^ 2\). The paper is well written and organized.

Reviewer: A.Klíč (Praha)

##### MSC:

30G12 | Finely holomorphic functions and topological function theory |

30C75 | Extremal problems for conformal and quasiconformal mappings, other methods |

30F60 | Teichmüller theory for Riemann surfaces |

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\textit{A. Douady} and \textit{J. H. Hubbard}, Acta Math. 171, No. 2, 263--297 (1993; Zbl 0806.30027)

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##### References:

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