THE NUMBERS OF PERIODIC ORBITS HIDDEN AT FIXED POINTS OF THREE-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 the M-th iteration fM of f. Then one can define the number [Formula: see text], which can be interpreted to be the number of periodic orbits of f with period M hidden at the fixed point 0. For a 3 × 3 matrix A, of which the eigenvalues are all distinct primitive M-th roots of unity, we will give a sufficient and necessary condition for A such that for any holomorphic mapping f: Δ 3 → ℂ3 with f(0) = 0 and Df(0) = A, if 0 is an isolated fixed point of the M-th iteration fM, then [Formula: see text].