Extremal rate of convergence in continuous dynamics

TL;DR

This paper investigates semigroups of holomorphic self-maps of the upper half-plane that converge to their Denjoy--Wolff point at the slowest rate, especially focusing on the parabolic case with zero hyperbolic step.

Contribution

It introduces new characterizations of extremal convergence rates for semigroups in the parabolic case, expanding understanding of their infinitesimal generators and associated functions.

Findings

Characterizations via Herglotz representation

Conformality at the Denjoy--Wolff point

Analysis of the zero hyperbolic step case

Abstract

This paper deals with semigroups of holomorphic self-maps of the upper half-plane that exhibit an extremal (i.e. the slowest possible) rate of convergence to their Denjoy--Wolff point. The main novelty lies in the parabolic case of zero hyperbolic step. We provide several characterizations for such semigroups in terms of the Herglotz representation of their infinitesimal generators, the conformality at the Denjoy--Wolff point of a modification of their associated Koenigs function, and more.