Zalcman's lemma, Pinchuk's rescaling method, and Catlin's estimates revisited

We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk’s scaling method in C^2 and, incidentally, reprove the Catlin’s estimates for the Kobayashi metric in finite type domains.