On the monotonicity of the speeds for semigroups of holomorphic self-maps of the unit disk

TL;DR

This paper investigates the monotonicity properties of speeds in semigroups of holomorphic self-maps of the unit disk, proving that orthogonal speeds are strictly increasing and exploring related asymptotic behaviors.

Contribution

It establishes the strict increase of orthogonal speeds in boundary Denjoy-Wolff point semigroups and provides counterexamples for total speed monotonicity.

Findings

Orthogonal speed is strictly increasing for these semigroups.

Total speed may not be eventually increasing, countering previous conjectures.

Examples of semigroups with specific asymptotic speed behaviors are provided.

Abstract

We study semigroups $(\phi_t)_{t\geq 0}$ of holomorphic self-maps of the unit disk with Denjoy-Wolff point on the boundary. We show that the orthogonal speed of such semigroups is a strictly increasing function. This answers a question raised by F. Bracci, D. Cordella, and M. Kourou, and implies a domain monotonicity property for orthogonal speeds conjectured by Bracci. We give an example of a semigroup such that its total speed is not eventually increasing. We also provide another example of a semigroup having total speed of a certain asymptotic behavior, thus answering another question of Bracci.