Milnor's Lemma, Newton's Method, and Continued Fractions

Abstract

Let f be a quadratic map of the Riemann sphere S2 into itself. Such a map has three fixed points, counted with multiplicity. Let μ1, μ2 and μ3 be the multipliers of f at these points. Milnor proved (see [2, Lemma 3.1]) that μ1, μ2 and μ3 determine f up to holomorphic conjugacy and are subject only to the restriction μ1μ2μ3 - (μ1 + μ1 + μ1) + 2 = 0. In the present paper, we consider arbitrary regular maps from a projective line over an arbitrary field K into itself and give an elementary proof of the fact that two such maps coincide if the maps have the same collection of fixed points and equal multipliers at the corresponding points. We apply this result to demystify the long-known link between Newton's approximations and continued fractions (see [5] and [1]).