The numbers of periodic orbits hidden at fixed points of \(n\)-dimensional holomorphic mappings. II. (Q1029922)

Let \(f\) be a holomorphic mapping of the unit ball of \({\mathbb C}^n\) centered at \(0\) into \({\mathbb C}^n\), fixing the origin which we assume, for some integer \(M>1\), to be an isolated fixed point of both \(f\) and of \(f^M\). Denote by \(P_M(f,0)\) the local Dold index of \(f\) at \(0\). Then \({\mathcal O}(f,0)=P_M(f,0)/M\) can be seen as the number of hidden periodic orbits of period \(M\) at \(0\). The main result of this paper is a sufficient condition for \({\mathcal O}(f,0)\geq 2\). Namely, let \(m_1,\dots,m_n\) be distinct primes and assume \(Df(0)=\text{diag}(\lambda_1,\dots,\lambda_n)\) with \(\lambda_i\) a \(m_i\)-th primitive root of unity, for \(i=1,\dots,n\). Then, up to a biholomorphic coordinate transform, one can write \(f\) as \[ \Big( \lambda_i x_i +x_i\sum_{j=1}^na_{ij}x_j^{m_j}+\text{h.o.t.} \Big)_{i=1,\dots,n} \] with \(a_{ij}\in{\mathbb C}\). The main result of this paper states that \({\mathcal O}(f,0)\geq 2\) if the matrix \((a_{ij})_{i,j=1,\dots,n}\) is singular.