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

## TL;DR

This paper investigates the conditions under which petals of one-parameter semigroups in the unit disk are conformal, linking geometric and dynamic properties for hyperbolic and parabolic cases.

## Contribution

It provides a necessary and sufficient condition for petal conformality in hyperbolic cases and characterizes conformality in parabolic cases based on asymptotic behavior.

## Key findings

- Hyperbolic petals are conformal if and only if certain hyperbolic geometric conditions are met.
- Parabolic petals' conformality is characterized by the asymptotic behavior of the Koenigs function.
- The study links geometric, dynamic, and functional aspects of semigroup petals.

## 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.