On Critical Points of Proper Holomorphic Maps on the Unit Disk

We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post-composition with a Möbius transformation by its critical points. Bull. London Math. Soc. 30 (1998) 62–66 Stony Brook IMS Preprint #1996/7 June 1996 This note will give a brief proof of the following known theorem: Theorem 1. Let c1, · · · , cd be given (not necessarily distinct) points in the open unit disk D in the complex plane. Then there exists a unique proper holomorphic map f : D → D of degree d+ 1 normalized as f(0) = 0 and f(1) = 1 with critical points at the cj. It is to be understood that whenever some cj repeats m times, the corresponding map has local degree m+ 1 at cj. Since given any two points a and b with |a| < 1 and |b| = 1 there exists a unique conformal automorphism of the unit disk which maps a to 0 and b to 1, we have the following version of the uniqueness part of the above theorem: Corollary 1. Two proper holomorphic maps f, g : D → D have the same critical points, counted with multiplicity, if and only if f = τ ◦ g for some conformal automorphism τ of the unit disk. From the point of view of complex analysis, it is quite natural to ask questions about the dependence of such maps on their critical points, but this kind of question is also of some interest when one studies the parameter space for complex polynomial maps of a given degree. For example in [3], this theorem immediately implies that the space of all “critically marked” normalized Blaschke products of degree d+1 is a topological cell of real dimension 2d. Since these provide a model space for hyperbolic polynomial maps with a given postcritical pattern, the discussion of hyperbolic components with marked critical points would become much easier and more natural having known this theorem. The corresponding questions for the two other simply-connected Riemann surfaces (i.e., the complex plane and the Riemann sphere) have trivial and surprising answers. Every proper holomorphic map on the plane is a polynomial, so given a finite number of points in the plane one can always find a polynomial with corresponding critical points, and this polynomial is unique up to post-composition with a complex affine transformation. In the case of the Riemann sphere, however, both existence and uniqueness parts of the theorem are false. Every proper holomorphic map of degree d + 1 on the sphere is a rational map with 2d critical points, counted with multiplicity. So, for example, it is impossible to realize a single point with multiplicity 2 as the critical set of a rational map, since any such map would have degree 2 and local degree 3 near the double critical point. However, it can be shown that any 2d distinct points on the sphere can be realized as the critical set of a degree