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.
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.
@article{AFST_2012_6_21_S5_1033_0,
author = {Kevin M. Pilgrim},
title = {An algebraic formulation of {Thurston{\textquoteright}s} characterization of rational functions},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1033--1068},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {S5},
year = {2012},
doi = {10.5802/afst.1361},
mrnumber = {3088266},
zbl = {1272.37025},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1361/}
}
TY - JOUR AU - Kevin M. Pilgrim TI - An algebraic formulation of Thurston’s characterization of rational functions JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 1033 EP - 1068 VL - 21 IS - S5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1361/ DO - 10.5802/afst.1361 LA - en ID - AFST_2012_6_21_S5_1033_0 ER -
%0 Journal Article %A Kevin M. Pilgrim %T An algebraic formulation of Thurston’s characterization of rational functions %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 1033-1068 %V 21 %N S5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1361/ %R 10.5802/afst.1361 %G en %F AFST_2012_6_21_S5_1033_0
Kevin M. Pilgrim. An algebraic formulation of Thurston’s characterization of rational functions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, pp. 1033-1068. doi : 10.5802/afst.1361. https://afst.centre-mersenne.org/articles/10.5802/afst.1361/
[1] Buff (X.), Epstein (A.), Koch (S.), and Pilgrim (K. M.).— On Thurston’s pullback map. Complex dynamics, p. 561-583, A K Peters, Wellesley, MA (2009). | MR | Zbl
[2] Bartholdi (L.) and Nekrashevych (V.).— Thurston equivalence of topological polynomials. Acta Math. 197, p. 1-51 (2006). | MR | Zbl
[3] Bonk (M.) and Meyer (D.).— Expanding Thurston maps. arXiv:1009.3647v1 (2010).
[4] Brock (J.) and Margalit (D.).— Weil-Petersson isometries via the pants complex. Proc. Amer. Math. Soc. 1353, p. 795-803 (2007). | MR | Zbl
[5] Douady (A.) and Hubbard (J.).— A Proof of Thurston’s Topological Characterization of Rational Functions. Acta. Math. 171, p. 263-297 (1993). | MR | Zbl
[6] Dym (H.).— Linear algebra in action, volume 78 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2007). | MR | Zbl
[7] Farb (B.) and Margalit (D.).— A primer on mapping class groups. Princeton University Press, Princeton, NJ. xiv+472 pp. ISBN: 978-0-691-14794-9 (2012). | MR | Zbl
[8] Kelsey (G.).— Mapping schemes realizable by obstructed topological polynomials, Conformal Geometry and Dynamics (electronic) 16, p. 44-80 (2012). | MR | Zbl
[9] Kent (R.).— Skinning maps. Duke Math. J.1512, p. 279-336 (2010). | MR | Zbl
[10] Koch (S.).— A new link between Teichmüller theory and complex dynamics. PhD thesis, Cornell University (2008). http://www.math.harvard.edu/~kochs/papers.html.
[11] Lodge (R.).— The boundary values of Thurston’s pullback map. PhD thesis, Indiana University (2012). http://mypage.iu.edu/~rlodge/LodgeThesisFinal.pdf.
[12] Milnor (J.) and Thurston (W.).— On iterated maps of the interval. Springer Lecture Notes in Mathematics 1342 (1988). | MR | Zbl
[13] Nekrashevych (V.).— Self-similar groups, volume 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2005). | MR | Zbl
[14] Nekrashevych (V.).— Combinatorics of polynomial iterations. Complex dynamics, 169-214, A K Peters, Wellesley, MA (2009). | MR | Zbl
[15] Nielsen (J.).— Abbildungsklassen endlicher Ordnung. Acta Math. 75, p. 23-115 (1943). | MR | Zbl
[16] Pilgrim (K. M.).— Combinations of complex dynamical systems. Springer Lecture Notes in Mathematics 1827 (2003). | MR | Zbl
[17] Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications, Inventiones Mathematicae 189, No. 1, p. 111-142 (2012). | MR
[18] Thurston (W. P.).— On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19, p. 417-431 (1988). | MR | Zbl
[19] Wolpert (S.).— The Weil-Petersson metric geometry. Handbook of Teichmüller theory, vol. II. IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Zürich (2009). | MR | Zbl
Cited by Sources: