# Trajectories of semigroups of holomorphic functions and harmonic measure

## TL;DR

This paper establishes a connection between the trajectory slopes of semigroups of holomorphic functions and harmonic measure, enabling the construction of semigroups with prescribed slope intervals and analyzing backward trajectories near super-repulsive fixed points.

## Contribution

It introduces an explicit relation between trajectory slopes and harmonic measure, and demonstrates how to construct semigroups with arbitrary slope intervals and analyze backward trajectories.

## Key findings

- Constructed semigroups with any slope interval in [π/2, -π/2]
- Linked trajectory slopes to harmonic measure of planar domains
- Analyzed backward trajectories approaching super-repulsive fixed points

## Abstract

We give an explicit relation between the slope of the trajectory of a semigroup of holomorphic functions and the harmonic measure of the associated planar domain ${\varOmega}$. We use this to construct a semigroup whose slope is an arbitrary interval in $ [{\pi}/2,-{\pi}/2] $. The same method is used for the slope of a backward trajectory approaching a super-repulsive fixed point.