Fixed point indices and periodic points of holomorphic mappings

Let ${\mathrm{\Delta }}^n$ be the ball $\mathrm{\mid }x\mathrm{\mid }<1$ in the complex vector space $\mathbb{C}$, let $f:{\mathrm{\Delta }}^n\to {\mathbb{C}}^n$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $% 0=(0,…, 0)$isanisolatedfixedpointofboth$f$andthe$M$-thiteration$f^{M}$of$f$\mathrm{.}Thenforeachfactor$m$of$M,$ the origin is again an isolated fixed point of $f^m$ and the fixed point index ${\mu }_{f^m}(0)$ of $f^m$ at the origin is well defined, and so is the (local) Dold’s index (see [\ref{Do}]) at the origin:% \begin{equation*} P_{M}(f,0)=\sum_{\tau \subset P(M)}(-1)^{#\tau}\mu_{f^{M:\tau}}(0), \end{equation*}% where $P(M)$ is the set of all primes dividing $M,$ the sum extends over all subsets $\tau$ of $P(M)$, #\tauis the cardinal number of $\tau$ and $% M:\tau =M(\prod_{p\in \tau}p)^{-1}$\mathrm{.}$P_{M}(f,0)$canbeinterpretedtobethenumberofperiodicpointsofperiod$M$of$f$overlappedattheorigin:anyholomorphicmapping$% f_{1}:\Delta ^{n}\to $\mathbb{C}$^{n}$sufficientlycloseto$f$hasexactly$P_{M}(f,0)$distinctperiodicpointsofperiod$M$neartheorigin$%, $ provided that all the fixed points of $f_1^M$ near the origin are simple. Note that $f$ itself has no periodic point of period $M$ near the origin$\mathrm{.}$ According to M. Shub and D. Sullivan’s work [\ref{SS}], a necessary condition so that $P_M(f,0)\mathrm{\ne }0$ is that the linear part of $f$ at the origin has a periodic point of period $M\mathrm{.}$ The goal of this paper is to prove that this condition is sufficient as well for holomorphic mappings.