Fixed point indices and invariant periodic sets of holomorphic systems

Abstract

Abstract. This note presents a method to study center families of periodicorbits of complex holomorphic differential equations near singularities, basedon some iteration properties of fixed point indices. As an application of thismethod, we will prove Needham’s theorem in a more general version. 1. Fixed point indices of holomorphic mappingsLet C n be the complex vector space of dimension n, let U be an open set in C n and let f : U → C n be a holomorphic mapping. If p ∈ U is an isolated zero of f,say, there exists a neighborhood V with p ∈ V ⊂ V ⊂ U such that p is the uniquesolution of the equation f(x) = 0 in V. Then we can define the zero index of f atp byπ f (p) = #{x ∈ V ;f(x) = q},where q is a regular value of f such that |q| is small enough and # denotes thecardinality. π f (p) is well defined (see [9] or [17] for the detail).If f : U → C n is a holomorphic mapping and p is an isolated fixed point of f,then there is a ball B in U centered at p so that p is the unique fixed point of f inB, in other words, p is the unique zero of the mappingf −I : B → C