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Register a new userAngular extents and trajectory slopes in the theory of holomorphic semigroups in the unit disk
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We study relationships between the asymptotic behaviour of a non-elliptic semigroup of holomorphic self-maps of the unit disk and the geometry of its planar domain (the image of the Koenigs function). We establish a sufficient condition for the trajectories of the semigroup to converge to its Denjoy-Wolff point with a definite slope. We obtain as a corollary two previously known sufficient conditions.
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We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disc, the total speed, the orthogonal speed and the tangential speed and show how they are related and what can be inferred from those.
We show that the orthogonal speed of semigroups of holomorphic self-maps of the unit disc is asymptotically monotone in most cases. Such a theorem allows to generalize previous results of D. Betsakos and D. Betsakos, M. D. Contreras and S. Diaz-Madrigal and to obtain new estimates for the rate of convergence of orbits of semigroups.
We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$.
Let $(phi_t)$ be a semigroup of holomorphic self-maps of~$mathbb D$. In this note, we use an abstract approach to define the Konigs function of $(phi_t)$ and holomorphic models and show how to deduce the existence and properties of the infinitesimal generator of $(phi_t)$ from this construction.
Let $(phi_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Omega$ be the starlike at infinity domain image of the Koenigs function of $(phi_t)$. In this paper we completely characterize the type of convergence of the orbits of $(phi_t)$ to the Denjoy-Wolff point in terms of the shape of $Omega$. In particular we prove that the convergence is non-tangential if and only if the domain $Omega$ is `quasi-symmetric with respect to vertical axes. We also prove that such conditions are equivalent to the curve $[0,infty) i tmapsto phi_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $Omega$.
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