We use the concept of angular derivative and the hyperbolic metric in the unit disk D, to study the dynamical aspects of the equilibrium points belonging to ?D of some complex-analytic dynamical systems on D. Our results show a deep connection between the dynamical properties of those equilibrium points and the geometry of certain simply connected domains of C. As a consequence, and in the context of semigroups of analytic functions, we give some geometric insight to a well-known inequality of Cowen and Pommerenke about the angular derivative of an analytic function.
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