Orbits inside Fatou sets

Abstract

In this paper, we investigate the precise behavior of orbits inside attracting basins of rational functions on $\mathbb P^1$ and entire functions $f$ in $\mathbb{C}$. Let $R(z)$ be a rational function on $\mathbb P^1$, $\mathcal {A}(p)$ be the basin of attraction of an attracting fixed point $p$ of $R$, and $\Omega_i$ $ (i=1, 2, \cdots)$ be the connected components of $\mathcal{A}(p)$, and $\Omega_1$ contains $p.$ Let $p_0\in\Omega_1$ be close to $p.$ If at least one $\Omega_i$ is not simply connected, then there exists a constant $C$ so that for any $z_0\in \Omega_i$, there is a point $q\in \cup_k R^{-k}(p_0), k\geq0$ so that the Kobayashi distance $d_{\Omega_i}(z_0, q)\leq C.$ If all $\Omega_i$ are simply connected, then the result is the same as for polynomials and is treated in an earlier paper. For entire functions $f$, we generally can not have similar results as for rational functions. However, if $f$ has finitely many critical points, then similar results hold.