Local dynamics of holomorphic diffeomorphisms

Bracci, Filippo

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 3, pp. 609-636.

Abstract (VO)

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.

Riassunto

Questo è un sunto dello stato dell'arte della dinamica olomorfa complessa dai tempi di Poincaré ai giorni nostri. Sono inoltre indicate alcune nuove idee per mettere in relazione la dinamica discreta con la dinamica continua. È il testo della conferenza tenuta dall'autore nel XVII Convegno dell'UMI a Milano.

Bibliographie

[1] M. Abate, Diagonalization of non-diagonalizable discrete holomorphic dynamical systems, Amer. J. Math., 122 (2000), 757-781. | MR | Zbl

[2] M. Abate, The residual index and the dynamics of holomorphic maps tangent to the identity, Duke Math. J., 107, 1 (2001), 173-207. | fulltext mini-dml | MR | Zbl

[3] M. Abate, An introduction to hyperbolic dynamical systems, I.E.P.I., Pisa, 2001.

[4] M. Abate, Discrete local holomorphic dynamics, Proceedings of 13th Seminar on Analysis and its Applications, Isfahan, March 2003. | MR | Zbl

[5] M. Abate-F. Tovena, Parabolic Curves in $\mathbb{C}^3$, Abstr. Appl. Anal., 5 (2003), 275-294. | fulltext mini-dml | MR | Zbl

[6] M. Abate-F. Bracci-F. Tovena, Index theorems for holomorphic self-maps, Ann. of Math., 159 (2004), 890-935. | MR | Zbl

[7] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer, 1983. | MR | Zbl

[8] P. Baum-R. Bott, Singularities of holomorphic foliations, J. Diff. Geom., 7 (1972), 279-342. | fulltext mini-dml | MR | Zbl

[9] F. Bracci, The dynamics of holomorphic maps near curves of fixed points, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2 (2003), 493-520. | fulltext mini-dml | MR | Zbl

[10] F. Bracci, First order extensions of holomorphic foliations, Hokkaido Math. J. (to appear). | fulltext mini-dml | MR | Zbl

[11] F. Bracci-L. Molino, The dynamics near quasi-parabolic fixed points of holomorphic diffeomorphisms in $\mathbb{C}^2$, Amer. J. Math. (to appear). | MR | Zbl

[12] F. Bracci-T. Suwa, Residues for singular pairs and dynamics of biholomorphic maps of singular surfaces, Internet. J. Math. (to appear). | MR | Zbl

[13] F. Bracci-T. Suwa, Residues for holomorphic foliations of singular pairs, Adv. Geom. (to appear). | MR | Zbl

[14] F. Bracci-F. Tovena, Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces, Math. Z., 242, 3 (2002), 481-490. | MR | Zbl

[15] F. E. Brochero-Martinez, Groups of germs of analytic diffeomorphisms in $(\mathbb{C}^2,0)$, J. Dynam. Control Systems, 9, 1 (2003), 1-32. | MR | Zbl

[16] A. D. Bryuno, Analytic form of differential equations, Trans. Moscow Math. Soc., 25 (1971), 131-288; 26 (1972), 199-239. | Zbl

[17] M. Brunella, Birational geometry of foliations, First Latin American Congress of Math., IMPA, Rio de Janeiro, Brazil 2000. | MR | Zbl

[18] C. Camacho, On the local structure of conformal mappings and holomorphic vector fields in $\mathbb{C}^2$, Asterisque 59-60 (1978), 83-94. | MR | Zbl

[19] C. Camacho-P. Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. (2), 115 (1982), 579-595. | MR | Zbl

[20] J. C. Canille Martins, Holomorphic flows in $\mathbb{C}^3,0$ with resonances, Trans. Amer. Math. Soc., 329 (1992), 825-837. | MR | Zbl

[21] J. Cano, Construction of invariant curves for singular holomorphic vector fields, Proc. Amer. Math. Soc., 125 (9) (1997), 2649-2650. | MR | Zbl

[22] L. Carleson-T. W. Gamelin, Complex dynamics, Springer, 1993. | MR | Zbl

[23] F. Degli Innocenti, Dinamica di germi di foliazioni e diffeomorfismi olomorfi vicino a curve singolari, Tesi di Laurea, Firenze, 2003.

[24] F. Dumortier-P. R. Rodrigues-R. Roussarie, Germs of diffeomorphisms in the plane, Lecture Notes in Math., 902, Springer-Verlag, 1981. | MR | Zbl

[25] J. Écalle, 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. | fulltext mini-dml | MR | Zbl

[26] J. Écalle. Iteration and analytic classification of local diffeomorphisms of $\mathbb{C}^n$, Iteration theory and its functional equations, Lect. Notes in Math. 1163, Springer-Verlag, Berlin, 1985, pp. 41-48. | MR | Zbl

[27] P. Fatou, Substitutions analytiques et equations fonctionelles de deux variables, Ann. Sc. Ec. Norm. Sup. (1924), 67-142. | fulltext mini-dml | Jbk 50.0692.01

[28] X. Gomez-Mont-I. Luengo, Germs of holomorphic vector fields in $\mathbb{C}^3$ without a separatrix, Invent. Math., 109 (1992), 211-219. | MR | Zbl

[29] M. Hakim, Attracting domains for semi-attractive transformations of $\mathbb{C}^p$, Publ. Mat., 38 (1994), 479-499. | MR | Zbl

[30] M. Hakim, Analytic transformations of $(\mathbb{C}^p, 0)$ tangent to the identity, Duke Math. J., 92 (1998), 403-428. | fulltext mini-dml | MR | Zbl

[31] M. Hirsch-C. Pugh-M. Shub, Invariant manifolds, Lecture Notes in Math., 583, Springer-Verlag, 1977. | MR | Zbl

[32] Yu.S. Il'Yashenko, Nonlinear Stokes phenomena, In Nonlinear Stokes phenomena, Adv. Soviet Math. 14, Amer. Math. Soc., Providence, (1993), 1-55. | MR | Zbl

[33] D. Lehmann, Résidus des sous-variétés invariants d'un feuilletage singulier, Ann. Inst. Fourier, Grenoble, 41, 1 (1991), 211-258. | fulltext mini-dml | MR | Zbl

[34] D. Lehmann-T. Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Diff. Geom., 42, 1 (1995), 165-192. | fulltext mini-dml | MR | Zbl

[35] S. Marmi, An introduction to small divisors problems, I.E.P.I. Pisa, 1999.

[36] J. Martinet-J.-P. Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sci. Éc. Norm. Sup., 16 (1983), 571-621. | fulltext mini-dml | MR | Zbl

[37] J.-F. Mattei-R. Moussu, Holonomie et intégrales première, Ann. Sci. École Norm. Sup. (4), 13 (1980), 469-523. | fulltext mini-dml | MR | Zbl

[38] Y. Nishimura, Automorphisms analytiques admettant des sous-variétés de points fixés attractives dans la direction transversale, J. Math. Kyoto Univ., 23-2 (1983), 289-299. | fulltext mini-dml | MR | Zbl

[39] J. Palis-F. Takens, Topological equivalence of normally hyperbolic dynamical systems, Topology, 16 (1977), 335-345. | MR | Zbl

[40] R. Perez-Marco, Non-linearizable holomorphic dynamics having an uncountable number of symmetries, Invent. Math., 199 (1995), 67-127. | MR | Zbl

[41] R. Pérez-Marco, Fixed points and circle maps, Acta Math., 179 (1997), 243-294. | MR | Zbl

[42] R. Perez-Marco-J.-C. Yoccoz, Germes de feuilletages holomorphes à holonomie prescrite, Astérisque, 222 (1994), 345-371. | MR | Zbl

[43] J. Pöschel, On invariant manifolds of complex analytic mappings near fixed points, Exp. Math., 4 (1986), 97-109. | MR | Zbl

[44] L. Reich, Das Typenproblem bei formal-biholomorphien Abbildungen mit anziehendem Fixpunkt, Math. Ann., 179 (1969), 227-250. | MR | Zbl

[45] L. Reich, Normalformen biholomorpher Abbildungen mit anziehendem Fixpunkt, Math. Ann., 180 (1969), 233-255. | MR | Zbl

[46] M. Rivi, Parabolic manifolds for semi-attractive holomorphic germs, Michigan Math. J., 49, 2 (2001), 211-241. | fulltext mini-dml | MR | Zbl

[47] M. Sebastiani, Sur l'existence de séparatrices locales des feuilletages des surfaces, An. Acad. Bras. Ci., 69 (2) (1997), 159-162. | MR | Zbl

[48] J. H. Shapiro, Composition operators and classical function theory, Springer UTX 1993. | MR | Zbl

[49] A. A. Shcherbakov, Topological classification of germs of conformal mappings with identity linear part, Moscow Univ. Math. Bull., 37 (1982), 60-65. | Zbl

[50] T. Suwa Indices of vector fields and residues of singular holomorphic foliations, Hermann, Paris, 1998. | MR | Zbl

[51] T. Ueda, Analytic transformations of two complex variables with parabolic fixed points, Preprint 1997.

[52] T. Ueda, Local structure of analytic transformations of two complex variables, I, J. Math. Kyoto Univ., 26-2 (1986), 233-261. | fulltext mini-dml | MR | Zbl

[53] T. Ueda, Local structure of analytic transformations of two complex variables, II, J. Math. Kyoto Univ., 31-3 (1991), 695-711. | fulltext mini-dml | MR | Zbl

[54] S. M. Voronin, Analytic classification of germs of conformal maps $(\mathbb{C}, 0 ) \to (\mathbb{C}, 0 )$ with identity linear part, Funct. Anal. Appl., 15 (1981), 1-17. | MR | Zbl

[55] B. J. Weickert, Attracting basins for automorphisms of $\mathbb{C}^2$, Invent. Math., 132 (1998), 581-605. | MR | Zbl

[56] H. Wu, Complex stable manifolds of holomorphic diffeomorphisms, Indiana Univ. J. Math., 42 (1993), 1349-1358. | MR | Zbl