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 Ufimsk. Mat. Zh., 2020, Volume 12, Issue 2, Pages 71–86 (Mi ufa517)

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

P. A. Shaikhullina

Chelyabinsk State University, Br. Kashiriny str. 129, 454001, Chelyabinsk, Russia

Abstract: 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.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00739_à The work is supported by RFBR (grant no. 17-01-00739a).

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English version:
Ufa Mathematical Journal, 2020, 12:2, 72–87 (PDF, 475 kB); https://doi.org/10.13108/2020-12-2-72

Bibliographic databases:

UDC: 517.938
MSC: 34M35, 34M40

Citation: P. A. Shaikhullina, “Sectorial normalization of simplest germs of semi-hyperbolic maps in a half-neighborhood”, Ufimsk. Mat. Zh., 12:2 (2020), 71–86; Ufa Math. J., 12:2 (2020), 72–87

Citation in format AMSBIB
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