# American Institute of Mathematical Sciences

2013, 33(5): 1773-1807. doi: 10.3934/dcds.2013.33.1773

## Formal Poincaré-Dulac renormalization for holomorphic germs

 1 Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy 2 Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy

Received  April 2012 Revised  July 2012 Published  December 2012

We shall describe an alternative approach to a general renormalization procedure for formal self-maps, originally suggested by Chen-Della Dora and Wang-Zheng-Peng, giving formal normal forms simpler than the classical Poincaré-Dulac normal form. As example of application we shall compute a complete list of normal forms for bi-dimensional superattracting germs with non-vanishing quadratic term; in most cases, our normal forms will be the simplest possible ones (in the sense of Wang-Zheng-Peng). We shall also discuss a few examples of renormalization of germs tangent to the identity, revealing interesting second-order resonance phenomena.
Citation: Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1773-1807. doi: 10.3934/dcds.2013.33.1773
##### References:
 [1] M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity,, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1. [2] M. Abate, Discrete holomorphic local dynamical systems,, in, 1998, (2010), 1. [3] M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity,, Discrete Contin. Dyn. Syst. Suppl (2005), Suppl (2005), 1. [4] M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields,, J. Differential Equations, 251 (2011), 2612. doi: 10.1016/j.jde.2011.05.031. [5] A. Algaba, E. Freire and E. Gamero, Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies,, Rocky Mountain J. Math. 29 (1999), 29 (1999), 13. doi: 10.1216/rmjm/1181071677. [6] A. Algaba, E. Freire, E. Gamero and C. Garcia, Quasi-homogeneous normal forms,, J. Comput. Appl. Math. 150 (2003), 150 (2003), 193. doi: 10.1016/S0377-0427(02)00660-X. [7] V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations,", Springer Verlag, (1988). [8] A. Baider, Unique normal forms for vector fields and Hamiltonians,, J. Differential Equations, 78 (1989), 33. doi: 10.1016/0022-0396(89)90074-0. [9] A. Baider and R. Churchill, Unique normal forms for planar vector fields,, Math. Z., 199 (): 303. [10] A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form,, J. Differential Equations, 99 (1992), 205. doi: 10.1016/0022-0396(92)90022-F. [11] G. R. Belitskii, Invariant normal forms of formal series,, Functional Anal. Appl., 13 (1979), 46. [12] G. R. Belitskii, Normal forms relative to a filtering action of a group,, Trans. Moscow Math. Soc., 40 (1979), 3. [13] F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms,, J. Eur. Math. Soc. , (). [14] A. D. Brjuno, Analytic form of differential equations. I,, Trans. Moscow Math. Soc. 25 (1971), 25 (1971), 131. [15] A. D. Brjuno, Analytic form of differential equations. II,, Trans. Moscow Math. Soc. 26 (1972), 26 (1972), 199. [16] H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case,, in, 898, (1980), 54. [17] H. Cartan, "Cours de calcul différentiel,", Hermann, (1977). [18] G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point,, Numer. Algorithms, 22 (1999), 213. [19] G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems,, J. Differential Equations, 166 (2000), 79. [20] J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux,", Publ. Math. Orsay, 85-05, (1985), 85. [21] J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$,, in, 1163, (1984), 41. [22] E. Fischer, Über die differentiationsprozesse der algebra,, J. für Math., 148 (1917), 1. [23] G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems,, Cubo, 9 (2007), 1. [24] A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point,, Z. Angew. Math. Phys., 39 (1988), 713. doi: 10.1007/BF00948732. [25] F. Ichikawa, On finite determinacy of formal vector fields,, Invent. Math. 70 (1982/83), 70 (): 45. [26] F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane,, Tokyo J. Math. 8 (1985), 8 (1985), 463. doi: 10.3836/tjm/1270151227. [27] H. Kokubu, H. Oka and D. Wang, Linear grading function and further reduction of normal forms,, J. Differential Equations, 132 (1996), 293. doi: 10.1006/jdeq.1996.0181. [28] E. Lombardi and L. Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation,, Ann. Sci. Éc. Norm. Supér. 43 (2010), 43 (2010), 659. [29] D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings,, J. Differential Equations, 247 (2009), 685. [30] J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag,, Berlin, (2003). [31] J. Murdock, Hypernormal form theory: Foundations and algorithms,, J. Differential Equations, 205 (2004), 424. [32] J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms,, J. Differential Equations, 238 (2007), 234. [33] J. Raissy, Torus actions in the normalization problem,, J. Geom. Anal. 20 (2010), 20 (2010), 472. [34] J. Raissy, Brjuno conditions for linearization in presence of resonances,, in, (2010), 201. [35] H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition,, Ergodic Theory Dynam. Systems, 22 (2002), 1551. [36] J. A. Sanders, Normal form theory and spectral sequences,, J. Differential Equations, 192 (2003), 536. [37] D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps,, C. R. Math. Acad. Sci. Paris, 343 (2006), 657. [38] D. Wang, M. Zheng and J. Peng, Further reduction of normal forms and unique normal forms of smooth maps,, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 18 (2008), 803. doi: 10.1142/S0218127408020665. [39] P. Yu and Y. Yuan, The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 219.

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##### References:
 [1] M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity,, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1. [2] M. Abate, Discrete holomorphic local dynamical systems,, in, 1998, (2010), 1. [3] M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity,, Discrete Contin. Dyn. Syst. Suppl (2005), Suppl (2005), 1. [4] M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields,, J. Differential Equations, 251 (2011), 2612. doi: 10.1016/j.jde.2011.05.031. [5] A. Algaba, E. Freire and E. Gamero, Hypernormal forms for equilibria of vector fields. Codimension one linear degeneracies,, Rocky Mountain J. Math. 29 (1999), 29 (1999), 13. doi: 10.1216/rmjm/1181071677. [6] A. Algaba, E. Freire, E. Gamero and C. Garcia, Quasi-homogeneous normal forms,, J. Comput. Appl. Math. 150 (2003), 150 (2003), 193. doi: 10.1016/S0377-0427(02)00660-X. [7] V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations,", Springer Verlag, (1988). [8] A. Baider, Unique normal forms for vector fields and Hamiltonians,, J. Differential Equations, 78 (1989), 33. doi: 10.1016/0022-0396(89)90074-0. [9] A. Baider and R. Churchill, Unique normal forms for planar vector fields,, Math. Z., 199 (): 303. [10] A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form,, J. Differential Equations, 99 (1992), 205. doi: 10.1016/0022-0396(92)90022-F. [11] G. R. Belitskii, Invariant normal forms of formal series,, Functional Anal. Appl., 13 (1979), 46. [12] G. R. Belitskii, Normal forms relative to a filtering action of a group,, Trans. Moscow Math. Soc., 40 (1979), 3. [13] F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms,, J. Eur. Math. Soc. , (). [14] A. D. Brjuno, Analytic form of differential equations. I,, Trans. Moscow Math. Soc. 25 (1971), 25 (1971), 131. [15] A. D. Brjuno, Analytic form of differential equations. II,, Trans. Moscow Math. Soc. 26 (1972), 26 (1972), 199. [16] H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case,, in, 898, (1980), 54. [17] H. Cartan, "Cours de calcul différentiel,", Hermann, (1977). [18] G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point,, Numer. Algorithms, 22 (1999), 213. [19] G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems,, J. Differential Equations, 166 (2000), 79. [20] J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux,", Publ. Math. Orsay, 85-05, (1985), 85. [21] J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$,, in, 1163, (1984), 41. [22] E. Fischer, Über die differentiationsprozesse der algebra,, J. für Math., 148 (1917), 1. [23] G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems,, Cubo, 9 (2007), 1. [24] A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point,, Z. Angew. Math. Phys., 39 (1988), 713. doi: 10.1007/BF00948732. [25] F. Ichikawa, On finite determinacy of formal vector fields,, Invent. Math. 70 (1982/83), 70 (): 45. [26] F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane,, Tokyo J. Math. 8 (1985), 8 (1985), 463. doi: 10.3836/tjm/1270151227. [27] H. Kokubu, H. Oka and D. Wang, Linear grading function and further reduction of normal forms,, J. Differential Equations, 132 (1996), 293. doi: 10.1006/jdeq.1996.0181. [28] E. Lombardi and L. Stolovitch, Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation,, Ann. Sci. Éc. Norm. Supér. 43 (2010), 43 (2010), 659. [29] D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings,, J. Differential Equations, 247 (2009), 685. [30] J. Murdock, "Normal Forms And Unfoldings For Local Dynamical Systems," Springer Verlag,, Berlin, (2003). [31] J. Murdock, Hypernormal form theory: Foundations and algorithms,, J. Differential Equations, 205 (2004), 424. [32] J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms,, J. Differential Equations, 238 (2007), 234. [33] J. Raissy, Torus actions in the normalization problem,, J. Geom. Anal. 20 (2010), 20 (2010), 472. [34] J. Raissy, Brjuno conditions for linearization in presence of resonances,, in, (2010), 201. [35] H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition,, Ergodic Theory Dynam. Systems, 22 (2002), 1551. [36] J. A. Sanders, Normal form theory and spectral sequences,, J. Differential Equations, 192 (2003), 536. [37] D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps,, C. R. Math. Acad. Sci. Paris, 343 (2006), 657. [38] D. Wang, M. Zheng and J. Peng, Further reduction of normal forms and unique normal forms of smooth maps,, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 18 (2008), 803. doi: 10.1142/S0218127408020665. [39] P. Yu and Y. Yuan, The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue,, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 8 (2001), 219.
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