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

Abstract

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 $f^{M}.$