Extremal rate of convergence in continuous dynamics

TL;DR

This work provides a comprehensive extremal-rate theory for parabolic semigroups of zero hyperbolic step in the upper half-plane with Denjoy–Wolff point at infinity. It develops a triad of equivalent characterizations: (i) Herglotz data of the infinitesimal generator with $\alpha=0$, finite $\int s^2 \,d\mu(s)$, and $\beta=\int s \,d\mu(s)$; (ii) conformality of the square-root modification $\sqrt{h-c}$ of the Koenigs function at infinity; and (iii) a sharp hyperbolic-distance asymptotic $d_{\mathbb{H}}(i,\phi_t(z)) - \frac{1}{4}\log t$. It additionally shows that extremality forces orthogonal (non-tangential) convergence and yields explicit asymptotics for $\phi_t(z)$, together with precise norm-growth results for induced composition-operator semigroups on Hardy and Bergman spaces. Collectively, these results bridge complex dynamics, geometric function theory, and operator theory, providing a complete description of extremal-rate behavior in this zero-hyperbolic-step regime and translating dynamical properties into concrete operator-theoretic consequences in the disk.

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.

Theorems & Definitions (44)

• Theorem A
• Theorem B
• Theorem C
• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• Definition 3.4
• Lemma 4.1