The dynamics of holomorphic maps near curves of fixed points

Bracci, Filippo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 493-520.

Let $M$ be a two-dimensional complex manifold and $f:M\to M$ a holomorphic map. Let $S\subset M$ be a curve made of fixed points of $f$, i.e. $\mathrm{Fix}\left(f\right)=S$. We study the dynamics near $S$ in case $f$ acts as the identity on the normal bundle of the regular part of $S$. Besides results of local nature, we prove that if $S$ is a globally and locally irreducible compact curve such that $S·S<0$ then there exists a point $p\in S$ and a holomorphic $f$-invariant curve with $p$ on the boundary which is attracted by $p$ under the action of $f$. These results are achieved introducing and studying a family of local holomorphic foliations related to $f$ near $S$.

[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 (2001), 173-207. | MR | Zbl

[3] P. Baum - R. Bott, Singularities of holomorphic foliations, J. Differential Geom. 7 (1972), 279-342. | MR | Zbl

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

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

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

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

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

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

[10] 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. | Zbl

[11] R. C. Gunning - H. Rossi, “Analytic functions of several complex variables", Prentice-Hall, 1965. | MR | Zbl

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

[13] H. B. Laufer, “Normal two-dimensional singularities", Ann. Math. Stud. 71, 1971. | MR | Zbl

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

[15] D. Lehmann - T. Suwa, Residues of holomorphic vector fields relative to singular invariant subvarieties, J. Differential Geom. 42 (1995), 165-192. | MR | Zbl

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

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

[18] T. Suwa, Indices of holomorphic vector fields relative to invariant curves on surfaces, Proc. Amer. Math. Soc. 123 (1995), 2989-2997. | MR | Zbl

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

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

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