Semigroups of holomorphic functions; rectifiability and Lipschitz properties of the orbits

TL;DR

This paper investigates the geometric properties of orbits generated by semigroups of holomorphic functions in the unit disk, establishing rectifiability and Lipschitz conditions based on hyperbolic geometry.

Contribution

It provides new criteria linking hyperbolic geometry to the Lipschitz regularity of orbits, including necessary and sufficient conditions for backward orbits.

Findings

All orbits are rectifiable.

Forward orbits are Lipschitz curves.

Characterization of backward orbits as Lipschitz curves.

Abstract

Let $(\phi_t)$ be a semigroup of holomorphic functions in the unit disk. We prove that all its orbits are rectifiable and that its forward orbits are Lipschitz curves. Moreover, we find a necessary and sufficient condition in terms of hyperbolic geometry so that a backward orbit is a Lipschitz curve. We further explore the Lipschitz condition for forward orbits lying on the unit circle and then for semigroups of holomorphic functions in general simply connected domains.