Sectorial normalization of simplest germs of semi-hyperbolic maps in a half-neighborhood

P. A. Shaikhullina

Ufa mathematical journal, Tome 12 (2020) no. 2, pp. 72-87

Résumé

We consider a problem on analytic classification of semi-hyperbolic maps on the plane for an example of the simplest class of germs, namely, the class of germs that are formally equivalent to $\mathsf{F}_{\lambda}$, which is the unit time shift along the vector field $x^2\frac{\partial}{\partial x}+{\lambda}y\frac{\partial}{\partial y},~\lambda\in\mathbb{R}_+$). A key step in the classification is an analytic normalization of the germs on sectorial domains forming a cut neighbourhood of the origin $(\mathbb{C}^2,0)\backslash\{x=0\}$. For this class, in the present work, we prove a theorem on a sectorial analytic normalization in the half-neighbourhood invariant with respect to $\mathsf{F}_{\lambda}^{-1}$. We also show that a formal normalizing change of the coordinates is asymptotic for the constructed sectorial normalizing change.

Détail

Keywords: analytic classification, semi-hyperbolic maps, sectorial normalization.

Bibliographie

[1] H. Poincare, “Mémoire sur les courbes définies par une équation différentielle I”, J. Math. Pures et Appl., 7 (1881), 375–422

[2] H. Poincare, “Mémoire sur les courbes définies par une équation différentielle II”, J. Math. Pures et Appl., 8 (1882), 251–296

[3] H. Poincare, “Sur les courbes définies par les équations différentielles III”, J. Math. Pures et Appl., 1 (1885), 167–244

[4] H. Poincare, “Sur les courbes définies par les équations différentielles IV”, J. Math. Pures et Appl., 2 (1886), 151–218

[5] H. Poincare, “Sur les problém des trois corps et les équations de la dinamique”, Acta Math., XIII (1890), 1–271 | MR

[6] C. L. Siegel, Vorlesungen uber Himmelsmechanik, Springer-Verlag, Berlin–Gottingen–Heidelberg, 1957 | MR

[7] A. D. Bryuno, “The analytic form of differential equations”, Trudy Mosk. Matem. Obsch., 25, 1971, 119–262 (in Russian) | Zbl

[8] A. D. Bryuno, “The analytic form of differential equations”, Trudy Mosk. Matem. Obsch., 26, 1972, 199–239 (in Russian) | Zbl

[9] V. I. Arnold, “Supplementary chapters to the theory of ordinary differential equations”, Nauka, M., 1978 (in Russian)

[10] J. C. Yoccoz, “Linearisation des germes de diffeomorphismes holomorphes de $(\mathbb{C},0)$”, C. R. Acad. Sci. Paris Ser. I Math., 306:1 (1988), 55–58 | MR | Zbl

[11] S. M. Voronin, “Analytic classification of germs of conformal mappings $(\mathbb{C},0)\to (\mathbb{C},0)$ with identity linear part”, Funct. Anal. Appl., 15:1 (1981), 1–13 | DOI | MR | Zbl

[12] J. Ecalle, Sur les fonctions resurgentes, Publ. Math. d'Orsay, Orsay, 1981

[13] J. Martinet, J. P. Ramis, “Probléme de modules pour des équations différentielles non linéaires du premier ordre”, Inst. Hautes Etudes Sci. Publ. Math., 55 (1982), 63–164 | DOI | MR | Zbl

[14] J. Martinet, J. P. Ramis, “Classification analytique desequations différentielles non linéaires resonnantes du premier ordre”, Ann. Sci. Ecole norm. supér., 16:4 (1983), 571–621 | DOI | MR | Zbl

[15] Y. S. Il'yashenko, “Nonlinear Stokes phenomena”, Nonlinear Stokes phenomena, Adv. in Sov. Math., 13, 1992, 1–55

[16] S. M. Voronin, “Darboux-Whitney's Problem and Related Questions”, Nonlinear Stokes Phenomena, Adv. in Sov. Math., 13, 1992, 139–223 | MR

[17] Yu. S. Ilyashenko, S. Yu. Yakovenko, Analytic theory of differential equatinos, v. 1, MCCME, M., 2013 (in Russian)

[18] S. M. Voronin, “Analytic classification of germs of holomorphic mapping with non-isolated equilibria and constant multiplications and applications”, Vestnik ChelGU, 5 (1999), 12–30 (in Russian)

[19] P. A. Shaikhullina, “Formal normalization of generic germs for semihyperbolic maps”, Matem. Zamet. SVFU, 22:4 (2015), 79–90 (in Russian) | Zbl

[20] S. M. Voronin, P. A. Fomina, “Sectorial normalization of germs of semi-hyperbolic mappings”, Vestnik ChelGU, 16 (2013), 94–113 (in Russian)

[21] T. Ueda, “Local structure of analytic transformations of two complex variables, I”, J. Math. Kyoto Univ., 26:2 (1986), 233–261 | DOI | MR | Zbl

[22] T. Ueda, “Local structure of analytic transformations of two complex variables, II”, J. Math. Kyoto Univ., 31:3 (1991), 695–711 | DOI | MR | Zbl

[23] B. V. Shabat, “Introduction into complex analysis”, Nauka, M., 1985 (in Russian)

[24] P. A. Shaikhullina, “On solution to the simplest functional equation in curvilinear-band-type domain”, Matem. Zamet. SVFU, 24:4 (2017), 87–95 (in Russian) | MR | Zbl