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Register a new userAsymptotic monotonicity of the orthogonal speed and rate of convergence for semigroups of holomorphic self-maps of the unit disc
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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.
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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.
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 $f$ 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 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.
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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