Matings and the other side of the dictionary

John Hubbard

Annales de la faculté des sciences de Toulouse Mathématiques (2012)

  • Volume: 21, Issue: S5, page 1139-1147
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Hubbard, John. "Matings and the other side of the dictionary." Annales de la faculté des sciences de Toulouse Mathématiques 21.S5 (2012): 1139-1147. <http://eudml.org/doc/250999>.

@article{Hubbard2012,
abstract = {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.},
author = {Hubbard, John},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {rational maps; polynomials; mating; manifold; foliation; Kleinian groups},
language = {eng},
month = {12},
number = {S5},
pages = {1139-1147},
publisher = {Université Paul Sabatier, Toulouse},
title = {Matings and the other side of the dictionary},
url = {http://eudml.org/doc/250999},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Hubbard, John
TI - Matings and the other side of the dictionary
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 1139
EP - 1147
AB - 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.
LA - eng
KW - rational maps; polynomials; mating; manifold; foliation; Kleinian groups
UR - http://eudml.org/doc/250999
ER -

References

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  9. Rees (M.).— A partial description of parameter space of rational maps of degree two: Part I, Acta Math. 168, p. 11-87 (1992). Zbl0774.58035MR1149864
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  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). Zbl1185.37111MR2508255
  12. Thurston (W.).— Hyperbolic structures on 3-manifolds, I: Deformation of acylindrical manifolds, Annals of Math 124, p. 203-246 (1986). Zbl0668.57015MR855294
  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). Zbl0756.58024MR1182664
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  16. Wada (M.).— Mating Siegel Quadratic Polynomials, JAMS 14, p. 25-78 (2000). MR1800348

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