Fixed Point Indices and Invariant Periodic Sets of Holomorphic Systems

This note presents a method to study center families of periodic orbits of complex holomorphic differential equations near singularities, based on some iteration properties of fixed point indices. As an application of this method, we will prove Needham’s theorem in a more general version. 1. Fixed point indices of holomorphic mappings Let C be the complex vector space of dimension n, let U be an open set in C and let f : U → C 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 unique solution of the equation f(x) = 0 in V . Then we can define the zero index of f at p by πf (p) = #{x ∈ V ; f(x) = q}, where q is a regular value of f such that |q| is small enough and # denotes the cardinality. πf (p) is well defined (see [9] or [17] for the detail). If f : U → C 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 in B, in other words, p is the unique zero of the mapping f − I : B → C, which puts each x ∈ B into f(x) − x, and then the fixed point index of f at p is well defined by μf (p) = πf−I(p). Fixed point indices of holomorphic mappings have the geometric properties stated in the following lemmas (see [16] and [17]). We will denote by ∆ a ball in C centered at the origin. Lemma 1. Let f : ∆ → C be a holomorphic mapping such that 0 ∈ ∆ is an isolated fixed point of f. Then μf (0) ≥ 1, and the equality holds if and only if the Jacobian matrix f (0) of f at 0 has no eigenvalue equal to 1. 2000 Mathematics Subject Classification. 32H50, 32M25, 37C25.