Méthodes de changement d’échelles en analyse complexe
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, p. 427-483

We discuss several rescaling methods in complex analysis and geometry and apply them to get elementary proofs of some classical results. The Bloch principle plays an important role in our approach and yields to a somewhat unified point of view.

Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe  et la seconde moitié du xxe  siècle.

@article{AFST_2006_6_15_3_427_0,
     author = {Berteloot, Fran\c cois},
     title = {M\'ethodes de changement d'\'echelles en analyse complexe},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 15},
     number = {3},
     year = {2006},
     pages = {427-483},
     doi = {10.5802/afst.1127},
     mrnumber = {2246412},
     zbl = {1123.37019},
     language = {fr},
     url = {http://www.numdam.org/item/AFST_2006_6_15_3_427_0}
}
Berteloot, François. Méthodes de changement d’échelles en analyse complexe. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 3, pp. 427-483. doi : 10.5802/afst.1127. http://www.numdam.org/item/AFST_2006_6_15_3_427_0/

[1] Arnold, V. Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Editions MIR Moscou - Editions du Globe Paris (1996) | MR 898218 | Zbl 0455.34001

[2] Bargmann, D. Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems, Tome 19 (1999) no. 3, pp. 553-558 | MR 1695942 | Zbl 0942.37033

[3] Bedford, E.; Pinchuk, S. Domains in C 2 with non-compact automorphism group, Math. USSR Sbornik, Tome 63 (1989), pp. 141-151 | MR 937803 | Zbl 0668.32029

[4] Bedford, E.; Pinchuk, S. Convex domains with non-compact automorphism group, J. Geometric Anal., Tome 1 (1991), pp. 165-191 | MR 1120679 | Zbl 0733.32014

[5] Bell, S. The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc., Tome 270 (1982) no. 2, pp. 685-691 | MR 645338 | Zbl 0482.32007

[6] Bell, S.; Ligocka, E. A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math., Tome 57 (1980) no. 3, pp. 283-289 | MR 568937 | Zbl 0411.32010

[7] Bergweiler, W. Rescaling principles in function theory, Analysis and its applications, Allied Publ., New Delhi (2001), pp. 11-29 ((Chennai, 2000)) | MR 1893221 | Zbl 0993.30018

[8] Berteloot, F. Attraction de disques analytiques et continuité Höldérienne d’applications holomorphes propres, Topics in Compl.Anal., Banach Center Publ. (1995), pp. 91-98 | MR 1341379 | Zbl 0831.32012

[9] Berteloot, F. Characterization of models in C 2 by their automorphism group, Int. J. Math., Tome 5 (1994) no. 5, pp. 619-634 | MR 1297410 | Zbl 0817.32010

[10] Berteloot, F. Principe de Bloch et estimations de la métrique de Kobayashi des domaines de C 2 , J. Geom. Anal., Tome 13 (2003) no. 1, pp. 29-37 | MR 1967034 | Zbl 1040.32011

[11] Berteloot, F.; Cœuré, G. Domaines de C 2 , pseudoconvexes et de type fini ayant un groupe non-compact d’automorphismes, Ann. Inst. Fourier Grenoble, Tome 41 (1991) no. 1, pp. 77-86 | Numdam | Zbl 0711.32016

[12] Berteloot, F.; Dupont, C. Une caractérisation des exemples de Lattès par leur mesure de Green (Comment. Math. Helv. (à paraître)) | Zbl 1079.37039

[13] Berteloot, F.; Duval, J. Une démonstration directe de la densité des cycles répulsifs dans l’ensemble de Julia, Complex analysis and geometry, Progr. Math., Basel, Tome 188 (2000), p. 221-222 ((Paris, 1997)) | Zbl 1073.37522

[14] Berteloot, F.; Duval, J. Sur l’hyperbolicité de certains complémentaires, L’Enseignement Mathématique, Tome 47 (2001), pp. 253-267 | MR 1876928 | Zbl 1009.32015

[15] Berteloot, F.; Loeb, J. J. Une caractérisation géométrique des exemples de Lattès de P k , Bull. Soc. Math. Fr., Tome 129 (2001) no. 2, pp. 175-188 | Numdam | MR 1871293 | Zbl 0994.32026

[16] Berteloot, F.; Mayer, V. Rudiments de dynamique holomorphe, Société Mathématique de France, EDP Sciences, Les Ulis (Cours Spécialisés) Tome 7 (2001) | MR 1973050 | Zbl 1051.37019

[17] Briend, J.-Y.; Duval, J. Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de P k , Acta Math., Tome 182 (1999) no. 2, pp. 143-157 | MR 1710180 | Zbl 1144.37436 | Zbl 01541209

[18] Briend, J.-Y.; Duval, J. Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k , Publ. Math. Inst. Hautes Études Sci., Tome 93 (2001), pp. 145-159 | Numdam | MR 1863737 | Zbl 1010.37004

[19] Brody, R. Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., Tome 235 (1978), pp. 213-219 | MR 470252 | Zbl 0416.32013

[20] Byun, J.; Gaussier, H.; Kim, K.-T. Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group, J. Geom. Anal., Tome 12 (2002) no. 4, pp. 581-599 | MR 1916860 | Zbl 1039.32003

[21] Catlin, D. Estimates of Invariant metrics on pseudoconvex domains of dimension two, Math. Z., Tome 200 (1989), pp. 429-466 | MR 978601 | Zbl 0661.32030

[22] Christ, M. C irregularity of the ¯-Neumann problem for worm domains, J. Amer. Math. Soc., Tome 9 (1996) no. 4, pp. 1171-1185 | MR 1370592 | Zbl 0945.32022

[23] Coupet, B. Precise regularity up to the boundary of proper holomorphic mappings, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Tome (4) 20 (1993) no. 3, pp. 461-482 | Numdam | MR 1256077 | Zbl 0812.32011

[24] Coupet, B.; Gaussier, H.; Sukhov, A. Regularity of CR maps between convex hypersurfaces of finite type, Proc. Amer. Math. Soc., Tome 127 (1999) no. 11, pp. 3191-3200 | MR 1610940 | Zbl 0951.32026

[25] Coupet, B.; Pinchuk, S.; Sukhov, A. On boundary rigidity and regularity of holomorphic mappings, Int. J. Math., Tome 7 (1996) no. 5, pp. 617-643 | MR 1411304 | Zbl 0952.32011

[26] Coupet, B.; Sukhov, A. On CR mappings between pseudoconvex hypersurfaces of finite type in C 2 , Duke Math. J. Vol., Tome 88 (1997) no. 2, pp. 281-304 | MR 1455521 | Zbl 0895.32007

[27] Demailly, J.-P. Variétés hyperboliques et équations différentielles algébriques, Gaz. Math., Tome 73 (1997), pp. 3-23 | MR 1462789 | Zbl 0901.32019

[28] Demailly, J.-P.; Elgoul, J. Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math., Tome 122 (2000) no. 3, pp. 515-546 | MR 1759887 | Zbl 0966.32014

[29] Dethloff, G.; Schumacher, G.; Wong, P. M. On the hyperbolicity of the complements of curves in algebraic surfaces, Duke Math. J., Tome 78 (1995), pp. 193-212 | MR 1328756 | Zbl 0847.32028

[30] Diederich, K.; Fornaess, J. E. Proper holomorphic maps onto pseudoconvex domains with real analytic boundary, Ann. Math., Tome 110 (1979), pp. 575-592 | MR 554386 | Zbl 0394.32012

[31] Diederich, K.; Pinchuk, S. Proper holomorphic maps in dimension 2 extend, Indiana. Math. J., Tome 44 (1995), pp. 1089-1126 | MR 1386762 | Zbl 0857.32015

[32] Dinh, T. C.; Sibony, N. Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), Tome 82 (2003) no. 4, pp. 367-423 | MR 1992375 | Zbl 1033.37023

[33] Dixon, P. G.; Esterle, J. Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon, Bull. Amer. Math. Soc. (N.S.), Tome 15 (1986) no. 2, pp. 127-187 | MR 854551 | Zbl 0608.32008

[34] Efimov, A. M. A generalization of the Wong-Rosay theorem for the unbounded case, Sb. Math., Tome 186 (1995) no. 7, pp. 967-976 | MR 1355455 | Zbl 0865.32020

[35] Eremenko, A. A Picard type theorem for holomorphic curves, Period. Math. Hungar., Tome 38 (1999) no. 1-2, pp. 39-42 | MR 1721476 | Zbl 0940.32010

[36] Fefferman, C. The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., Tome 26 (1974), pp. 1-65 | MR 350069 | Zbl 0289.32012

[37] Fornaess, J. E.; Sibony, N. Construction of P.S.H. functions on weakly pseudoconvex domains, Duke Math. J., Tome 58 (1989), pp. 633-656 | MR 1016439 | Zbl 0679.32017

[38] Fornaess, J. E.; Sibony, N. Complex Dynamics in higher dimensions, Complex potential theory (Montréal, PQ, 1993), Kluwer Acad. Publ. (NATO ASI series Math. and Phys. Sci.) Tome 439 (1994), pp. 131-186 | MR 1332961 | Zbl 0811.32019

[39] Fornaess, J. E.; Sibony, N. Complex Dynamics in higher dimensions II, Ann. of Math. Studies, Tome 137 (1995) (Princeton Univ. Press, Princeton, NJ) | MR 1369137 | Zbl 0847.58059

[40] Forstneric, F. An elementary proof of Fefferman’s theorem, Exposition. Math., Tome 10 (1992) no. 2, pp. 135-149 | MR 1164529 | Zbl 0759.32018

[41] Forstneric, F.; Rosay, J.-P. Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann., Tome 110 (1987), pp. 239-252 | MR 919504 | Zbl 0644.32013

[42] Françoise, J.-P. Géométrie analytique et systèmes dynamiques, Presses Universitaires de France, Paris (1995) (Cours de troisième cycle) | MR 1620294

[43] Frankel, S. Complex geometry of convex domains that cover varieties, Acta Math., Tome 163 (1989) no. 1-2, pp. 109-149 | MR 1007621 | Zbl 0697.32016

[44] Gaussier, H. Characterization of models for convex domains (preprint) | MR 1460422

[45] Graham, I. Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary, Trans. Amer. Math. Soc., Tome 207 (1975), pp. 219-240 | MR 372252 | Zbl 0305.32011

[46] Gromov, M. Foliated plateau problem, part II : harmonic maps of foliations, GAFA, Tome 1 (1991) no. 3, pp. 253-320 | MR 1118731 | Zbl 0768.53012

[47] Henkin, G. An analytic polyhedron is not biholomorphic to a strictly pseudoconvex domain, Dokl. Akad. Nauk SSSR, Tome 210 (1973), pp. 1026-1029 | MR 328125 | Zbl 0288.32015

[48] Isaev, A.; Krantz, S. Domains with non-compact automorphism group : a survey, Adv. Math., Tome 146 (1999) no. 1, pp. 1-38 | MR 1706680 | Zbl 1040.32019

[49] Jonsson, M.; Varolin, D. Stable manifolds of holomorphic diffeomorphisms, Invent. Math., Tome 149 (2002) no. 2, pp. 409-430 | MR 1918677 | Zbl 1048.37047

[50] Kim, K.-T.; Krantz, S. Some new results on domains in complex space with non-compact automorphism group, J. Math. Anal. Appl., Tome 281 (2003) no. 2, pp. 417-424 | MR 1982663 | Zbl 1035.32019

[51] Kobayashi, G. Hyperbolic complex spaces, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften, Tome 318 (1998) | MR 1635983 | Zbl 0917.32019

[52] Landau, E. Uber di Blochste Konstante und zwei verwandte Weltkonstanten, Math. Z., Tome 30 (1929), pp. 608-634 | JFM 55.0770.03 | MR 1545082

[53] Lang, S. Introduction to complex hyperbolic manifolds, Springer Verlag (1987) | MR 886677 | Zbl 0628.32001

[54] Ligocka, E. Some remarks on extension of biholomorphic mappings, Analytic functions, Springer, Berlin, Proc. Seventh Conf., Kozubnik, 1979 (Lecture Notes in Math.) (1980), pp. 350-363 | MR 577466 | Zbl 0458.32008

[55] Lohwater, A. J.; Pommerenke, Ch. On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I (1973) no. 550 | MR 338381 | Zbl 0275.30027

[56] Pinchuk, S. On proper holomorphic mappings of strictly pseudoconvex domains, Sib. Math. J., Tome 15 (1974), pp. 909-917 | Zbl 0303.32016

[57] Pinchuk, S. The scaling method ans holomorphic mappings, Proc. Sympos. Pure Math., Tome 52 (1991) no. 1, pp. 151-161 | Zbl 0744.32013

[58] Pinchuk, S.; Khasanov, S. Asymptotically holomorphic functions and their applications, Math. USSR-Sb., Tome 62 (1989) no. 2, pp. 541-550 | MR 933702 | Zbl 0663.32006

[59] Range, R. Holomorphic functions and integral representations in several complex variables, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 108 (1986) | MR 847923 | Zbl 0591.32002

[60] Ros, A. The Gauss map of minimal surfaces, Differential Geometry-Valencia (2001), pp. 235-250 | MR 1922054 | Zbl 1028.53008

[61] Rosay, J.-P. Sur une caractérisation de la boule parmi les domaines de C n par son groupe d’automorphismes, Ann. Inst. Fourier, Tome 29 (1979) no. 4, pp. 91-97 | Numdam | MR 558590 | Zbl 0402.32001

[62] Rosay, J.-P.; Rudin, W. Holomorphic maps from C n to C n , Trans. Amer. Math. Soc., Tome 310 (1988) no. 1, pp. 47-86 | MR 929658 | Zbl 0708.58003

[63] Ruelle, D. Elements of differentiable dynamics and bifurcation theory, Academic Press, Inc., Boston, MA (1989) | MR 982930 | Zbl 0684.58001

[64] Schwick, W. Repelling periodic points in the Julia set, Bull. London Math. Soc., Tome 29 (1997) no. 3, pp. 314-316 | MR 1435565 | Zbl 0878.30020

[65] Sibony, N. A class of hyperbolic manifolds, Ann. Math. Studies (1981) no. 100, pp. 357-372 | MR 627768 | Zbl 0476.32033

[66] Sibony, N. Dynamique des applications rationnelles de P k , Panor. Synthèses, Soc. Math. France, Paris, Dynamique et géométrie complexes (Lyon, 1997) (1999), pp. 97-185 | MR 1760844 | Zbl 1020.37026

[67] Siu, Y. T.; Yeung, S. K. Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., Tome 124 (1996) no. 1-3, pp. 573-618 | MR 1369429 | Zbl 0856.32017

[68] Stensones, B. A proof of the Michael conjecture (1998) (Preprint)

[69] Stensones, B. Fatou-Bieberbach domains with C -smooth boundary, Ann. of Math. (2), Tome 145 (1997) no. 2, pp. 365-377 | MR 1441879 | Zbl 0883.32020

[70] Sternberg, S. Local contractions and a theorem of Poincaré, Amer. J. Math., Tome 79 (1957), pp. 809-824 | MR 96853 | Zbl 0080.29902

[71] Sukhov, A. On boundary regularity of holomorphic mappings, Mat. Sb., Tome 185 (1994), pp. 131-142 | MR 1317303 | Zbl 0843.32016

[72] Webster, S. On the reflection principle in several complex variables, Proc. Amer. Math. Soc., Tome 71 (1978) no. 1, pp. 26-28 | MR 477138 | Zbl 0626.32019

[73] Wong, B. Characterization of the unit ball in C n by its automorphism group, Invent. Math., Tome 41 (1977) no. 3, pp. 253-257 | MR 492401 | Zbl 0385.32016

[74] Zalcman, L. Normal families : new perspectives, Bull. Amer. Math. Soc., Tome 35 (1998), pp. 215-230 | MR 1624862 | Zbl 1037.30021

[75] Zalcman, L. A heuristic principle in complex function theory, Amer. Math. Monthly, Tome 82 (1975) no. 8, pp. 813-817 | MR 379852 | Zbl 0315.30036