Formal Poincaré-Dulac renormalization for holomorphic germs

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

Formal Poincaré-Dulac renormalization for holomorphic germs

Marco Abate ^1, and Jasmin Raissy ^2,

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

Abstract
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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.

Keywords: Poincaré-Dulac normal form, superattracting germs, renormalization, tangent to the identity maps., formal transformation.

Mathematics Subject Classification: Primary: 37G05, 37F99, 32H5.

Citation: Marco Abate, Jasmin Raissy. Formal Poincaré-Dulac renormalization for holomorphic germs. Discrete and Continuous Dynamical Systems, 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), 1-14.
[2]	M. Abate, Discrete holomorphic local dynamical systems, in "Holomorphic Dynamical Systems" (G.Gentili, J. Guénot and G. Patrizio, eds.), Lect. Notes in Math. 1998, Springer, Berlin, 2010, pp. 1-55.
[3]	M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10.
[4]	M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 251 (2011), 2612-2684. 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), 13-45. 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), 193-216. doi: 10.1016/S0377-0427(02)00660-X.
[7]	V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988.
[8]	A. Baider, Unique normal forms for vector fields and Hamiltonians, J. Differential Equations, 78 (1989), 33-52. doi: 10.1016/0022-0396(89)90074-0.
[9]	A. Baider and R. Churchill, Unique normal forms for planar vector fields, Math. Z., 199 (988), 303-310.
[10]	A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244. doi: 10.1016/0022-0396(92)90022-F.
[11]	G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67.
[12]	G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46.
[13]	F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms, J. Eur. Math. Soc. arXiv:0912.0428v2.
[14]	A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288.
[15]	A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239.
[16]	H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in "Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980)", Lecture Notes in Math. 898, Springer, Berlin, 1981, pp. 54-74.
[17]	H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977.
[18]	G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230.
[19]	G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.
[20]	J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux," Publ. Math. Orsay, 85-05, Université de Paris-Sud, Orsay, 1985.
[21]	J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$, in "Iteration Theory And Its Functional Equations (Lochau, 1984)", Lect. Notes in Math., 1163, Springer-Verlag, Berlin, 1985, pp. 41-48.
[22]	E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78.
[23]	G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11.
[24]	A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys., 39 (1988), 713-732. doi: 10.1007/BF00948732.
[25]	F. Ichikawa, On finite determinacy of formal vector fields, Invent. Math. 70 (1982/83), 45-52.
[26]	F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane, Tokyo J. Math. 8 (1985), 463-472. 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), 293-318. 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), 659-718.
[29]	D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709.
[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), 424-465.
[32]	J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256.
[33]	J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524.
[34]	J. Raissy, Brjuno conditions for linearization in presence of resonances, in "Asymptotics In Dynamics, Geometry And PDE's; Generalized Borel Summation, Vol. I" (O. Costin, F. Fauvet, F. Menous and D. Sauzin, eds.), Edizioni Della Normale, Pisa, 2010, pp. 201-218.
[35]	H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems, 22 (2002), 1551-1573.
[36]	J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552.
[37]	D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps, C. R. Math. Acad. Sci. Paris, 343 (2006), 657-660.
[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), 803-825. 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), 219-249.

show all references

References:

[1]	M. Abate, Holomorphic classification of $2$-dimensional quadratic maps tangent to the identity, Sūkikenkyūsho Kōkyūroku, 1447 (2005), 1-14.
[2]	M. Abate, Discrete holomorphic local dynamical systems, in "Holomorphic Dynamical Systems" (G.Gentili, J. Guénot and G. Patrizio, eds.), Lect. Notes in Math. 1998, Springer, Berlin, 2010, pp. 1-55.
[3]	M. Abate and F. Tovena, Formal classification of holomorphic maps tangent to the identity, Discrete Contin. Dyn. Syst. Suppl (2005), 1-10.
[4]	M. Abate and F. Tovena, Poincaré-Bendixson theorems for meromorphic connections and holomorphic homogeneous vector fields, J. Differential Equations, 251 (2011), 2612-2684. 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), 13-45. 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), 193-216. doi: 10.1016/S0377-0427(02)00660-X.
[7]	V. I. Arnold, "Geometrical Methods In The Theory Of Ordinary Differential Equations," Springer Verlag, New York, 1988.
[8]	A. Baider, Unique normal forms for vector fields and Hamiltonians, J. Differential Equations, 78 (1989), 33-52. doi: 10.1016/0022-0396(89)90074-0.
[9]	A. Baider and R. Churchill, Unique normal forms for planar vector fields, Math. Z., 199 (988), 303-310.
[10]	A. Baider and J. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244. doi: 10.1016/0022-0396(92)90022-F.
[11]	G. R. Belitskii, Invariant normal forms of formal series, Functional Anal. Appl., 13 (1979), 46-67.
[12]	G. R. Belitskii, Normal forms relative to a filtering action of a group, Trans. Moscow Math. Soc., 40 (1979), 3-46.
[13]	F. Bracci and D. Zaitsev, Dynamics of one-resonant biholomorphisms, J. Eur. Math. Soc. arXiv:0912.0428v2.
[14]	A. D. Brjuno, Analytic form of differential equations. I, Trans. Moscow Math. Soc. 25 (1971), 131-288.
[15]	A. D. Brjuno, Analytic form of differential equations. II, Trans. Moscow Math. Soc. 26 (1972), 199-239.
[16]	H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in "Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980)", Lecture Notes in Math. 898, Springer, Berlin, 1981, pp. 54-74.
[17]	H. Cartan, "Cours de calcul différentiel," Hermann, Paris, 1977.
[18]	G. T. Chen and J. Della Dora, Normal forms for differentiable maps near a fixed point, Numer. Algorithms, 22 (1999), 213-230.
[19]	G. T. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.
[20]	J. Écalle, "Les Fonctions Résurgentes. Tome III: L'Équation Du Pont Et La Classification Analytique Des Objects Locaux," Publ. Math. Orsay, 85-05, Université de Paris-Sud, Orsay, 1985.
[21]	J. Écalle, Iteration and analytic classification of local diffeomorphisms of $\mathbbC^v$, in "Iteration Theory And Its Functional Equations (Lochau, 1984)", Lect. Notes in Math., 1163, Springer-Verlag, Berlin, 1985, pp. 41-48.
[22]	E. Fischer, Über die differentiationsprozesse der algebra, J. für Math., 148 (1917), 1-78.
[23]	G. Gaeta, Further reduction of Poincaré-Dulac normal forms in symmetric systems, Cubo, 9 (2007), 1-11.
[24]	A. Giorgilli and A. Posilicano, Estimates for normal forms of differential equations near an equilibrium point, Z. Angew. Math. Phys., 39 (1988), 713-732. doi: 10.1007/BF00948732.
[25]	F. Ichikawa, On finite determinacy of formal vector fields, Invent. Math. 70 (1982/83), 45-52.
[26]	F. Ichikawa, Classification of finitely determined singularities of formal vector fields on the plane, Tokyo J. Math. 8 (1985), 463-472. 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), 293-318. 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), 659-718.
[29]	D. Malonza and J. Murdock, An improved theory of asymptotic unfoldings, J. Differential Equations, 247 (2009), 685-709.
[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), 424-465.
[32]	J. Murdock and J. A. Sanders, A new transvectant algorithm for nilpotent normal forms, J. Differential Equations, 238 (2007), 234-256.
[33]	J. Raissy, Torus actions in the normalization problem, J. Geom. Anal. 20 (2010), 472-524.
[34]	J. Raissy, Brjuno conditions for linearization in presence of resonances, in "Asymptotics In Dynamics, Geometry And PDE's; Generalized Borel Summation, Vol. I" (O. Costin, F. Fauvet, F. Menous and D. Sauzin, eds.), Edizioni Della Normale, Pisa, 2010, pp. 201-218.
[35]	H. Rüssmann, Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition, Ergodic Theory Dynam. Systems, 22 (2002), 1551-1573.
[36]	J. A. Sanders, Normal form theory and spectral sequences, J. Differential Equations, 192 (2003), 536-552.
[37]	D. Wang, M. Zheng and J. Peng, Further reduction of normal forms of formal maps, C. R. Math. Acad. Sci. Paris, 343 (2006), 657-660.
[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), 803-825. 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), 219-249.

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