Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Michael's problem and the Poincaré-Fatou-Bieberbach phenomenon


Authors: P. G. Dixon and J. Esterle
Journal: Bull. Amer. Math. Soc. 15 (1986), 127-187
MSC (1985): Primary 32H25; Secondary 46H05, 32-03, 01A55, 01A60
MathSciNet review: 854551
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. N. H. Abel, Détermination d'une fonction au moyen d'une équation qui ne contient qu'une seule variable, Oeuvres complètes II, Christiania (1881), 36-39.
  • 2. Lars V. Ahlfors, The theory of meromorphic curves, Acta Soc. Sci. Fennicae. Nova Ser. A. 3 (1941), no. 4, 31. MR 0004309
  • 3. Mohamed Akkar, Sur la structure des algèbres topologiques localement multiplicativement convexes, C. R. Acad. Sci. Paris Sér. A 279 (1974), 941–944 (French). MR 0361778
  • 4. H. Alexander, Explicit embedding of the (punctured) disc into C2, Notices Amer. Math. Soc. 24 (1977), A 472.
  • 5. G. R. Allan, Embedding the algebra of formal power series in a Banach algebra, Proc. London Math. Soc. (3) 25 (1972), 329–340. MR 0305071
  • 6. N. U. Arakeljan, Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187–1206 (Russian). MR 0170017
  • 7. N. U. Arakelian and P. M. Gauthier, On tangential approximation by holomorphic functions, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 17 (1982), no. 6, 421–441, 501 (English, with Russian and Armenian summaries). MR 693003
  • 8. Richard Arens, Dense inverse limit rings, Michigan Math. J 5 (1958), 169–182. MR 0105034
  • 9. V. I. Arnold, Chapitres supplémentaires a la théorie des équations différentielles, Editions de Moscou, 1978.
  • 10. Hyman Bass, Edwin H. Connell, and David Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287–330. MR 663785, 10.1090/S0273-0979-1982-15032-7
  • 11. L. Bieberbach, Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des R, S. B. Preuss Akad. Wiss. 14/15 (1933), 476-479.
  • 12. G. R. Belickiĭ, Normalnye formy, invarianty i lokalnye otobrazheniya, “Naukova Dumka”, Kiev, 1979 (Russian). MR 537763
  • 13. Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. MR 0027863
  • 14. Emile Borel, Sur les zéros des fonctions entières, Acta Math. 20 (1897), no. 1, 357–396 (French). MR 1554885, 10.1007/BF02418037
  • 15. Nicolas Bourbaki, Espaces vectoriels topologiques. Chapitres 1 à 5, New edition, Masson, Paris, 1981 (French). Éléments de mathématique. [Elements of mathematics]. MR 633754
  • 16. N. Bourbaki, Topologie générale, Chapitre II, Hermann, Paris, 1960.
  • 17. A. D. Bruno, Analytic forms of differential equations, Trudy Moskov. Mat. Obshch. 25 (1971), 120-162; English transl. in Trans. of the Moscow Math. Soc. 25 (1971), 131-288.
  • 18. Dan Burns, Curvatures of Monge-Ampère foliations and parabolic manifolds, Ann. of Math. (2) 115 (1982), no. 2, 349–373. MR 647810, 10.2307/1971395
  • 19. César Camacho, On the local structure of conformal mappings and holomorphic vector fields in 𝐶², Journées Singulières de Dijon (Univ. Dijon, Dijon, 1978) Astérisque, vol. 59, Soc. Math. France, Paris, 1978, pp. 3, 83–94. MR 542732
  • 20. César Camacho and Paulo Sad, Topological classification and bifurcations of holomorphic flows with resonances in 𝐶², Invent. Math. 67 (1982), no. 3, 447–472. MR 664115, 10.1007/BF01398931
  • 21. James Carlson and Phillip Griffiths, A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. (2) 95 (1972), 557–584. MR 0311935
  • 22. H. Cartan, Sur les systèmes de fonctions holomorphes à variétés lacunaires et leurs applications, Ann. Sci. École Norm. Sup. 45 (1928), 256-346.
  • 23. H. Cartan, Sur les fonctions de deux variables complexes. Les transformations d’un domaine borné 𝐷 en un domaine intérieur à 𝐷, Bull. Soc. Math. France 58 (1930), 199–219 (French). MR 1504964
  • 24. Shiing Shen Chern, Differential geometry: its past and its future, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 41–53. MR 0428217
  • 25. B. Chevreau and J. Esterle, Normal forms for germs of invertible analytic mappings (C , 0) → ( C , 0) (in preparation)
  • 26. Dennis Clayton, A reduction of the continuous homomorphism problem for 𝐹-algebras, Rocky Mountain J. Math. 5 (1975), 337–344. MR 0390771
  • 27. John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
  • 28. D. Couty, Sur l’image d’une fonction entière de deux variables complexes, Proc. Edinburgh Math. Soc. (2) 27 (1984), no. 3, 327–331 (French). MR 800224, 10.1017/S0013091500022471
  • 29. H. G. Dales, A discontinuous homomorphism from 𝐶(𝑋), Amer. J. Math. 101 (1979), no. 3, 647–734. MR 533196, 10.2307/2373803
  • 30. Burgess Davis, Picard’s theorem and Brownian motion, Trans. Amer. Math. Soc. 213 (1975), 353–362. MR 0397900, 10.1090/S0002-9947-1975-0397900-8
  • 31. Burgess Davis, Brownian motion and analytic functions, Ann. Probab. 7 (1979), no. 6, 913–932. MR 548889
  • 32. P. G. Dixon, Scalar homomorphisms on algebras of infinitely long polynomials with an application to automatic continuity theory, J. London Math. Soc. (2) 19 (1979), no. 3, 488–496. MR 540065, 10.1112/jlms/s2-19.3.488
  • 33. P. G. Dixon and D. H. Fremlin, A remark concerning multiplicative functionals on LMC algebras, J. London Math. Soc. (2) 5 (1972), 231–232. MR 0315447
  • 34. Klas Diederich and Nessim Sibony, Strange complex structures on Euclidean space, J. Reine Angew. Math. 311/312 (1979), 397–407. MR 549981
  • 35. Adrien Douady, Systèmes dynamiques holomorphes, Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 39–63 (French). MR 728980
  • 36. Jacques Dufresnoy, Théorie nouvelle des familles complexes normales. Applications à l’étude des fonctions algébroïdes, Ann. Sci. École Norm. Sup. (3) 61 (1944), 1–44 (French). MR 0014469
  • 37. H. Dulac, Recherche sur les points singuliers des équations différentielles, J. École Polytechnique (2) 9 (1904), 1-125.
  • 38. H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
  • 39. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • 40. J. Écalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9) 54 (1975), 183–258 (French). MR 0499882
  • 41. P. M. Elizarov and Yu. S. Il′yashenko, Remarks on orbital analytic classification of germs of vector fields, Mat. Sb. (N.S.) 121(163) (1983), no. 1, 111–126 (Russian). MR 699741
  • 42. Jean Esterle, Injection de semi-groupes divisibles dans des algèbres de convolution et construction d’homomorphismes discontinus de \cal𝐶(𝐾), Proc. London Math. Soc. (3) 36 (1978), no. 1, 59–85 (French). MR 0482218
  • 43. J. Esterle, Elements for a classification of commutative radical Banach algebras, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 4–65. MR 697578
  • 44. Jean Esterle, Problème de Michael et fonctions entières de plusieurs variables complexes, Complex analysis (Toulouse, 1983) Lecture Notes in Math., vol. 1094, Springer, Berlin, 1984, pp. 65–83 (French). MR 773104, 10.1007/BFb0099156
  • 45. Jean Esterle, Mittag-Leffler methods in the theory of Banach algebras and a new approach to Michael’s problem, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 107–129. MR 769501, 10.1090/conm/032/769501
  • 46. P. Fatou, Sur certaines fonctions uniformes de deux variables, C. R. Acad. Sci. Paris 175 (1922), 1030-1033.
  • 47. J.-P. Françoise, Sur les formes normales de champs de vecteurs, Boll. Un. Mat. Ital. A (5) 17 (1980), no. 1, 60–66 (French, with English summary). MR 562114
  • 48. W. H. Fuchs, Théorie de l'approximation des fonctions d'une variable complexe, Presses de l'Université de Montréal, Montréal, 1968.
  • 49. P. M. Gauthier and Ngô van Quê, Problème de surjectivité des applications holomorphes, Ann. Scuola Norm. Sup. Pisa (3) 27 (1973), 555–559 (1974) (French). MR 0379907
  • 50. Mark L. Green, Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89–103. MR 0308433, 10.1090/S0002-9947-1972-0308433-6
  • 51. Phillip A. Griffiths, Entire holomorphic mappings in one and several complex variables, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. The fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974; Annals of Mathematics Studies, No. 85. MR 0447638
  • 52. Lawrence Gruman, Value distribution for holomorphic maps in 𝐶ⁿ, Math. Ann. 245 (1979), no. 3, 199–218. MR 553341, 10.1007/BF01673507
  • 53. Lawrence Gruman, Ensembles exceptionnels pour les applications holomorphes dans 𝐶ⁿ, P. Lelong-P. Dolbeault-H. Skoda analysis seminar, 1981/1983, Lecture Notes in Math., vol. 1028, Springer, Berlin, 1983, pp. 125–162 (French). MR 774975, 10.1007/BFb0071681
  • 54. Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • 55. S. Hitotomatu, Proposed problems and solutions, Sûgaku, Math. Soc. Japan 10 (1958), 195. (Japanese)
  • 56. Lars Hörmander, An introduction to complex analysis in several variables, Second revised edition, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematical Library, Vol. 7. MR 0344507
  • 57. James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
  • 58. G. Julia, Mémoire sur l'itération des fractions rationnelles, Journal de Math. 8 Sér. 1 (1918), 47-245.
  • 59. G. Julia, Sur quelques problèmes relatifs à itération des fractions rationnelles, C. R. Acad. Sci. Paris Ser. A 168 (1919), 147-148.
  • 60. Masahide Kato, A generalization of Bieberbach’s example, Proc. Japan Acad. 50 (1974), 329–333. MR 0407325
  • 61. K. Kodaira, Holomorphic mappings of polydiscs into compact complex manifolds, J. Differential Geometry 6 (1971/72), 33–46. MR 0301228
  • 62. M. Koenigs, Recherches sur les intégrales de certaines équations fonctionnelles, Ann. École Norm. Sup. 1 (1882), 3-41.
  • 63. M. Kuczma, Functional equations, P. W. N., Warsaw, 1968.
  • 64. Henry B. Laufer, Imbedding annuli in 𝐶², J. Analyse Math. 26 (1973), 187–215. MR 0346189
  • 65. S. Lattès, Sur les formes réduites des transformations ponctuelles à deux variables, C. R. Acad. Sci. Paris Ser. A 152 (1911), 1566-1569.
  • 66. S. Lattès, Sur l'itération des substitutions rationnelles à deux variables, C. R. Acad. Sci. Paris Ser. A 166 (1918), 151-153.
  • 67. L. Leau, Étude sur les équations fonctionnelles à une ou plusieurs variables, Ann. Fac. Sci. Toulouse Math. 11 (1897), 1-110.
  • 68. Bernard Malgrange, Travaux d’Écalle et de Martinet-Ramis sur les systèmes dynamiques, Bourbaki Seminar, Vol. 1981/1982, Astérisque, vol. 92, Soc. Math. France, Paris, 1982, pp. 59–73 (French). MR 689526
  • 69. R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343
  • 70. Jean Martinet, Normalisation des champs de vecteurs holomorphes (d’après A.-D. Brjuno), Bourbaki Seminar, Vol. 1980/81, Lecture Notes in Math., vol. 901, Springer, Berlin-New York, 1981, pp. 55–70 (French). MR 647488
  • 71. Jean Martinet and Jean-Pierre Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 63–164 (French). MR 672182
  • 72. Jean Martinet and Jean-Pierre Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 4, 571–621 (1984) (French). MR 740592
  • 73. Ernest A. Michael, Locally multiplicatively-convex topological algebras, Mem. Amer. Math. Soc., No. 11 (1952), 79. MR 0051444
  • 74. T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math. 340 (1983), 140–212. MR 691964
  • 75. R. Nevanlinna, Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Gauthiers-Villars, Paris, 1929.
  • 76. Yasuichiro Nishimura, Automorphismes analytiques admettant des sous-variétés de points fixés attractives dans la direction transversale, J. Math. Kyoto Univ. 23 (1983), no. 2, 289–299 (French). MR 706166
  • 77. Yasuichiro Nishimura, Applications holomorphes injectives de 𝐶² dans lui-même qui exceptent une droite complexe, J. Math. Kyoto Univ. 24 (1984), no. 4, 755–761 (French). MR 775985
  • 78. T. Nishino, Proposed problems and solutions, Sûgaku, Math. Soc. Japan 20 (1968), 125 (Japanese).
  • 79. Toshio Nishino, Le théorème de Borel et le théorème de Picard, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 14, 667–668 (French, with English summary). MR 770459
  • 80. Emile Picard, Sur une classe de transcendantes nouvelles, Acta Math. 18 (1894), no. 1, 133–154 (French). MR 1554853, 10.1007/BF02418277
  • 81. Emile Picard, Sur une classe de transcendantes nouvelles, Acta Math. 23 (1900), no. 1, 333–337 (French). Second mémoire. MR 1554921, 10.1007/BF02418677
  • 82. E. Picard, Sur une classe de surfaces algébriques dont les coordonnées s’expriment par des fonctions uniformes de deux paramètres, Bull. Soc. Math. France 28 (1900), 17–25 (French). MR 1504358
  • 83. E. Picard, Sur certaines équations fonctionnelles et sur une classe de surfaces algébriques, C. R. Acad. Sci. Paris 139 (1904), 5-9.
  • 84. H. Poincaré, Sur les propriétés des fonctions définies par les équations aux différences partielles, Thèse, Gauthier-Villars, Paris, 1879.
  • 85. H. Poincaré, Sur une classe nouvelle de transcendantes uniformes, J.de Math. 6 (1890), 313-365.
  • 86. M. Rajoelina, Eléments de Cohen et fonctions extérieures de l'algèbre du disque, J. Functional Analysis (to appear).
  • 87. L. Reich, Das Typenproblem bei formal-biholomorphen Abbildungen mit anziehendem Fixpunkt, Math. Ann. 179 (1969), 227–250 (German). MR 0237485
  • 88. L. Reich, Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt, Math. Ann. 180 (1969), 233–255 (German). MR 0243101
  • 89. L. I. Ronkin, Introduction to the theory of entire functions of several variables, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 44. MR 0346175
  • 90. Alice Roth, Approximationseigenschaften und Strahlengrenzwerte meromorpher und ganzer Funktionen, Comment. Math. Helv. 11 (1938), no. 1, 77–125 (German). MR 1509593, 10.1007/BF01199693
  • 91. W. Rudin, Private communication.
  • 92. H. Russmann, On the convergence of power series transformations of analytic mappings near a fixed point into a normal form (preprint) Inst. Hautes Études Sci. Publ. Math. I.H.E.S. 1977.
  • 93. A. Sadullaev, On Fatou’s example, Mat. Zametki 6 (1969), 437–441 (Russian). MR 0414944
  • 94. Martin Schottenloher, Michael problem and algebras of holomorphic functions, Arch. Math. (Basel) 37 (1981), no. 3, 241–247. MR 637767, 10.1007/BF01234351
  • 95. E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math. Ann. 2 (1870), no. 2, 317–365 (German). MR 1509664, 10.1007/BF01444024
  • 96. E. Schröder, Über iterierte Funktionen, Math. Ann. 3 (1871), 296-322.
  • 97. A. A. Shcherbakov, Germs of mappings that are analytically inequivalent to their formal normal form, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 94–95 (Russian). MR 659181
  • 98. Nessim Sibony and Pit Mann Wong, Some remarks on the Casorati-Weierstrass theorem, Ann. Polon. Math. 39 (1981), 165–174. MR 617458
  • 99. Carl Ludwig Siegel, Iteration of analytic functions, Ann. of Math. (2) 43 (1942), 607–612. MR 0007044
  • 100. Henri Skoda, Fibrés holomorphes à base et à fibre de Stein, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 19, A1199–A1202. MR 0437802
  • 101. Jean-Luc Stehlé, Plongements du disque dans 𝐶², Séminaire Pierre Lelong (Analyse), Année 1970–1971, Springer, Berlin, 1972, pp. 119–130. Lecture Notes in Math., Vol. 275 (French). MR 0397012
  • 102. Shlomo Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math. 79 (1957), 809–824. MR 0096853
  • 103. Shlomo Sternberg, Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech. 10 (1961), 451–474. MR 0133400
  • 104. Wilhelm Stoll, The characterization of strictly parabolic manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 1, 87–154. MR 577327
  • 105. Dennis P. Sullivan and William P. Thurston, Extending holomorphic motions, Acta Math. 157 (1986), no. 3-4, 243–257. MR 857674, 10.1007/BF02392594
  • 106. Al Vitter, The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), no. 1, 89–104. MR 0432924
  • 107. S. M. Voronin, Analytic classification of germs of conformal mappings (𝐶,0)→(𝐶,0), Funktsional. Anal. i Prilozhen. 15 (1981), no. 1, 1–17, 96 (Russian). MR 609790
  • 108. Stuart Sui Sheng Wang, A Jacobian criterion for separability, J. Algebra 65 (1980), no. 2, 453–494. MR 585736, 10.1016/0021-8693(80)90233-1
  • 109. H. Wu, An 𝑛-dimensional extension of Picard’s theorem, Bull. Amer. Math. Soc. 75 (1969), 1357–1361. MR 0254273, 10.1090/S0002-9904-1969-12422-5
  • 110. Hung-hsi Wu, The equidistribution theory of holomorphic curves, Annals of Mathematics Studies, No. 64, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. MR 0273070
  • 111. H. Wu, Mappings of Riemann surfaces (Nevanlinna theory), Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 480–532. MR 0237772
  • 112. M. G. Zaĭdenberg, Picard’s theorems and hyperbolicity, Sibirsk. Mat. Zh. 24 (1983), no. 6, 44–55 (Russian). MR 731042
  • 113. Eduard Zehnder, A simple proof of a generalization of a theorem by C. L. Siegel, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Springer, Berlin, 1977, pp. 855–866. Lecture Notes in Math., Vol. 597. MR 0461575
  • 114. Fouad Zouakia, Semi-groupes réels dans les algèbres de Banach commutatives, J. London Math. Soc. (2) 36 (1987), no. 3, 543–552 (French). MR 918644, 10.1112/jlms/s2-36.3.543

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 32H25, 46H05, 32-03, 01A55, 01A60

Retrieve articles in all journals with MSC (1985): 32H25, 46H05, 32-03, 01A55, 01A60


Additional Information

DOI: http://dx.doi.org/10.1090/S0273-0979-1986-15463-7