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.
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Betsakos, D. On the Asymptotic Behavior of the Trajectories of Semigroups of Holomorphic Functions. J Geom Anal 26, 557–569 (2016). https://doi.org/10.1007/s12220-015-9562-1
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DOI: https://doi.org/10.1007/s12220-015-9562-1