Interior Dynamics of Fatou Sets

TL;DR

This paper analyzes the internal dynamics of Fatou sets, specifically the behavior of points within attracting basins of polynomial maps, establishing bounds on distances to preimages using Kobayashi metrics.

Contribution

It provides a new bound on the Kobayashi distance between points in attracting basins and their preimages, deepening understanding of Fatou set dynamics.

Findings

Existence of a uniform constant C for points in attracting basins.

Bound on Kobayashi distance between points and preimages.

Characterization of orbit behavior inside Fatou components.

Abstract

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let $f$ be a holomorphic polynomial of degree $m\geq2$ 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 $d_{\Omega_i}(z_0, q)\leq C$, where $d_{\Omega_i}$ is the Kobayashi distance on $\Omega_i.$