The numbers of periodic orbits hidden at fixed points of n-dimensional holomorphic mappings

Abstract

Let M be a positive integer and let f be a holomorphic mapping from a ball Δn={x∈ℂn:∣x∣<δ} into ℂn such that the origin 0 is an isolated fixed point of both f and fM, the Mth iteration of f. Then one can define the number 𝒪M(f,0), interpreted as the number of periodic orbits of f with period M that are hidden at the fixed point 0. For an n×n matrix A whose eigenvalues are all the same primitive Mth root of unity, we give a sufficient and necessary condition on A such that for any holomorphic mapping f:Δn→ℂn with f(0)=0 and Df(0)=A, if 0 is an isolated fixed point of the Mth iteration fM , then 𝒪M (f,0)≥2.