Orbits inside Fatou sets

TL;DR

This paper studies the behavior of orbits within attracting basins of rational functions and entire functions, revealing conditions under which points can be approximated by preimages with bounded Kobayashi distance.

Contribution

It extends understanding of orbit behavior in Fatou sets, especially for rational functions with non-simply connected basins and entire functions with finitely many critical points.

Findings

Existence of a uniform bound for Kobayashi distance in non-simply connected basins.

Results align with polynomial cases when all basins are simply connected.

Similar orbit approximation results hold for entire functions with finitely many critical points.

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.