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Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann. (French) Zbl 0559.58020

L’auteur propose une méthode de construction de fractions rationnelles ayant une orbite dense sur la sphère de Riemann. Il donne des nouveaux démonstrations des théorèmes de Arnold et Siegel.
Reviewer: K.Sibirskij

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
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[1] ARNOLD (V. I.) . - On the mappings of the circumference onto itself , Translations A.M.S., vol. 46, 2nd series, p. 213-284. Zbl 0152.41905 · Zbl 0152.41905
[2] BROLIN (H.) . - Invariant sets under iteration of rational functions , Arkiv för Mathematik, vol. 6, 1966 , p. 103-144. MR 33 #2805 | Zbl 0127.03401 · Zbl 0127.03401 · doi:10.1007/BF02591353
[3] BRJUNO (A. D.) . - Analytical form of differential equations , Trans. Moscow Math. Soc., vol. 25, 1971 , p. 131-288. MR 51 #13365 | Zbl 0272.34018 · Zbl 0272.34018
[4] BAKER (I. N.) . - An entire function which has wandering domains , J. Austral. Math. Soc., vol. 22, 1976 , p. 173-176. MR 54 #7777 | Zbl 0335.30001 · Zbl 0335.30001
[5] CHERRY (T. M.) . - A singular case of iteration of analytic functions: A contribution to small divisor problem , in Nonlinear Problems of Engineering, Academic Press, New York, 1964 , p. 29-50. MR 31 #2383 | Zbl 0143.29401 · Zbl 0143.29401
[6] CHOQUET (G.) . - Lectures on analysis , vol. I, Benjamin Inc., 1969 .
[7] CREMER (H.) . - Zum Zentrumproblem , Math. Ann., vol. 98, 1928 , p. 151-163. JFM 53.0303.04 · JFM 53.0303.04
[8] CREMER (H.) . - Über die Schrödersche funktionalgleichung und das Schwarscher Eckenabbildungsproblem , Ber. Math. Phys. Klasse der Sächs Akad. Wiss. Leipzig, vol. 84, 1932 , p. 291-324. Zbl 0007.16902 | JFM 58.0327.02 · Zbl 0007.16902
[9] DOUADY (A.) . - Systèmes dynamiques holomorphes , Séminaire N. Bourbaki, n^\circ 599, vol. 1982-1983, Astérisque, vol. 104-105, S.M.F., 1983 , p. 39-63. Numdam | MR 85h:58090 | Zbl 0532.30019 · Zbl 0532.30019
[10] FATHI (A.) et HERMAN (M.) . - Existence de difféomorphismes minimaux , Astérisque, vol. 49, 1977 , p. 37-59. MR 58 #2889 | Zbl 0374.58010 · Zbl 0374.58010
[11] FATOU (P.) . - Mémoires sur les équations fonctionnelles , Bull. S.M.F., vol. 47, 1919 , p. 161-271 et vol. 48, 1920 , p. 33-94, p. 208-304. Numdam | JFM 47.0921.02 · JFM 47.0921.02
[12] HELSON (H.) et SARASON (D.) . - Past and future , Math. Scand., vol. 21, 1967 , p. 5-16. MR 38 #5282 | Zbl 0241.60029 · Zbl 0241.60029
[13] HERMAN (M.) . - Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations , Pub. Math. I.H.E.S., vol. 49, 1979 , p. 5-233. Numdam | MR 81h:58039 | Zbl 0448.58019 · Zbl 0448.58019 · doi:10.1007/BF02684798
[14] LANG (S.) . - Elliptic functions , Addison-Wesley, 1973 . MR 53 #13117 | Zbl 0316.14001 · Zbl 0316.14001
[15] LATTÈS (S.) . - Sur l’itération des substitutions rationnelles et les fonctions de Poincaré , Note aux C.R. Acad. Sc. Paris, t. 166, 1918 , p. 26-28. JFM 46.0522.01 · JFM 46.0522.01
[16] MAÑ;É (R.) , SAD (P.) et SULLIVAN (D.) . - On the dynamics of rational maps , Ann. scient. Éc. norm. sup., 4e série, t. 16, 1983 , p. 193-217. Numdam | MR 85j:58089 | Zbl 0524.58025 · Zbl 0524.58025
[17] NEHARI (Z.) . - Conformal mappings , Dover Pub., New York, 1975 . MR 51 #13206 · Zbl 0342.30012
[18] REES (M.) . - Ergodic rational maps with dense critical forward orbit , Preprint, Univ. of Minnesota, 1983 , à paraître dans Erg. Th. Dyn. syst. Zbl 0553.58008 · Zbl 0553.58008 · doi:10.1017/S0143385700002467
[19] RÜSSMANN (H.) . - Kleine Nenner, II : Bemerkungen zur Newtonschen Methode , Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl, 1972 , p. 1-20. MR 46 #8407 | Zbl 0255.30003 · Zbl 0255.30003
[20] RÜSSMANN (H.) . - Über die Iteration analytischer Funktionen , J. Math. Mech., vol. 17, 1967 , p. 523-532. Zbl 0186.47704 · Zbl 0186.47704
[21] RUDIN (W.) . - Real and complex analysis , McGraw Hill, New York, 1978 . Zbl 0142.01701 · Zbl 0142.01701
[22] SAMUEL (P.) . - Théories algébriques des nombres , Hermann, Paris, 1971 .
[23] SIEGEL (C. L. C.) . - Iteration of analytic functions , Ann. Math., vol. 43, 1942 , p. 607-612. MR 4,76c | Zbl 0061.14904 · Zbl 0061.14904 · doi:10.2307/1968952
[24] SULLIVAN (D.) . - Quasi-conformal homeomorphisms and dynamics (I) , solution of the Fatou-Julia problem on wandering domains, I.H.E.S., Preprint, 1982 , à paraître aux Ann. Math. Zbl 0589.30022 · Zbl 0589.30022 · doi:10.2307/1971308
[25] SULLIVAN (D.) . - Quasi-conformal homeomorphisms and dynamics (III) , topological conjugacy classes of analytic endomorphisms, I.H.E.S., Preprint, 1983 .
[26] YOCCOZ (J. C.) . - C1-conjugaison des difféomorphismes du cercle , preprint I.M.P.A., Rio de Janeiro, à paraître dans Proc. Symp. Dynamical Systems, Rio de Janeiro, 1981 , Springer, Lect. Notes in Math., n^\circ 1007, 1983 , p. 815-827. Zbl 0537.57015 · Zbl 0537.57015
[27] YOCCOZ (J. C.) . - Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne , preprint I.M.P.A., Rio de Janeiro, 1982 , Ann. scient. Ec. norm. sup., 4e série, t. 17, 1984 , p. 339-365. Numdam | Zbl 0595.57027 · Zbl 0595.57027
[28] ZEHNDER (E.) . - A simple proof of generalization of a theorem by C. L. Siegel , Lect. Notes in Math., vol. 597, Springer-Verlag, 1977 , p. 855-866. MR 57 #1560 | Zbl 0361.32007 · Zbl 0361.32007
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