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 $q\in {\cup }_k{R}^{-k}(p_0),k\ge 0$ so that the Kobayashi distance $d_{{\Omega }_i}(z_0,q)\le C.$ 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.

arXiv admin note: text overlap with arXiv:2208.00546