On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions

Abstract

Let $\{\phi _t\}_{t\ge 0}$ be a semigroup of holomorphic self-maps of the unit disk. We assume that the Denjoy–Wolff point of the semigroup is the point 1; so 1 is the unique attractive boundary fixed point of the semigroup. We further assume that for all $t\ge 0$, $\phi _t^\prime (1)=1$ (angular derivative), namely the semigroup is parabolic. We disprove a conjecture of Contreras and Díaz-Madrigal on the asymptotic behavior of the trajectories $\gamma _z(t)=\phi _t(z)$, as $t\rightarrow +\infty$. We also prove that if the boundary of the associated planar domain is contained in a half-strip, then all the trajectories of the semigroup converge to 1 radially.