Slope problem for trajectories of holomorphic semigroups in the unit disc

TL;DR

This paper resolves a long-standing open problem by showing that not all trajectories of parabolic semigroups in the unit disk have a common slope at the Denjoy-Wolff point.

Contribution

It provides a negative answer to whether all trajectories share a common slope, clarifying the behavior of parabolic semigroup trajectories.

Findings

Not all trajectories tend to the Denjoy-Wolff point with a definite slope.

The slope behavior of trajectories can vary, counter to previous conjectures.

The result refutes a ten-year-old open problem.

Abstract

It has been an open problem for about ten years whether every trajectory of a parabolic one-parameter semigroup in the unit disk tends to the Denjoy-Wolff point with a definite (and common for all trajectories) slope. In this paper, we give the negative answer to this question.