The angular derivative problem for petals of one-parameter semigroups in the unit disk

Abstract

We study the angular derivative problem for petals of one-parameter semigroups of holomorphic self-maps of the unit disk. For hyperbolic petals, we prove a necessary and sufficient condition for the conformality of the petal in terms of the intrinsic hyperbolic geometry of the petal and the backward dynamics of the semigroup. For parabolic petals, we characterize conformality of the petal in terms of the asymptotic behaviour of the Koenigs function at the Denjoy–Wolff point.