Dynamics inside Fatou sets in higher dimensions

Abstract

In this paper, we investigate the behavior of orbits inside attracting basins in higher dimensions. Suppose $F(z,w)=(P(z),Q(w))$, where $P(z),Q(w)$ are two polynomials of degree $m_1,m_2\ge 2$ on $\mathbb{C}$, $P(0)=Q(0)=0,$ and $0<|P^{\prime }(0)|,|Q^{\prime }(0)|<1.$ Let $\Omega$ be the immediate attracting basin of $F(z,w)$. Then there is a constant $C$ such that for every point $(z_0,w_0)\in \Omega$, there exists a point $(\tilde{z},\tilde{w})\in {\cup }_k{F}^{-k}(0,0),k\ge 0$ so that $d_{\Omega }((z_0,w_0),(\tilde{z},\tilde{w}))\le C,d_{\Omega }$ is the Kobayashi distance on $\Omega$. However, for many other cases, this result is invalid.