| We are going to talk about a problem left in [2] by G. Y.Zhang.Let Δn denote the unit open ball|x|<1in the complex vector space Cn.Then let f:Δn→Cn be a holomorphic map,and M be a natural number.Suppose the origin0=(0,...,0) is an isolated fixed point of both f and its M-th iteration fM.Then for each factor m of M,the origin is also an isolated fixed point of fm.So we can define the fixed point index of fm at the origin,that is μfm(0) Then we can define the local Dold indice at the origin as follows: Here, P(M) is the set of prime factors of M,and the sum extends over all subsets τ of P(M),#τ is the cardinal number of τ,and M:τ=M(Πp)-1.PM(f,0) can be interpreted as the number of periodic points of period M of f overlapped at the origin:any holomorphic map f1:Δn→Cn sufficiently close to f has exactly PM (f,0) distinct periodic points of perod M near the origin,provided that all the fixed points of f1M near the origin are simple.So the number OM (f,0)=PM (f,0)/M can be interpreted as the number of periodic orbits of M of f overlapped at the origin. According to Shub-Sullivan [4], Chow-Mallet-Paret-Yorke [5],and G. Y.Zhang [1]:OM(f,0)≥1if and only if the linear part of at the origin has a periodic point of period M.In [2] by GYZhang,the author supposes Df(0) has m1-th,...,mn-th roots of unity as its eigenvalues, where m1,...,mn are distinct prime numbers, M=m1...mn.Then he gives a sufficient condition to conclude OM (f,0)≥2.Finally,G.Y.Zhang asks:Is the sufficient condition also a necessary one?This is what I try to solve.The second question I considered is:how the area-length ratio of a special covering surface over the unit sphere varies.This question arises from the classical inequality in Ahlfors’theory of covering surfaces. According to Ahlfors,suppose f:Δ→S is a holomorphic map,here A denotes the unit closed disc in the complex plane, S is the Riemann sphere, a1, a2,a3are distinct points of S.Then there exists a positive constant h=h(a1,a2,a3) such that A(f,Δ)≤hL(f,(?)Δ) for the above f.Here A(f,Δ) is the spherical area of(Δ), and L(f,(?)Δ) is the spherical length of f((?)Δ).The computation of sup-is connected with my second question. |
