Stable manifolds of biholomorphisms in \(\mathbb{C}^n\) asymptotic to formal curves. (English) Zbl 1523.32035
Summary: Given a germ of biholomorphism \(F\in \operatorname{Diff}(\mathbb{C}^n,0)\) with a formal invariant curve \(\Gamma\) such that the multiplier of the restricted formal diffeomorphism \(F|_\Gamma\) is a root of unity or satisfies \(|(F|_\Gamma )^{\prime }(0)|<1\), we prove that either \(\Gamma\) is contained in the set of periodic points of \(F\) or there exists a finite family of stable manifolds of \(F\) where all the orbits are asymptotic to \(\Gamma\) and whose union eventually contains every orbit asymptotic to \(\Gamma \). This result generalizes to the case where \(\Gamma\) is a formal periodic curve.
{© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}
{© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.}
MSC:
| 32M25 | Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
References:
| [1] | M.Abate and F.Tovena, Parabolic curves in \(\mathbb{C}^3\), Abstr. Appl. Anal.1 (2003), no. 5, 275-294. · Zbl 1018.37027 |
| [2] | M.Arizzi and J.Raissy, On Écalle‐Hakim’s theorems in holomorphic dynamics. Frontiers in complex dynamics, Princeton Mathematical Series 51, Princeton University Press, Princeton, NJ, 2014, pp. 387-449. · Zbl 1359.37001 |
| [3] | W.Balser, Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, New York, 2000. · Zbl 0942.34004 |
| [4] | M. A.Barkatou, An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system, Appl. Algebra Engrg. Comm. Comput.8 (1997), no. 1, 1-23. · Zbl 0867.65034 |
| [5] | G.Binyamini, Finiteness properties of formal Lie group actions, Transform. Groups20 (2015), no. 4, 939-952. · Zbl 1361.37030 |
| [6] | B.Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier (Grenoble)42 (1992), no. 3, 517-540. · Zbl 0759.34003 |
| [7] | C.Camacho and P.Sad, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math.115 (1982), 579-595. · Zbl 0503.32007 |
| [8] | F.Cano, R.Moussu, and J.‐P.Rolin, Non‐oscillating integral curves and valuations, J. Reine Angew. Math.582 (2005), 107-141. · Zbl 1078.32020 |
| [9] | F.Cano, R.Moussu, and F.Sanz, Pinceaux de courbes intégrales d’un champ de vecteurs analytique, Astérisque297 (2004), 1-34. · Zbl 1101.34017 |
| [10] | F.Cano, C.Roche, and M.Spivakovsky, Reduction of singularities of three‐dimensional line foliations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM108 (2014), no. 1, 221-258. · Zbl 1346.14038 |
| [11] | D.Cerveau and A. N.Lins, Codimension two holomorphic foliations, J. Differential Geom.113 (2019), no. 3, 385-416. · Zbl 1432.37076 |
| [12] | D.Cerveau and J.‐F.Mattei, Formes intégrables holomorphes singulières, Astérisque, vol. 97, Société Mathématique de France, Paris, 1982, 193 pp. · Zbl 0545.32006 |
| [13] | J.Dugundji, Topology. Series in Advanced Mathematics. Allyn and Bacon, Boston, MA, 1966. · Zbl 0144.21501 |
| [14] | N.Dunford and J. T.Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402 |
| [15] | J.Écalle, Théorie itérative: introduction à la théorie des invariants holomorphes, J. Math. Pures Appl. (9)54 (1975), 183-258. · Zbl 0285.26010 |
| [16] | X.Gómez‐Mont and I.Luengo, Germs of holomorphic vector fields in \(\mathbb{C}^3\) without a separatrix, Invent. Math.109 (1992), no. 2, 211-219. · Zbl 0774.32019 |
| [17] | R. C.Gunning, Introduction to holomorphic functions of several variables, Vol. 2, Wadsworth, AMS, Florence, KY, 1990. · Zbl 0699.32001 |
| [18] | M.Hakim, Analytic transformations of \((\mathbb{C}^p,0)\) tangent to the identity, Duke Math. J.92 (1998), no. 2, 403-428. · Zbl 0952.32012 |
| [19] | M.Hakim, Transformations tangent to the identity. Stable pieces of manifolds, Prépublication d’Orsay numéro 30, Orsay, 1997. |
| [20] | M.Hukuhara, T.Kimura, and T.Matuda, Equations différentielles ordinaires du premier ordre dans le champ complexe. The Mathematical Society of Japan, Tokyo, 1961. · Zbl 0101.30002 |
| [21] | Y.Ilyashenko and S.Yakovenko, Lectures on analytic differential equations, Grad. Stud. Math., vol. 86, AMS, Providence, RI, 2008. · Zbl 1186.34001 |
| [22] | L.López‐Hernanz, J.Raissy, J.Ribón, and F.Sanz Sánchez, Stable manifolds of two‐dimensional biholomorphisms asymtptic to formal curves, Int. Math. Res. Not. IMRN (2021), no. 17, 12847-12887. · Zbl 1487.32159 |
| [23] | L.López‐Hernanz and F.Sanz Sánchez, Parabolic curves of diffeomorphisms asymptotic to formal invariant curves, J. Reine Angew. Math.739 (2018), 277-296. · Zbl 1412.32015 |
| [24] | M.Martelo and J.Ribón, Derived length of solvable groups of local diffeomorphisms, Math. Ann.358 (2014), no. 3‐4, 701-728. · Zbl 1328.37028 |
| [25] | J.Martinet and J.‐P.Ramis, Classification analytique des équations differentielles non linéaires résonnantes du premier ordre, Ann. Sci. École Norm. Sup. (4)16 (1983), no. 4, 571-621 (1984). · Zbl 0534.34011 |
| [26] | A. S.deMedeiros, Singular foliations and differential \(p\)‐forms, Ann. Fac. Sci. Toulouse Math. (6)9 (2000), no. 3, 451-466. · Zbl 0997.58001 |
| [27] | A. L.Onishchik and E. B.Vinberg, Lie groups and algebraic groups, Springer, Berlin, 1990. · Zbl 0722.22004 |
| [28] | D.Panazzolo, Resolution of singularities of real‐analytic vector fields in dimension three, Acta Math.197 (2006), no. 2, 167-289. · Zbl 1112.37016 |
| [29] | R.Pérez‐Marco, Sur une question de Dulac et Fatou, C. R. Acad. Sci. Paris321 s. I (1995), 1045-1048. · Zbl 0847.58060 |
| [30] | R.Pérez‐Marco, Solution to Briot and Bouquet problem on singularities of differential equations, arXiv:1802.03630v4. |
| [31] | J. P.Ramis and Y.Sibuya, A new proof of multisummability of formal solutions of non linear meromorphic differential equations, Ann. Inst. Fourier33 (1994), 811-848. · Zbl 0812.34005 |
| [32] | J.Ribón, Families of diffeomorphisms without periodic curves, Michigan Math. J.53 (2005), no. 2, 243-256. · Zbl 1094.37029 |
| [33] | J.Ribón, Embedding smooth and formal diffeomorphisms through the Jordan‐Chevalley decomposition, J. Differ. Equ.253 (2012), no. 12, 3211-3231. · Zbl 1359.37043 |
| [34] | J.Ribón, Finite dimensional groups of local diffeomorphisms, Israel J. Math.227 (2018), no. 1, 289-329. · Zbl 1403.32010 |
| [35] | J.Ribón, The solvable length of groups of local diffeomorphisms, J. Reine Angew. Math.752 (2019), 105-139. · Zbl 1432.32024 |
| [36] | K.Saito, On a generalization of de Rham lemma, Ann. Inst. Fourier (Grenoble)26 (1976), no. 2, vii, 165-170. · Zbl 0338.13009 |
| [37] | H. L.Turrittin, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math.93 (1955), 27-66. · Zbl 0064.33603 |
| [38] | T.Ueda, Local structure of analytic transformations of two complex variables, I, J. Math. Kyoto Univ.26‐2 (1986), 233-261. · Zbl 0611.32001 |
| [39] | R. J.Walker, Algebraic curves, Dover Publications, Inc., New York, 1950. · Zbl 0039.37701 |
| [40] | W.Wasow, Asymptotic expansions for ordinary differential equations, Intersciencie, New York, 1965 (re‐edited Dover Publications Inc. 1987). · Zbl 0169.10903 |
| [41] | X.Zhang, The embedding flows of \(C^\infty\) hyperbolic diffeomorphisms, J. Differ. Equ.250 (2011), no. 5, 2283-2298. · Zbl 1221.37101 |
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