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 — the existence of a finite angular derivative in the sense of Carathéodory and the weaker property of angle preservation — in terms of the non-tangential asymptotic behavior of the hyperbolic distortion of the map. 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.