A proof of Thurston’s topological characterization of rational functions. (English) Zbl 0806.30027

From 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.
Reviewer: A.Klíč (Praha)


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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[1] [A]Ahlfors, L.,Lectures on Quasiconformal Mappings. Van Nostrand, 1966. · Zbl 0138.06002
[2] [B]Beardon, A. F.,The Geometry of Discrete Groups. Springer-Verlag, 1983. · Zbl 0528.30001
[3] [BFH]Bielefeld, B., Fischer, Y. &Hubbard, J. H., The classification of critically preperiodic polynomials as dynamical systems.J. Amer. Math. Soc., 5 (1992), 721–762. · Zbl 0784.58031
[4] [G]Gunning, R.,Lectures on Riemann Surfaces. Mathematical Notes, Princeton University, Press, 1966. · Zbl 0175.36801
[5] [H]Hubbard, J. H.,Sur les sections analytiques de la courbe universelle de Teichmüller. Mem. Amer. Math. Soc., 166, 1976. · Zbl 0318.32020
[6] [H-M]Hubbard, J. H. &Masur, H., Quadratic differentials and foliations.Acta Math., 142 (1979), 221–274. · Zbl 0415.30038
[7] [J]Jenkins, J. A., On the existence of certain general extremal metrics.Ann. of Math., 66 (1957), 440–453. · Zbl 0082.06301
[8] [O]Ohtsuka, M.,Dirichlet Problem, Extremal Length and Prime Ends. Van Nostrand Reinhold Math. Stud., 22, 1970.
[9] [R]Royden, H., Automorphisms and isometries of Teichmüller space, inAdvances in the Theory of Riemann Surfaces, pp. 369–383. Ann. of Math. Stud., 66. Princeton University Press, 1971.
[10] [S]Strebel, K.,On Quadratic Differentials and Extremal Quasiconformal Mappings. Lecture Notes, University of Minnesota, 1967.
[11] [T1]Thurston, W.,Lecture Notes. Princeton University.
[12] [T2]Thurston, W.,Lecture Notes. CBMS Conference, University of Minnesota at Duluth, 1983.
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