Fixed point indices and periodic points of holomorphic mappings
| dc.creator | Zhang, Guang Yuan | |
| dc.date | 2005-11-10 | |
| dc.date | 2006-04-16 | |
| dc.date.accessioned | 2026-07-07T06:51:04Z | |
| dc.date.available | 2026-07-07T06:51:04Z | |
| dc.description | Let $Δ^{n}$ be the ball $|x|<1$ in the complex vector space $\mathbb{C}% ^{n}$, let $f:Δ^{n}\to \mathbb{C}^{n}$ 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 $f^{M}$ of $f$. Then for each factor $m$ of $M,$ the origin is again an isolated fixed point of $f^{m}$ and the fixed point index $μ_{f^{m}}(0)$ 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_{τ\subset P(M)}(-1)^{#τ}μ_{f^{M:τ}}(0), \end{equation*}% where $P(M)$ is the set of all primes dividing $M,$ the sum extends over all subsets $τ$ of $P(M)$, $#τ$is the cardinal number of $τ$ and $% M:τ=M(\prod_{p\in τ}p)^{-1}$. $P_{M}(f,0)$ can be interpreted to be the number of periodic points of period $M$ of $f$ overlapped at the origin: any holomorphic mapping $% f_{1}:Δ^{n}\to \mathbb{C}^{n}$ sufficiently close to $f$ has exactly $P_{M}(f,0)$ distinct periodic points of period $M$ near the origin$%, $ 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$.$ According to M. Shub and D. Sullivan's work [\ref{SS}], a necessary condition so that $P_{M}(f,0)\neq 0$ 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. | |
| dc.description | 26 pages | |
| dc.identifier | https://arxiv.org/abs/math/0511250 | |
| dc.identifier | http://arxiv.org/abs/math/0511250 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/104866 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Complex Variables | |
| dc.subject | 32H50, 37C25 | |
| dc.title | Fixed point indices and periodic points of holomorphic mappings | |
| dc.type | text |