Twisted matings and equipotential gluings

Xavier Buff[1]; Adam L. Epstein[2]; Sarah Koch[3]

  • [1] Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
  • [2] Mathematics institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • [3] Department of Mathematics, Science Center, 1 Oxford Street, Harvard University, Cambridge MA 02138, United States

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

  • Volume: 21, Issue: S5, page 995-1031
  • ISSN: 0240-2963

Abstract

top
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.

How to cite

top

Buff, Xavier, Epstein, Adam L., and Koch, Sarah. "Twisted matings and equipotential gluings." Annales de la faculté des sciences de Toulouse Mathématiques 21.S5 (2012): 995-1031. <http://eudml.org/doc/251004>.

@article{Buff2012,
abstract = {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.},
affiliation = {Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France; Mathematics institute, University of Warwick, Coventry CV4 7AL, United Kingdom; Department of Mathematics, Science Center, 1 Oxford Street, Harvard University, Cambridge MA 02138, United States},
author = {Buff, Xavier, Epstein, Adam L., Koch, Sarah},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {12},
number = {S5},
pages = {995-1031},
publisher = {Université Paul Sabatier, Toulouse},
title = {Twisted matings and equipotential gluings},
url = {http://eudml.org/doc/251004},
volume = {21},
year = {2012},
}

TY - JOUR
AU - Buff, Xavier
AU - Epstein, Adam L.
AU - Koch, Sarah
TI - Twisted matings and equipotential gluings
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2012/12//
PB - Université Paul Sabatier, Toulouse
VL - 21
IS - S5
SP - 995
EP - 1031
AB - 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.
LA - eng
UR - http://eudml.org/doc/251004
ER -

References

top
  1. Buff (X.), Epstein (A.), Koch (S.) & Pilgrim (K.).— On Thurston’s Pullback Map Complex Dynamics, Families and Friends, D. Schleicher, AK Peters (2009). Zbl1180.37065MR2508269
  2. Bartholdi (L.) & Nekrashevych (V.).— Thurston equivalence of topological polynomials, Acta Math. 197/1, p. 1-51 (2006). Zbl1176.37020MR2285317
  3. Douady (A.), & Hubbard (J.H.).— A proof of Thurston’s topological characterization of rational functions, Acta Math., Dec. (1993). Zbl0806.30027MR1251582
  4. Hubbard (J. H.).— Teichmüller theory, volume I, Matrix Editions (2006). MR2245223
  5. Hubbard (J. H.).— Teichmüller theory, volume II, Matrix Editions, to appear. Zbl06594695MR2245223
  6. Hubbard (J. H.) & Koch (S.).— An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, submitted. Zbl1318.32019
  7. Keel (S.).— Intersection theory of moduli space of stable n -pointed curves of genus zero, Trans AMS. 330/2, p. 545-574 (1992). Zbl0768.14002MR1034665
  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). Zbl0592.14011MR799252
  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). Zbl0343.14008MR437541
  10. Knudsen (F.).— The projectivity of the moduli space of stable curves II: The stacks M g , n , Math. Scand. 52, p. 161-199 (1983). Zbl0544.14020MR702953
  11. Koch (S.).— Teichmüller theory and critically finite endomorphisms, submitted. Zbl1310.32016
  12. Milnor (J.).— Pasting together Julia sets: a worked example of mating, Experimen. Math. 13, p. 55-92 (2004). Zbl1115.37051MR2065568
  13. 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
  14. Selinger (N.).— Thurston’s pullback map on the augmented Teichmüller space and applications, To appear in Inventiones mathematicae. Zbl1298.37033
  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). Zbl1062.37039MR1765095

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.