In the theory of rational maps an important role is played by matings. These are probably the best understood of all rational functions, but they are bizarre, and involve gluing dendrites together to get spheres carrying Peano curves. In the theory of Kleinian groups, there is a parallel construction, the construction of double limits, that is central to Thurston’s hyperbolization theorem for 3-manifolds that fiber over the circle with pseudo-Anosov monodromy. It also involves gluing dendrites and Peano curves. Clearly these two constructions form one entry of the Sullivan dictionary. This article attempts to spell out the similarities and differences.
Les accouplements forment une classe essentielle d’applications rationelles, sans doute celle qui est la mieux comprise. Mais elle fait intervenir des objets bizarres : recollements de dendrites, courbes de Peano, etc. La construction analogue pour les groupes Kleiniens est celle des limites doubles. Cette construction est essentielle pour l’hyperbolisation des variétés de dimension trois fibres sur le cercle. Ces deux constructions se correspondent par le dictionnaire de Sullivan. Cet article essaie de montrer les similitudes et les différences.
@article{AFST_2012_6_21_S5_1139_0,
author = {John Hubbard},
title = {Matings and the other side of the dictionary},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1139--1147},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {S5},
year = {2012},
doi = {10.5802/afst.1364},
mrnumber = {3088269},
zbl = {1283.37052},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1364/}
}
TY - JOUR AU - John Hubbard TI - Matings and the other side of the dictionary JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2012 SP - 1139 EP - 1147 VL - 21 IS - S5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1364/ DO - 10.5802/afst.1364 LA - en ID - AFST_2012_6_21_S5_1139_0 ER -
%0 Journal Article %A John Hubbard %T Matings and the other side of the dictionary %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2012 %P 1139-1147 %V 21 %N S5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1364/ %R 10.5802/afst.1364 %G en %F AFST_2012_6_21_S5_1139_0
John Hubbard. Matings and the other side of the dictionary. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 21 (2012) no. S5, pp. 1139-1147. doi : 10.5802/afst.1364. https://afst.centre-mersenne.org/articles/10.5802/afst.1364/
[1] Brock (J.), Canary (R.), Minsky (Y.).— The Classification of Kleinian Surface Groups, II: the Ending Lamination Conjecture, arXiv:math/0412006. | MR
[2] Hubbard (J. H.).— Teichmüller theory, vol. I, Matrix Editions (2006). | MR
[3] Hubbard (J. H.).— Teichmüller theory, vol. II, Matrix Editions, to appear. | MR
[4] McMullen (C.).— Iteration on Teichmüller space, Invent.math. 99, p. 425-454 (1990). | MR | Zbl
[5] McMullen (C.).— Rational maps and Kleinian groups, In Proceedings of the International Congress of Mathematicians (Kyoto, 1990), p. 889-900. Springer-Verlag (1991). | MR | Zbl
[6] Minsky (Y.).— The Classification of Kleinian Surface Groups, I. Models and Bounds, Ann. Math., 171(1), p. 1-107 (2010). | MR | Zbl
[7] Mahan (Mj).— Cannon-Thurston Maps for Kleinian Groups, arXiv:math arxiv:1002.0996. | MR
[8] Otal (J.-P.).— The hyperbolization theorem for fibered 3-manifolds, Translated from the 1996 French original by Leslie D. Kay. SMF/AMS Texts and Monographs, 7. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris (2001). | MR | Zbl
[9] 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
[10] Sullivan (D.).— Quasiconformal Homeomorphisms and Dynamics, I. Solution of the Fatou-Julia problem on wandering domains, Annals of math 122, p. 401-418 (1985). | MR | Zbl
[11] Thurston (W.).— On the geometry and Dynamics of iterated Rational Maps, in Complex Dynamics: Families and Friends, Schleicher ed., AK Peters, p. 3-130 (2009). | MR | Zbl
[12] Thurston (W.).— Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Annals of Math 124, p. 203-246 (1986). | MR | Zbl
[13] Thurston (W.).— Hyperbolic structures on 3-manifolds, II: Surface Groups and 3-Manifolds that fiber over the Circle, arXiv:math/9801045.
[14] Lei (T.).— Matings of quadratic polynomials, Ergod. th. and dyn. syst., vo. 12, p. 589-620 (1992). | MR | Zbl
[15] Wada (M.).— Opti, http://delta.math.sci.osaka-u.ac.jp/OPTi
[16] Wada (M.).— Mating Siegel Quadratic Polynomials, JAMS 14, p. 25-78 (2000). | MR
Cited by Sources: