Local Behaviour of the Trajectories and the φ-Metric

Abstract

This chapter is devoted to the investigation of the trajectory structure and the φ-metric near the critical points. The treatment in Schaeffer and Spencer [1] and Jenkins [3] is based on the theory of differential equations. Here, special conformal parameters will be introduced in terms of which the representation of the quadratic differential becomes particularly simple. This is achieved by computing the integral Ф and expressing it in simple terms. Differentiation and squaring then gives the representation of φ with respect to the new parameters. The procedure is particularly simple if there are no logarithmic terms, which is always the case if n is positive, or negative and odd (Strebel [7], Pfluger [1], Jensen [1]).