Interior Dynamics of Fatou Sets

Abstract

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let $f$ be a holomorphic polynomial of degree $m\ge 2$ in $\mathbb{C}$, $\mathcal{A}(p)$ be the basin of attraction of an attracting fixed point $p$ of $f$, and ${\Omega }_i(i=1,2,\cdots )$ be the connected components of $\mathcal{A}(p)$. We prove that there is a constant $C$ so that for every point $z_0$ inside any ${\Omega }_i$, there exists a point $q\in {\cup }_k{f}^{-k}(p)$ inside ${\Omega }_i$ such that $d_{{\Omega }_i}(z_0,q)\le C$, where $d_{{\Omega }_i}$ is the Kobayashi distance on ${\Omega }_i.$