Quasi-geodesics and backward orbits under semigroups of holomorphic functions

TL;DR

This paper investigates the geometric properties of backward orbits under semigroups of holomorphic functions in the unit disk, establishing their quasi-geodesic nature in hyperbolic geometry and Euclidean properties.

Contribution

It proves that regular backward orbits are quasi-geodesics in hyperbolic distance and identifies Euclidean properties of these orbits, advancing understanding of their geometric behavior.

Findings

Backward orbits are quasi-geodesics in hyperbolic geometry.

Backward orbits satisfy specific Euclidean properties.

The results deepen the geometric understanding of holomorphic semigroup dynamics.

Abstract

We explore two properties of backward orbits under semigroups of holomorphic self-maps in the unit disk. First, we prove that regular backward orbits are quasi-geodesics for the hyperbolic distance of the unit disk. Then, we show that backward orbits satisfy a useful property, this time in Euclidean terms.