2005/11/10 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/0511250
26 pages
arxiv created 2006/04/16 · arxiv updated 2009/12/01
Let Δn be the ball |x|<1 in the complex vector space ℂ% n, let f:Δn→ ℂn be a holomorphic mapping and let M be a positive integer. Assume that the origin % 0=(0,..., 0) is an isolated fixed point of both f and the M-th iteration fM of f. Then for each factor m of M, the origin is again an isolated fixed point of fm and the fixed point index μfm(0) of fm at the origin is well defined, and so is the (local) Dold's index (see [\refDo]) at the origin:% PM(f,0)=∑τ⊂ P(M)(-1)#τμfM:τ(0),% where P(M) is the set of all primes dividing M, the sum extends over all subsets τ of P(M), #τis the cardinal number of τ and % M:τ=M(∏p∈ τp)-1. PM(f,0) can be interpreted to be the number of periodic points of period M of f overlapped at the origin: any holomorphic mapping % f1:Δn→ ℂn sufficiently close to f has exactly PM(f,0) distinct periodic points of period M near the origin%, provided that all the fixed points of f1M near the origin are simple. Note that f itself has no periodic point of period M near the origin. According to M. Shub and D. Sullivan's work [\refSS], a necessary condition so that PM(f,0)≠ 0 is that the linear part of f at the origin has a periodic point of period M. The goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings.