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Let $mathbb D$ be the unit disc in $mathbb C$ and let $f:mathbb D to mathbb C$ be a Riemann map, $Delta=f(mathbb D)$. We give a necessary and sufficient condition in terms of hyperbolic distance and horocycles which assures that a compactly divergent sequence ${z_n}subset Delta$ has the property that ${f^{-1}(z_n)}$ converges orthogonally to a point of $partial mathbb D$. We also give some applications of this to the slope problem for continuous semigroups of holomorphic self-maps of $mathbb D$.
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-Madri
We investigate the relationship between quasisymmetric and convergence groups acting on the circle. We show that the Mobius transformations of the circle form a maximal convergence group. This completes the characterization of the Mobius group as a m
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 attach copies of the circle to points of a countable dense subset $D$ of a separable metric space $X$ and construct an earring space $E(X,D)$. We show that the fundamental group of $E(X,D)$ is isomorphic to a subgroup of the Hawaiian earring gro
Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset generaliz