Dynamic extension of the Julia-Wolff-Carathéodory theorem (Q2770839)

We study the asymptotic behavior of a one-parameter semigroup \(\{{\mathcal S}(t)\}\) of holomorphic self-mappings of the unit disk \(\Delta\) by using a version of the Julia-Wolff-Carathédory theorem in terms of the infinitesimal generator \(f=-{\partial{\mathcal S}(t) \over\partial t}|_{t=0^+}\). Namely, we establish that \(f\) has no null point in \(\Delta\) if and only if there is a boundary point \(\tau\in \partial\Delta\) such that the angular limit of \(f\) at \(\tau\) is zero and the angular derivative \(\beta={\mathcal L}f'(\tau)\) exists (finitely) with \(\text{Re} \beta\geq 0\). Moreover, in this case \(\beta\) is actually real and \(\tau\) is a unique globally attractive point of the semigroup \(\{{\mathcal S}(t)\}\).