Geometric description of the classification of holomorphic semigroups (Q2790267)

Let \(\mathbb D\) be the complex unit disk and \((\phi_t)\) a continuous semigroup of holomorphic functions \(\phi_t:\mathbb D\to\mathbb D\), so that there exists a point \(\tau\in\mathbb D\bigcup\partial\mathbb D\) such that, for every \(z\in\mathbb D\), we have \(\lim_{t\to\infty}\phi_t(z)=\tau\), which is the Denjoy-Wolff point of the semigroup. The author considers parabolic semigroups of holomorphic self-maps of \(\mathbb D\) with the Denjoy-Wolff point 1, the Koenigs function \(h\) (i.e., \(h:\mathbb D\to\mathbb C\), \(h(0)=0\), \(\phi_t(z)=h^{-1}(h(z)+t)\) for \(t\geq 0,\,z\in\mathbb D\)) and the associated planar domain \(\Omega=h(\mathbb D)\). It is known that the Koenigs function is a unique conformal mapping, which is a very useful representation of the semigroup \(\phi\), and that \(\Omega\) is convex in the positive direction. A theorem of Contreras and Díaz-Madrigal states that a semigroup with the Denjoy-Wolff point 1 is hyperbolic if \(\Omega\) is contained in a horizontal strip; otherwise, it is parabolic. The author shows that a parabolic semigroup \((\phi_t)_{t\geq 0}\) is of hyperbolic step if and only if \(\Omega\) is contained in a horizontally half-plane, and if this is the case, then it has trajectories that converge to 1 strongly tangentially (i.e., the semigroup is of finite shift) if and only if \(h\) is conformal at 1.