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 $m_1, m_2\geq2$ on $\mathbb{C}$, $P(0)=Q(0)=0,$ and $0<|P'(0)|, |Q'(0)|<1.$ 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 $(\tilde{z}, \tilde{w})\in \cup_k F^{-k}(0, 0), k\geq0$ so that $d_\Omega\big((z_0, w_0), (\tilde{z}, \tilde{w})\big)\leq C, d_\Omega$ is the Kobayashi distance on $\Omega$. However, for many other cases, this result is invalid.