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 Diaz-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.