An algebraic formulation of Thurston’s characterization of rational functions

Pilgrim, Kevin M.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. S5, pp. 1033-1068.

À la suite de Douady-Hubbard et de Bartholdi-Nekrashevych, nous donnons une formulation algébrique de la caractérisation de Thurston des fractions rationnelles. Les techniques développées sont appliquées à l’étude de la dynamique sur l’ensemble des classes d’homotopie de courbes simples qui est induite par une fraction rationnelle. Le théorème de finitude qui en résulte donne de nouvelles informations à propos de la dynamique globale sur l’espace de Teichmüller de l’application introduite dans le théorème de caractérisation de Thurston.

Following Douady-Hubbard and Bartholdi-Nekrashevych, we give an algebraic formulation of Thurston’s characterization of rational functions. The techniques developed are applied to the analysis of the dynamics on the set of free homotopy classes of simple closed curves induced by a rational function. The resulting finiteness results yield new information on the global dynamics of the pullback map on Teichmüller space used in the proof of the characterization theorem.

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