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Register a new userOn the Konigs function of semigroups of holomorphic self-maps of the unit disc
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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.
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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.
Let $f$ be a univalent self-map of the unit disc. We introduce a technique, that we call {sl semigroup-fication}, which allows to construct a continuous semigroup $(phi_t)$ of holomorphic self-maps of the unit disc whose time one map $phi_1$ is, in a sense, very close to $f$. The semigrup-fication of $f$ is of the same type as $f$ (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of $f$ with a given multiplier and the corresponding set for $phi_1$. Moreover, in case $f$ (and hence $phi_1$) has no interior fixed points, the slope of the orbits converging to the Denjoy-Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of $f$ converge non-tangentially to the Denjoy-Wolff point if and only if the Koenigs domain of $f$ is almost symmetric with respect to vertical lines.
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 $2$.
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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