Un outil crucial pour l’étude des fractions rationnelles post-critiquement finies est la caractérisation topologique des fractions rationnelles due à Thurston. La démonstration de ce théorème repose sur l’itération d’un endomorphisme holomorphe d’un certain espace de Teichmüller. Le graphe de cet endomorphisme revêt une correspondance au niveau de l’espace des modules. Dans des cas favorables, cette correspondance est le graphe d’une application qui peut être utilisée pour étudier les accouplements. Nous illustrons ceci par un exemple : nous étudions l’auto-accouplement de la basilique.
One crucial tool for studying postcritically finite rational maps is Thurston’s topological characterization of rational maps. This theorem is proved by iterating a holomorphic endomorphism on a certain Teichmüller space. The graph of this endomorphism covers a correspondence on the level of moduli space. In favorable cases, this correspondence is the graph of a map, which can be used to study matings. We illustrate this by way of example: we study the mating of the basilica with itself.
@article{AFST_2012_6_21_S5_995_0,
author = {Xavier Buff and Adam L. Epstein and Sarah Koch},
title = {Twisted matings and equipotential gluings},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {995--1031},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {S5},
year = {2012},
doi = {10.5802/afst.1360},
mrnumber = {3088265},
zbl = {06167099},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1360/}
}
TY - JOUR AU - Xavier Buff AU - Adam L. Epstein AU - Sarah Koch TI - Twisted matings and equipotential gluings JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 995 EP - 1031 VL - 21 IS - S5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1360/ DO - 10.5802/afst.1360 LA - en ID - AFST_2012_6_21_S5_995_0 ER -
%0 Journal Article %A Xavier Buff %A Adam L. Epstein %A Sarah Koch %T Twisted matings and equipotential gluings %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 995-1031 %V 21 %N S5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1360/ %R 10.5802/afst.1360 %G en %F AFST_2012_6_21_S5_995_0
Xavier Buff; Adam L. Epstein; Sarah Koch. Twisted matings and equipotential gluings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. S5, pp. 995-1031. doi : 10.5802/afst.1360. https://afst.centre-mersenne.org/articles/10.5802/afst.1360/
[1] Buff (X.), Epstein (A.), Koch (S.) Pilgrim (K.).— On Thurston’s Pullback Map Complex Dynamics, Families and Friends, D. Schleicher, AK Peters (2009). | MR | Zbl
[2] Bartholdi (L.) & Nekrashevych (V.).— Thurston equivalence of topological polynomials, Acta Math. 197/1, p. 1-51 (2006). | MR | Zbl
[3] Douady (A.), Hubbard (J.H.).— A proof of Thurston’s topological characterization of rational functions, Acta Math., Dec. (1993). | MR | Zbl
[4] Hubbard (J. H.).— Teichmüller theory, volume I, Matrix Editions (2006). | MR
[5] Hubbard (J. H.).— Teichmüller theory, volume II, Matrix Editions, to appear. | MR
[6] Hubbard (J. H.) & Koch (S.).— An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, submitted.
[7] Keel (S.).— Intersection theory of moduli space of stable -pointed curves of genus zero, Trans AMS. 330/2, p. 545-574 (1992). | MR | Zbl
[8] Kirwan (F. C.).— Desingularizations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math. (2) 122, no. 1, p. 41-85 (1985). | MR | Zbl
[9] Knudsen (F.) & Mumford (D.).— The projectivity of the moduli space of stable curves I: Preliminaries on “det” and “Div”, Math. Scand. 39, p. 19-55 (1976). | MR | Zbl
[10] Knudsen (F.).— The projectivity of the moduli space of stable curves II: The stacks , Math. Scand. 52, p. 161-199 (1983). | MR | Zbl
[11] Koch (S.).— Teichmüller theory and critically finite endomorphisms, submitted.
[12] Milnor (J.).— Pasting together Julia sets: a worked example of mating, Experimen. Math. 13, p. 55-92 (2004). | MR | Zbl
[13] Rees (M.).— A partial description of parameter space of rational maps of degree two: Part I, Acta Math. 168, p. 11-87 (1992). | MR | Zbl
[14] Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications, To appear in Inventiones mathematicae.
[15] Shishikura (M.) & Lei (T.).— On a theorem of M. Rees for matings of polynomials, London Math. Soc. Lect. Note 274, Ed. Tan Lei, Cambridge Univ. Press, p. 289-305 (2000). | MR | Zbl
Cité par Sources :