Analytic flows on the unit disk: angular derivatives and boundary fixed points

Manuel D. Contreras and Santiago Díaz-Madrigal

Vol. 222 (2005), No. 2, 253–286

DOI: 10.2140/pjm.2005.222.253

Abstract

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.

Keywords

angular derivative, fixed points, planar vector fields, semigroups of analytic functions

Mathematical Subject Classification 2000

Primary: 30C20, 30D05, 37F99

Secondary: 30F45, 37E35

Milestones

Received: 31 March 2004

Revised: 4 November 2005

Accepted: 9 November 2004

Published: 1 December 2005