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

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.