Ghys-like models for Lavaurs and simple entire maps
HTML articles powered by AMS MathViewer
- by Arnaud Chéritat
- Conform. Geom. Dyn. 10 (2006), 227-256
- DOI: https://doi.org/10.1090/S1088-4173-06-00141-X
- Published electronically: September 26, 2006
- PDF | Request permission
Abstract:
We provide a new geometric construction of pre-models (à la Ghys) for Lavaurs maps, from which we deduce that their Siegel disk is a Jordan curve running through a critical point, which was not known before. The construction turns out to work also for a class of entire maps, very specific, nonetheless including cases where no pre-models were known.References
- A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360 [Br]Br A.D. Brjuno, Analytical form of differential equations, Trans. Mosc. Math. Soc. 25 (191), 131–288. [C]C A. Chéritat, Recherche d’ensembles de Julia de mesure de Lebesgue positive, Thèse, université Paris-sud, France, 2001.
- Adrien Douady, Disques de Siegel et anneaux de Herman, Astérisque 152-153 (1987), 4, 151–172 (1988) (French). Séminaire Bourbaki, Vol. 1986/87. MR 936853
- Adrien Douady, Does a Julia set depend continuously on the polynomial?, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 91–138. MR 1315535, DOI 10.1090/psapm/049/1315535
- Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590 [DHu]O A. Douady, J.H. Hubbard, Étude dynamique des polynômes complexes, Publications mathématiques d’Orsay, France, 1984–85.
- Peter G. Doyle, Random walk on the Speiser graph of a Riemann surface, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 371–377. MR 752802, DOI 10.1090/S0273-0979-1984-15315-1
- A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102, DOI 10.5802/aif.1318
- Lukas Geyer, Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3661–3683. MR 1837254, DOI 10.1090/S0002-9947-01-02662-9 [Gh]Gh E. Ghys, Transformations holomorphes au voisinage d’une courbe de Jordan, C.R. Acad. Sc. Paris, t. 289 (1984), 383–388. [He1]He1 M. Herman, Conjuguaison quasi-symétrique des difféomorphismes du cercle et applications aux disques singuliers de Siegel, Manuscrit, 1986. [He2]He2 M. Herman, Conjugaison quasi symétrique des homéomorphismes du cercle à des rotations, Manuscrit, 1987. [L]L P. Lavaurs, Systèmes dynamiques holomorphes : explosion de points périodiques paraboliques, Ths̀e, Université Paris-Sud, France, 1989.
- Curtis T. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets, Acta Math. 180 (1998), no. 2, 247–292. MR 1638776, DOI 10.1007/BF02392901
- Welington de Melo and Sebastian van Strien, One-dimensional dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 25, Springer-Verlag, Berlin, 1993. MR 1239171, DOI 10.1007/978-3-642-78043-1
- Carsten Lunde Petersen, Local connectivity of some Julia sets containing a circle with an irrational rotation, Acta Math. 177 (1996), no. 2, 163–224. MR 1440932, DOI 10.1007/BF02392621
- Lei Tan (ed.), The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, vol. 274, Cambridge University Press, Cambridge, 2000. MR 1765080
- C. L. Petersen and S. Zakeri, On the Julia set of a typical quadratic polynomial with a Siegel disk, Ann. of Math. (2) 159 (2004), no. 1, 1–52. MR 2051390, DOI 10.4007/annals.2004.159.1
- Burton Rodin, Intrinsic rotations of simply connected regions, Complex Variables Theory Appl. 2 (1984), no. 3-4, 319–326. MR 743955, DOI 10.1080/17476938408814052
- James T. Rogers Jr., Is the boundary of a Siegel disk a Jordan curve?, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 284–287. MR 1160003, DOI 10.1090/S0273-0979-1992-00324-5
- Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 7044, DOI 10.2307/1968952
- Mitsuhiro Shishikura, Bifurcation of parabolic fixed points, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 325–363. MR 1765097
- Grzegorz Świątek, On critical circle homeomorphisms, Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), no. 2, 329–351. MR 1654840, DOI 10.1007/BF01237654
- Michael Yampolsky and Saeed Zakeri, Mating Siegel quadratic polynomials, J. Amer. Math. Soc. 14 (2001), no. 1, 25–78. MR 1800348, DOI 10.1090/S0894-0347-00-00348-9 [Yo]Yo J.C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, Vol. 231, Société Mathématique de France, 1995.
Bibliographic Information
- Arnaud Chéritat
- Affiliation: Laboratoire Émile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France
- Received by editor(s): June 1, 2005
- Published electronically: September 26, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn. 10 (2006), 227-256
- MSC (2000): Primary 37F40, 37F50; Secondary 37F10
- DOI: https://doi.org/10.1090/S1088-4173-06-00141-X
- MathSciNet review: 2261050