No abstract

Abstract. We prove that evolution families on complex complete hyperbolic manifolds are in one to one correspondence with certain semicomplete non-autonomous holomorphic vector fields, providing the solution to a very general Loewner type differential equation on manifolds.

We use the concept of angular derivative and the hyperbolic metric in the unit disk ‫,ބ‬ to study the dynamical aspects of the equilibrium points belonging to ‫ބ∂‬ of some complex-analytic dynamical systems on ‫.ބ‬ Our results show a deep connection between the dynamical properties of those equilibrium points and the geometry of certain simply connected domains of ‫.ރ‬ 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.

In this paper we introduce a general version of the notion of Loewner chains which comes from the new and unified treatment, given in [5], of the radial and chordal variant of the Loewner differential equation, which is of special interest in geometric function theory as well as for various developments it has given rise to, including the famous Schramm-Loewner evolution. In this very general setting, we establish a deep correspondence between these chains and the evolution families introduced in [5]. Among other things, we show that, up to a Riemann map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we also prove that these chains are solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner-Kufarev PDE.

We analyze the relationship between boundary fixed points of semigroups of analytic functions and boundary critical points of their infinitesimal generators. As a consequence, we show two new inequalities for analytic self-maps of the unit disk. The first one is about angular derivatives at fixed points of functions belonging to semigroups of analytic functions. The second one deals with angular derivatives at contact points of arbitrary analytic functions from the unit disk into itself.