Local dynamics of holomorphic diffeomorphisms

Filippo Bracci

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 3, page 609-636
  • ISSN: 0392-4041


This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.

How to cite


Bracci, Filippo. "Local dynamics of holomorphic diffeomorphisms." Bollettino dell'Unione Matematica Italiana 7-B.3 (2004): 609-636. <http://eudml.org/doc/195600>.

abstract = {This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.},
author = {Bracci, Filippo},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {609-636},
publisher = {Unione Matematica Italiana},
title = {Local dynamics of holomorphic diffeomorphisms},
url = {http://eudml.org/doc/195600},
volume = {7-B},
year = {2004},

AU - Bracci, Filippo
TI - Local dynamics of holomorphic diffeomorphisms
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/10//
PB - Unione Matematica Italiana
VL - 7-B
IS - 3
SP - 609
EP - 636
AB - This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.
LA - eng
UR - http://eudml.org/doc/195600
ER -


  1. ABATE, M., Diagonalization of non-diagonalizable discrete holomorphic dynamical systems, Amer. J. Math., 122 (2000), 757-781. Zbl0966.32018MR1771573
  2. ABATE, M., The residual index and the dynamics of holomorphic maps tangent to the identity, Duke Math. J., 107, 1 (2001), 173-207. Zbl1015.37035MR1815255
  3. ABATE, M., An introduction to hyperbolic dynamical systems, I.E.P.I., Pisa, 2001. 
  4. ABATE, M., Discrete local holomorphic dynamics, Proceedings of 13th Seminar on Analysis and its Applications, Isfahan, March 2003. Zbl1072.37036MR2114495
  5. ABATE, M.- TOVENA, F., Parabolic Curves in C 3 , Abstr. Appl. Anal., 5 (2003), 275-294. Zbl1018.37027MR1981266
  6. ABATE, M.- BRACCI, F.- TOVENA, F., Index theorems for holomorphic self-maps, Ann. of Math., 159 (2004), 890-935. Zbl1056.37025MR2081441
  7. ARNOLD, V. I., Geometrical methods in the theory of ordinary differential equations, Springer, 1983. Zbl0507.34003MR695786
  8. BAUM, P.- BOTT, R., Singularities of holomorphic foliations, J. Diff. Geom., 7 (1972), 279-342. Zbl0268.57011MR377923
  9. BRACCI, F., The dynamics of holomorphic maps near curves of fixed points, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 493-520. Zbl1170.32311MR2020858
  10. BRACCI, F., First order extensions of holomorphic foliations, Hokkaido Math. J. (to appear). Zbl1082.32021MR2073011
  11. BRACCI, F.- MOLINO, L., The dynamics near quasi-parabolic fixed points of holomorphic diffeomorphisms in C 2 , Amer. J. Math. (to appear). Zbl1063.37040MR2058387
  12. BRACCI, F.- SUWA, T., Residues for singular pairs and dynamics of biholomorphic maps of singular surfaces, Internet. J. Math. (to appear). Zbl1057.37045MR2072088
  13. BRACCI, F.- SUWA, T., Residues for holomorphic foliations of singular pairs, Adv. Geom. (to appear). Zbl1074.32009MR2110462
  14. BRACCI, F.- TOVENA, F., Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces, Math. Z., 242, 3 (2002), 481-490. Zbl1052.37037MR1985461
  15. BROCHERO-MARTINEZ, F. E., Groups of germs of analytic diffeomorphisms in C 2 , 0 , J. Dynam. Control Systems, 9, 1 (2003), 1-32. Zbl1017.37011MR1956442
  16. BRYUNO, A. D., Analytic form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239. Zbl0272.34018
  17. BRUNELLA, M., Birational geometry of foliations, First Latin American Congress of Math., IMPA, Rio de Janeiro, Brazil2000. Zbl1073.14022MR1948251
  18. CAMACHO, C., On the local structure of conformal mappings and holomorphic vector fields in C 2 , Asterisque59-60 (1978), 83-94. Zbl0415.30015MR542732
  19. CAMACHO, C.- SAD, P., Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2), 115 (1982), 579-595. Zbl0503.32007MR657239
  20. CANILLE MARTINS, J. C., Holomorphic flows in C 3 , 0 with resonances, Trans. Amer. Math. Soc., 329 (1992), 825-837. Zbl0746.58068MR1073776
  21. CANO, J., Construction of invariant curves for singular holomorphic vector fields, Proc. Amer. Math. Soc., 125 (9) (1997), 2649-2650. Zbl0890.34001MR1389507
  22. CARLESON, L.- GAMELIN, T. W., Complex dynamics, Springer, 1993. Zbl0782.30022MR1230383
  23. DEGLI INNOCENTI, F., Dinamica di germi di foliazioni e diffeomorfismi olomorfi vicino a curve singolari, Tesi di Laurea, Firenze, 2003. 
  24. DUMORTIER, F.- RODRIGUES, P. R.- ROUSSARIE, R., Germs of diffeomorphisms in the plane, Lecture Notes in Math., 902, Springer-Verlag, 1981. Zbl0502.58001MR653474
  25. ÉCALLE, J., Les fonctions résurgentes, Tome III: L'équation du pont et la classification analytiques des objects locaux, Publ. Math. Orsay, 85-5, Université de Paris-Sud, Orsay, 1985. Zbl0602.30029MR852210
  26. ÉCALLE, J.. Iteration and analytic classification of local diffeomorphisms of C n , Iteration theory and its functional equations, Lect. Notes in Math.1163, Springer-Verlag, Berlin, 1985, pp. 41-48. Zbl0602.58007MR829756
  27. FATOU, P., Substitutions analytiques et equations fonctionelles de deux variables, Ann. Sc. Ec. Norm. Sup. (1924), 67-142. JFM50.0692.01
  28. GOMEZ-MONT, X.- LUENGO, I., Germs of holomorphic vector fields in C 3 without a separatrix, Invent. Math., 109 (1992), 211-219. Zbl0774.32019MR1172688
  29. HAKIM, M., Attracting domains for semi-attractive transformations of C p , Publ. Mat., 38 (1994), 479-499. Zbl0832.58031MR1316642
  30. HAKIM, M., Analytic transformations of C p , 0 tangent to the identity, Duke Math. J., 92 (1998), 403-428. Zbl0952.32012MR1612730
  31. HIRSCH, M.- PUGH, C.- SHUB, M., Invariant manifolds, Lecture Notes in Math., 583, Springer-Verlag, 1977. Zbl0355.58009MR501173
  32. IL'YASHENKO, YU.S., Nonlinear Stokes phenomena, In Nonlinear Stokes phenomena, Adv. Soviet Math.14, Amer. Math. Soc., Providence, (1993), 1-55. Zbl0804.32011MR1206041
  33. LEHMANN, D., Résidus des sous-variétés invariants d'un feuilletage singulier, Ann. Inst. Fourier, Grenoble, 41, 1 (1991), 211-258. Zbl0727.57024MR1112198
  34. LEHMANN, D.- SUWA, T., Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Diff. Geom., 42, 1 (1995), 165-192. Zbl0844.32007MR1350698
  35. MARMI, S., An introduction to small divisors problems, I.E.P.I.Pisa, 1999. 
  36. MARTINET, J.- RAMIS, J.-P., Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Éc. Norm. Sup., 16 (1983), 571-621. Zbl0534.34011MR740592
  37. MATTEI, J.-F.- MOUSSU, R., Holonomie et intégrales première, Ann. Sci. École Norm. Sup. (4), 13 (1980), 469-523. Zbl0458.32005MR608290
  38. NISHIMURA, Y., Automorphisms analytiques admettant des sous-variétés de points fixés attractives dans la direction transversale, J. Math. Kyoto Univ., 23-2 (1983), 289-299. Zbl0545.32014MR706166
  39. PALIS, J.- TAKENS, F., Topological equivalence of normally hyperbolic dynamical systems, Topology, 16 (1977), 335-345. Zbl0391.58015MR474409
  40. PEREZ-MARCO, R., Non-linearizable holomorphic dynamics having an uncountable number of symmetries, Invent. Math., 199 (1995), 67-127. Zbl0862.58045MR1309972
  41. PÉREZ-MARCO, R., Fixed points and circle maps, Acta Math., 179 (1997), 243-294. Zbl0914.58027MR1607557
  42. PEREZ-MARCO, R.- YOCCOZ, J.-C., Germes de feuilletages holomorphes à holonomie prescrite, Astérisque, 222 (1994), 345-371. Zbl0809.32008MR1285395
  43. PÖSCHEL, J., On invariant manifolds of complex analytic mappings near fixed points, Exp. Math., 4 (1986), 97-109. Zbl0597.32011MR879908
  44. REICH, L., Das Typenproblem bei formal-biholomorphien Abbildungen mit anziehendem Fixpunkt, Math. Ann., 179 (1969), 227-250. Zbl0177.34403MR237485
  45. REICH, L., Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt, Math. Ann., 180 (1969), 233-255. Zbl0177.34501MR243101
  46. RIVI, M., Parabolic manifolds for semi-attractive holomorphic germs, Michigan Math. J., 49, 2 (2001), 211-241. Zbl1015.37037MR1852300
  47. SEBASTIANI, M., Sur l'existence de séparatrices locales des feuilletages des surfaces, An. Acad. Bras. Ci., 69 (2) (1997), 159-162. Zbl0887.57033MR1754036
  48. SHAPIRO, J. H., Composition operators and classical function theory, SpringerUTX1993. Zbl0791.30033MR1237406
  49. SHCHERBAKOV, A. A., Topological classification of germs of conformal mappings with identity linear part, Moscow Univ. Math. Bull., 37 (1982), 60-65. Zbl0508.30015
  50. SUWA, T.Indices of vector fields and residues of singular holomorphic foliations, Hermann, Paris, 1998. Zbl0910.32035MR1649358
  51. UEDA, T., Analytic transformations of two complex variables with parabolic fixed points, Preprint 1997. 
  52. UEDA, T., Local structure of analytic transformations of two complex variables, I, J. Math. Kyoto Univ., 26-2 (1986), 233-261. Zbl0611.32001MR849219
  53. UEDA, T., Local structure of analytic transformations of two complex variables, II, J. Math. Kyoto Univ., 31-3 (1991), 695-711. Zbl0756.32014MR1127094
  54. VORONIN, S. M., Analytic classification of germs of conformal maps C , 0 C , 0 with identity linear part, Funct. Anal. Appl., 15 (1981), 1-17. Zbl0463.30010MR609790
  55. WEICKERT, B. J., Attracting basins for automorphisms of C 2 , Invent. Math., 132 (1998), 581-605. Zbl0932.37028MR1625716
  56. WU, H., Complex stable manifolds of holomorphic diffeomorphisms, Indiana Univ. J. Math., 42 (1993), 1349-1358. Zbl0811.58009MR1266097

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