The numbers of periodic orbits of holomorphic mappings hidden at fixed points

Details The numbers of periodic orbits of holomorphic mappings hidden at fixed points The numbers of periodic orbits of holomorphic mappings hidden at fixed points

2006-12-02

arxiv.org/abs/math/0612049v1

Mathematics

Dynamical Systems

33 pages

Scientific paper

Let $\Delta ^{2}$ be a ball in the complex vector space $\mathbb{C}^{2}$ centered at the origin, let $f:\Delta ^{2}\to \mathbb{C}^{2}$ be a holomorphic mapping$,$ with $f(0)=0$, and let $M$ be a positive integer. If the origin 0 is an isolated fixed point of the $M$ th iteration $f^{M}$ of $f,$ then one can define the number $\mathcal{O}_{M}(f,0)$ of periodic orbits of $f$ with period $M$ hidden at the fixed point 0, which has the meaning: any holomorphic mapping $g:\Delta ^{2}\to \mathbb{C}^{2}$ sufficiently close to $f$ in a neighborhood of the origin has exactly $% \mathcal{O}_{M}(f,0)$ distinct periodic orbits with period $M$ near the origin, provided that all fixed points of $g^{M}$ near the origin are all simple. It is known that $\mathcal{O}_{M}(f,0)\geq 1$ iff the linear part of $f$ at the origin has a periodic point of period $M.$ This paper will continue to study the number $\mathcal{O}_{M}(f,0)$. We are interested in the condition for the linear part of $f$ at the origin such that $\mathcal{O}_{M}(f,0)\geq 2.$ For a $2\times 2$ matrix $A$ that is arbitrarily given, the goal of this paper is to give a necessary and sufficient condition for $A$,$ $such that $% \mathcal{O}_{M}(f,0)\geq 2$ for all holomorphic mappings $f:\Delta ^{2}\to \mathbb{C}^{2}$ such that $f(0)=0,$ $Df(0)=A$ and that the origin 0 is an isolated fixed point of $f^{M}.$

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