From an Iteration Formula to Poincaré’s Isochronous Center Theorem for Holomorphic Vector Fields

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center. 1. An iteration formula Let f be a holomorphic function germ at the origin in the complex plane with f(0) = 0. For each positive integer k, we denote by f the k-th iteration of f defined as f = f, f = f ◦f, ... , f = f ◦fk−1 inductively, which is a well defined holomorphic function germ at the origin. Assume that λ = f ′(0) is a primitive m-th root of unity, say, λ = 1, but λ = 1 for each positive integer j with j < m. Then it is interesting that there exists a positive integer r, such that the m-th iteration f has a power series expansion f(z) = z + a1z + a2z + . . . at the origin: all the terms of degrees from 2 to rm vanish! This can be proved by applying Rouché’s theorem (see [12]). In this section we generalize this formula to germs of higher dimensional mappings by using normal form theory. We denote by C the complex vector space and by ∆ a ball in C centered at the origin. Proposition 1 (Iteration Formula). Let f : ∆ → C be a holomorphic mapping given by f(z) = λz + o(|z|), z ∈ ∆, where λ is a primitive m-th root of unity. Then, in a neighborhood of the origin, (1.1) f(z) = z + o(|z|). In the proposition, z = (z1, . . . , zn) and the expression o(|z|) means that each component of the mapping f(z) − z is a power series in z1, . . . , zn consisting of terms of degree > m. Received by the editors November 23, 2005 and, in revised form, May 30, 2006. 2000 Mathematics Subject Classification. Primary 32H50, 32M25, 37C27.