Angular and unrestricted limits of one-parameter semigroups in the unit disk

TL;DR

This paper investigates the boundary behavior of one-parameter semigroups of holomorphic functions in the unit disk, establishing stronger results on angular and unrestricted limits at boundary points without restrictions on the Denjoy-Wolff point.

Contribution

It proves stronger versions of boundary limit results for semigroups without assuming the position of the Denjoy-Wolff point.

Findings

Elements have angular limits everywhere on the boundary.

Elements have unrestricted limits at all boundary fixed points.

Results hold without assumptions on the Denjoy-Wolff point.

Abstract

We study local boundary behaviour of one-parameter semigroups of holomorphic functions in the unit disk. Earlier under some addition condition (the position of the Denjoy - Wolff point) it was shown in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488] that elements of one-parameter semigroups have angular limits everywhere on the unit circle and unrestricted limits at all boundary fixed points. We prove stronger versions of these statements with no assumption on the position of the Denjoy - Wolff point. In contrast to many other problems, in the question of existence for unrestricted limits it appears to be more complicated to deal with the boundary Denjoy - Wolff point (the case not covered in [M.D.Contreras, S.Diaz-Madrigal and Ch.Pommerenke, Ann. Acad. Sci. Fenn. Math. 29(2004), No.2, 471-488]) than with all the other boundary fixed points of the semigroup.