Geometric description of the classification of holomorphic semigroups

Abstract:

We consider parabolic semigroups $(\phi _t)_{t\geq 0}$ of holomorphic self-maps of the unit disk $\mathbb {D}$ with Denjoy-Wolff point $1$, Koenigs function $h$ and associated planar domain $\Omega$. We give a geometric description of the classification of such semigroups: The semigroup is of positive hyperbolic step if and only if $\Omega$ is contained in a horizontal half-plane. Moreover, a semigroup of positive hyperbolic step has trajectories that converge to $1$ strongly tangentially (namely the semigroup is of finite shift) if and only if $h$ is conformal at $1$.