From the expression above, we can see? i(2j?1) (z) and? i(2j) (z) have conjugate values. Indeed, according to the properties of ? and? i above, we have 2l, any element ? := (? 1, ? n ) ? R m (cf. (2.1)), we have ? ? := (? ?(1) , . . . , ? ?(n) ) ? R ?(m) . It follows that if? im,? w ? is a resonant term in? im (w), then? im ,
, z 2l?1 , z 2l =z 2l?1 , z 2l+1 , . . . , z n ) and for m > 2l, the values (not the functions)? im (z 1 ,z 1 , . . . , z 2l?1 ,z 2l?1 , z 2l+1, Hence, for m 2l,? im and? i?(m) are a pair of conjugate functions of variables, vol.2
F q ) be a formal non-degenerate discrete integrable system of type (p, q) on R n at a common fixed point, say the origin 0. We assume that the family of its linear parts {A j x} is either projectively hyperbolic or infinitesimally integrable with a weakly non-resonant family of generators. Assume furthermore that the commuting family of real diffeomorphisms {? i } satisfies A j ? i = ? i A j ,
F q ? P ) satisfies assumption of Theorem 2.6, then P ?1 ? i P is of the form (2.6) with (2.7), for all i. Therefore, Hence, the family {P ?1 ? i P } is in Poincaré-Dulac normal form as it commutes with the family of its linear part {D j }Since the family (P ?1 ? 1 ,
Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol.60, 1989. ,
Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, vol.250, 1988. ,
Extended integrability and bi-Hamiltonian systems, Communications in mathematical physics, vol.196, issue.1, pp.19-51, 1998. ,
, Convergence to normal forms of integrable PDEs, p.29, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02094584
, Géométrie différentielle et singularités de systèmes dynamiques, vol.444, pp.138-139, 1986.
A Forgotten Theorem on Z k × R m -action Germs and Related Questions, Regular and Chaotic Dynamics, vol.18, issue.6, pp.742-773, 2013. ,
On the existence and uniqueness of the real logarithm of a matrix, Proceedings of the American Mathematical Society, vol.17, issue.5, pp.1146-1151, 1966. ,
Hamiltonian systems with Poisson commuting integrals, 1984. ,
Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Commentarii Mathematici Helvetici, vol.65, issue.1, pp.4-35, 1990. ,
The theory of matrices, 1959. ,
Real submanifolds of maximum complex tangent space at a CR singular point, I, Invent. math, vol.206, pp.293-377, 2016. ,
, Convergence of Birkhoff normal forms for integrable systems, vol.64, pp.412-461, 1989.
Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case, Mathematische Annalen, vol.292, issue.1, pp.411-444, 1992. ,
Local normal forms of smooth weakly hyperbolic integrable systems, Regular and Chaotic Dynamics, vol.21, issue.1, pp.18-23, 2016. ,
, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol.45
, , 2003.
Vey theorem in infinite dimensions and its application to KdV, Discrete Contin. Dyn. Syst, vol.27, issue.1, pp.1-24, 2010. ,
URL : https://hal.archives-ouvertes.fr/hal-00832747
Extension of Floquet's Theory to Nonlinear Periodic Differential Systems and Embedding Diffeomorphisms in Differential Flows, American Journal of Mathematics, vol.124, issue.1, pp.107-127, 2002. ,
Note sur l'intégration deséquations différentielles de la Dynamique, Journal de Mathématiques Pures et Appliquées, pp.137-138, 1855. ,
, Singular complete integrability, Publications Mathématiques de l'Institut des HautesÉtudes Scientifiques, vol.91, pp.133-210, 2000.
Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Annals of mathematics, pp.589-612, 2005. ,
Family of intersecting totally real manifolds of (C n , 0) and germs of holomorphic diffeomorphisms, vol.143, pp.247-263, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-01285115
, Integrable mappings, Uspekhi Mat. Nauk, vol.46, pp.3-45, 1991.
Growth and integrability in the dynamics of mappings, Comm. Math. Phys, vol.145, issue.1, pp.181-193, 1992. ,
Sur certains systemes dynamiques séparables, American journal of mathematics, vol.100, pp.591-614, 1978. ,
On differential equations in normal form, Mathematische Annalen, vol.291, issue.1, pp.293-314, 1991. ,
Analytic normalization of analytic integrable systems and the embedding flows, Journal of Differential Equations, vol.244, issue.5, pp.1080-1092, 2008. ,
Analytic integrable systems: Analytic normalization and embedding flows, Journal of Differential Equations, vol.254, issue.7, pp.3000-3022, 2013. ,
Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I, Functional Analysis and Its Applications, vol.13, pp.181-189, 1982. ,
, Convergence versus integrability in Birkhoff normal form, pp.141-156, 2005.
Non-degenerate singularities of integrable dynamical systems, Ergodic Theory and Dynamical Systems, vol.35, pp.994-1008, 2015. ,
URL : https://hal.archives-ouvertes.fr/hal-00999733
A conceptual approach to the problem of action-angle variables, Archive for Rational Mechanics and Analysis, vol.229, issue.2, pp.789-833, 2018. ,
, UMR CNRS 7351 Université de Nice Sophia-Antipolis E-mail address: kai.jiang@bicmr.pku.edu.cn CNRS and