Parabolic type semigroups: asymptotics and order of contact

TL;DR

This paper investigates the asymptotic behavior and local geometric properties of parabolic type semigroups acting on the unit disk and right half-plane, focusing on boundary behavior and rigidity properties.

Contribution

It provides new insights into the asymptotics, limit curvature, and order of contact of semigroup trajectories near the boundary Denjoy--Wolff point, establishing asymptotic rigidity results.

Findings

Characterization of asymptotic behavior of semigroup trajectories

Analysis of limit curvature and order of contact near boundary

Establishment of asymptotic rigidity properties for broad class of semigroups

Abstract

We study the asymptotic behavior of parabolic type semigroups acting on the unit disk as well as those acting on the right half-plane. We use the asymptotic behavior to investigate the local geometry of the semigroup trajectories near the boundary Denjoy--Wolff point. The geometric content includes, in particular, the asymptotes to trajectories, the so-called limit curvature, the order of contact, and so on. We then establish asymptotic rigidity properties for a broad class of semigroups of parabolic type.