On the scaling methods by Pinchuk and Frankel

The main purpose of this paper is to study two scaling methods developed respectively by Pinchuk and Frankel. We introduce first a continuously-varying global coordinate system, and give an alternative proof to the convergence of Pinchuk’s scaling sequence (and of our modification) on bounded domains with finite type boundaries in ${\mathbb{C}}^2$. Using this, we discuss the modification of the Frankel scaling sequence on nonconvex domains. We also observe that two modified scalings are equivalent.