Renormalization of planar analytic saddles
Let \(X=\lambda_1x_1\frac{\partial}{\partial x_1}+ \lambda_2x_2\frac{\partial}{\partial x_2}+O(|x|^2)\) be a planar analytic vector field near \(x=(0,0)\). It is assumed that the linear part of \(X\) has real eigenvalues \(\lambda_1\), \(\lambda_2\) whose ratio \(\eta =-\frac{\lambda_1}{\lambda_2}\) is a positive irrational number. In [the first author and \textit{P. De Maesschalck}, Discrete Contin. Dyn. Syst., Ser. B 10, No. 2--3, 377--400 (2008; Zbl 1153.37328)], it was shown that any analytic saddle can be conjugated analytically to a form `as close as desired' to the formal normal form. In the present paper, these conjugacies are iterated and renormalized. The iteration of this procedure is strongly connected with the diophantine nature of \(\eta\). The convergence of this process is established. The main result of the paper is the following theorem. Let \(\left(\frac{q_n}{p_n}\right)_{n\in\mathbb N}\) be continued fraction approximants of \(\eta\). Suppose that \[ \sum_{n\geq 0}\frac{|\ln (C_{n+1})|}{p_{n+1}+q_{n+1}}<\infty , \text{ where } C_{n+1}=\frac{|\lambda_1p_{n+1}+\lambda_2q_{n+1}|} {p_{n+1}+q_{n+1}+1}. \] Then the vector field \(X\) can be linearized to \(\lambda_1x_1\frac{\partial}{\partial x_1}+ \lambda_2x_2\frac{\partial}{\partial x_2}\) by means of a convergent transformation. From this result, the two-dimensional version of the classical linearization theorem of A. D. Bruno follows.
scientific article; zbMATH DE number 6100825
scientific article