The Numbers of Periodic Orbits of Holomorphic Mappings Hidden at Fixed Points

Let ∆ be a ball in the complex vector space C centered at the origin, let f : ∆ → C be a holomorphic mapping, with f(0) = 0, and let M be a positive integer. If the origin 0 is an isolated fixed point of the M th iteration f of f, then one can define the number OM (f, 0) of periodic orbits of f with period M hidden at the fixed point 0, which has the meaning: any holomorphic mapping g : ∆ → C sufficiently close to f in a neighborhood of the origin has exactly OM (f, 0) distinct periodic orbits with period M near the origin, provided that all fixed points of g near the origin are all simple. It is known that OM (f, 0) ≥ 1 iff the linear part of f at the origin has a periodic point of period M. This paper will continue to study the number OM (f, 0). We are interested in the condition for the linear part of f at the origin such that OM (f, 0) ≥ 2. For a 2 × 2 matrix A that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for A, such that OM (f, 0) ≥ 2 for all holomorphic mappings f : ∆ → C such that f(0) = 0, Df(0) = A and that the origin 0 is an isolated fixed point of f .