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

Abstract

Let $\Delta ^{n}$ be the ball $|x|< 1$ in the complex vector space ${\mathbb C} ^{n}$, let $f\colon \Delta ^{n}\rightarrow {\mathbb C}^{n}$ be a holomorphic mapping and let $M$ be a positive integer. Assume that the origin $0=(0,\ldots ,0)$ is an isolated fixed point of both $f$ and the $M$-th iteration $f^{M}$ of $f$. Then the (local) Dold index $P_{M}(f,0)$ at the origin is well defined, which can be interpreted to be the number of periodic points of period $M$ of $f$ hidden at the origin: any holomorphic mapping $f_{1}\colon \Delta ^{n}\rightarrow {\mathbb C}^{n}$ 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. Therefore, the number ${\mathcal O}_{M}(f,0)=P_{M}(f,0)/M$ can be understood to be the number of periodic orbits of period $M$ hidden at the fixed point. According to Shub-Sullivan [< i> A remark on the Lefschetz fixed point formula for differentiable maps< /i> , Topology < b> 13< /b> (1974), 189–191] and Chow-Mallet-Paret-Yorke [< i> A periodic orbit index which is a bifurcation invariant< /i> , Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 109–131], a necessary condition so that there exists at least one periodic orbit of period $M$ hidden at the fixed point, say, ${\mathcal O}_{M}(f,0)\geq 1$, is that the linear part of $f$ at the origin has a periodic point of period $M$. It is proved by the author in [< i> Fixed point indices and periodic points of holomorphic mappings< /i> , Math. Ann. < b> 337< /b> (2007), 401–433] that the converse holds true. In this paper, we continue to study the number ${\mathcal O}_{M}(f,0)$. We will give a sufficient condition such that ${\mathcal O}_{M}(f,0)\geq 2$, in the case that all eigenvalues of $Df(0)\ $are primitive $m_{1}$-th, $\ldots $, $m_{n}$-th roots of unity, respectively, and $m_{1},\ldots ,m_{n}$ are distinct primes with $M=m_{1}\ldots m_{n}$.