2006/12/02 by Guang Yuan Zhang, Zhang, Guang Yuan
Mathematics · #32H50 #37C25 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #math.CV #math.DS #msc:32H50 #msc:37C25
paper · pdf · doi:10.48550/arxiv.math/0612049
33 pages
arxiv created 2006/12/02 · arxiv updated 2009/12/01
Let Δ2 be a ball in the complex vector space ℂ2 centered at the origin, let f:Δ2→ ℂ2 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 fM 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:Δ2→ ℂ2 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 gM 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:Δ2→ ℂ2 such that f(0)=0, Df(0)=A and that the origin 0 is an isolated fixed point of fM.