Hyperbolic distortion and conformality at the boundary

Abstract

We characterize two classical types of conformality of a holomorphic self-map of the unit disk at a boundary point - existence of a finite angular derivative in the sense of Carath\'eodory and the weaker property of angle preservation - in terms of the non-tangential asymptotic behaviour of the hyperbolic distortion of the map. These characterizations are given purely with reference to the intrinsic metric geometry of the unit disk. In particular, we relate the classical Julia-Wolff-Carath\'eodory theorem with the case of equality in the Schwarz-Pick lemma at the boundary. We also provide an operator-theoretic characterization of the existence of a finite angular derivative based on Hilbert space methods. As an application we study the backward dynamics of discrete dynamical systems induced by holomorphic self-maps, and characterize the regularity of the associated pre-models in terms of a Blaschke-type condition involving the hyperbolic distortion along regular backward orbits.