A Fundamental Domain for V 3
[Un domaine fondamental pour V 3 ]
Mémoires de la Société Mathématique de France, no. 121 (2010) , 141 p.

Nous décrivons un domaine fondamental pour la surface de Riemann V 3,m qui paramétrise (à conjugaison près) l’ensemble des fonctions rationelles par le biais des points critiques énumérés, de manière à ce que le premier point critique ait une période de 3, et que le deuxième point critique ne soit pas envoyé sur le premier après m itérations ou moins. Cela nous fournit une description, à conjugaison topologique près, des dynamiques de toutes les composantes de type III en V 3 , et nous donne des indications sur un modèle topologique de V 3 , au même temps que l’ensemble des composantes hyperboliques qui y sont contenues.

We describe a fundamental domain for the punctured Riemann surface V 3,m which parametrises (up to Möbius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three, and such that the second critical point is not mapped in m iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to topological conjugacy, of all dynamics in all type III hyperbolic components in V 3 , and gives indications of a topological model for V 3 , together with the hyperbolic components contained in it.

@book{MSMF_2010_2_121__1_0,
     author = {Rees, Mary},
     title = {A {Fundamental} {Domain} for $V_{3}$},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {121},
     year = {2010},
     doi = {10.24033/msmf.433},
     mrnumber = {2768577},
     zbl = {1222.30001},
     language = {en},
     url = {http://archive.numdam.org/item/MSMF_2010_2_121__1_0/}
}
TY  - BOOK
AU  - Rees, Mary
TI  - A Fundamental Domain for $V_{3}$
T3  - Mémoires de la Société Mathématique de France
PY  - 2010
IS  - 121
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/MSMF_2010_2_121__1_0/
DO  - 10.24033/msmf.433
LA  - en
ID  - MSMF_2010_2_121__1_0
ER  - 
%0 Book
%A Rees, Mary
%T A Fundamental Domain for $V_{3}$
%S Mémoires de la Société Mathématique de France
%D 2010
%N 121
%I Société mathématique de France
%U http://archive.numdam.org/item/MSMF_2010_2_121__1_0/
%R 10.24033/msmf.433
%G en
%F MSMF_2010_2_121__1_0
Rees, Mary. A Fundamental Domain for $V_{3}$. Mémoires de la Société Mathématique de France, Série 2, no. 121 (2010), 141 p. doi : 10.24033/msmf.433. http://numdam.org/item/MSMF_2010_2_121__1_0/

[1] M. Aspenberg & M. Yampolsky« Mating non-renormalizable quadratic polynomials », Comm. Math. Phys. 287 (2009), p. 1–40. | MR | Zbl

[2] A. Avila, J. Kahn, M. Lyubich & W. Shen« Combinatorial rigidity for unicritical polynomials », Ann. of Math. 170 (2009), p. 783–797. | MR | Zbl

[3] B. H. Bowditch« Length bounds on curves arising from tight geodesics », Geom. Funct. Anal. 17 (2007), p. 1001–1042. | MR | Zbl

[4] —, « End invariants of hyperbolic 3-manifolds », preprint http://www.warwick.ac.uk/~masgak/preprints.html.

[5] —, « Geometric models for hyperbolic 3-manifolds », preprint http://www.warwick.ac.uk/~masgak/preprints.html.

[6] B. Branner & J. H. Hubbard« The iteration of cubic polynomials. I. The global topology of parameter space », Acta Math. 160 (1988), p. 143–206. | MR | Zbl

[7] —, « The iteration of cubic polynomials. II. Patterns and parapatterns », Acta Math. 169 (1992), p. 229–325. | MR | Zbl

[8] J. Brock, R. Canary & Y. Minsky« The classification of Kleinian surface groups II: The ending laminations conjecture », preprint http://www.math.yale.edu/users/yair/research, 2004. | MR

[9] J. Brock, B. K., R. Evans & J. Souto – in preparation.

[10] A. Douady & J. H. Hubbard« Études dynamiques des polynômes complexes, avec la collaboration de P. Lavaurs, Tan Lei, P. Sentenac. Parties I and II », Publications Mathématiques d’Orsay, 1985. | MR

[11] —, « A proof of Thurston’s topological characterization of rational functions », Acta Math. 171 (1993), p. 263–297. | MR | Zbl

[12] J. H. Hubbard« Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz », in Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, 1993, p. 467–511. | MR | Zbl

[13] J. Kahn« A priori bounds for some infinitely renormalizable quadratics: I. Bounded primitive combinatorics », preprint http://front.math.ucdavis.edu/math.DS/0609045.

[14] J. Kahn & M. Lyubich« A priori bounds for some infinitely renormalizable quadratics. II. Decorations », Ann. Sci. Éc. Norm. Supér. 41 (2008), p. 57–84. | MR | EuDML | Zbl | Numdam

[15] —, « Local connectivity of julia sets for unicritical polynomials », Ann. of Math. 170 (2009), p. 413–426. | Zbl

[16] —, « A priori bounds for some infinitely renormalizable quadratics. III. Molecules », in Complex dynamics (D. Schleicher & N. Selinger, éds.), A K Peters, 2009, p. 229–254. | Zbl

[17] —, « The quasi-additivity law in conformal geometry », Ann. of Math. 169 (2009), p. 561–593. | Zbl

[18] J. Kiwi« Rational laminations of complex polynomials », in Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) (M. Lyubich et al., éds.), Contemp. Math., vol. 269, Amer. Math. Soc., 2001, Volume arising from conference, SUNY at Stony Brook, 1998, p. 111–154. | Zbl

[19] —, « eal laminations and the topological dynamics of complex polynomials », Adv. Math. 184 (2004), p. 207–267. | MR

[20] —, « Combinatorial continuity in complex polynomial dynamics », Proc. London Math. Soc. 91 (2005), p. 215–248. | MR | Zbl

[21] —, « Puiseux series polynomial dynamics and iteration of complex cubic polynomials », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 1337–1404. | MR | EuDML | Zbl | Numdam

[22] J. Luo« Combinatorics and holomorphic dynamics: Captures, matings and Newton’s method », Thèse, Cornell University, 1995. | MR

[23] M. Lyubich« Dynamics of quadratic polynomials. III. Parapuzzle and SBR measures », Astérisque 261 (2000), p. 173–200. | MR | Zbl

[24] J. Milnor« Geometry and dynamics of quadratic rational maps », Experiment. Math. 2 (1993), p. 37–83. | MR | Zbl

[25] —, « Rational maps with two critical points », Exper. Math. 9 (2000), p. 481–522. | MR | Zbl

[26] Y. Minsky« The classification of Kleinian surface groups I », Ann. of Math. 171 (2010), p. 1–107. | MR | Zbl

[27] C. L. Petersen« Puzzles and Siegel disks », in Progress in holomorphic dynamics, Pitman Res. Notes Math. Ser., vol. 387, Longman, 1998, p. 50–85. | MR | Zbl

[28] M. Rees« Components of degree two hyperbolic rational maps », Invent. Math. 100 (1990), p. 357–382. | MR | EuDML | Zbl

[29] —, « A partial description of parameter space of rational maps of degree two. I », Acta Math. 168 (1992), p. 11–87. | MR | Zbl

[30] —, « A partial description of the parameter space of rational maps of degree two. II », Proc. London Math. Soc. 70 (1995), p. 644–690. | MR | Zbl

[31] —, « Views of parameter space: Topographer and Resident », Astérisque 288 (2003). | Zbl | Numdam

[32] —, « Multiple equivalent matings with the aeroplane polynomial », Ergod. Th. and Dynam. Syst. 30 (2010), p. 1239–1257, with erratum on p. 1259. | MR | Zbl

[33] —, « The ending laminations theorem direct from teichmuller geodesics », preprint arXiv:math/0404007.

[34] P. Roesch« Puzzles de Yoccoz pour les applications à allure rationnelle », Enseign. Math. 45 (1999), p. 133–168. | MR | Zbl

[35] —, « Holomorphic motions and puzzles (following m. shishikura) », in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, 2000, p. 117–131. | MR | Zbl

[36] —, « On local connectivity for the julia set of rational maps: Newton’s famous example », Ann. of Math. 168 (2008), p. 127–174. | Zbl

[37] J. Stimson« Degree two rational maps with a periodic critical point », Thèse, University of Liverpool, 1993.

[38] L. Tan« Matings of quadratic polynomials », Ergodic Theory Dynam. Systems 12 (1992), p. 589–620. | MR | Zbl

[39] W. P. Thurston« On the geometry and dynamics of iterated rational maps », in Complex dynamics (D. Schleicher & N. Selinger, éds.), A K Peters, 2009, p. 3–137. | MR | Zbl

[40] V. Timorin« External boundary of m 2 », in Proceedings of the Fields Institute dedicated to the 75th birthday of J. Milnor, Fields Institute Communications, vol. 53, AMS, 2006, p. 225–267. | Zbl

[41] —, « Topological regluing of holomorphic functions », Invent. Math. 179 (2010), p. 461–506. | MR | Zbl

[42] B. Wittner« On the bifurcation loci of rational maps of degree two », Thèse, Cornell University, 1988. | MR

Cité par Sources :