Transition functions and moduli of stability for 3-dimensional homogeneous vector-fields with a hyperbolic blowing-up

Abstract

In the first part of the paper we introduce the “Normal transition function” for saddle connections of planar diffeomorphisms. It is a positive multiple of the usual transition function, but in its definition we do not need C1 -linearizing coordinates. Among other nice properties, it is found to be analytic when the diffeomorphism is. The second part of the paper deals with the existence of a modulus of stability for germs on 3 of homogeneous vector fields with a hyperbolic blowing-up. We show that inside a specific class of examples the modulus occurs for a sufficiently high degree.