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

DOI: http://dx.doi.org/10.12775/TMNA.2009.006

#### 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 $M=m_{1}\ldots m_{n}$.

^{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 $M=m_{1}\ldots m_{n}$.

#### Keywords

Fixed point index; periodic point

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