Dynamics inside Fatou sets in higher dimensions

TL;DR

This paper studies the dynamics within Fatou sets in higher dimensions, showing that points in attracting basins are uniformly close to preimages of the origin under certain polynomial maps, with limitations in other cases.

Contribution

It establishes a uniform bound on the Kobayashi distance between points in attracting basins and preimages of the origin for specific polynomial maps in higher dimensions.

Findings

Points in the attracting basin are within a bounded Kobayashi distance of preimages of the origin.

The result holds for maps where $F(z,w)=(P(z),Q(w))$ with specific polynomial conditions.

The result does not extend to many other cases, indicating limitations of the established bound.

Abstract

In this paper, we investigate the behavior of orbits inside attracting basins in higher dimensions. Suppose $F(z, w)=(P(z), Q(w))$, where $P(z), Q(w)$ are two polynomials of degree $m_1, m_2\geq2$ on $\mathbb{C}$, $P(0)=Q(0)=0,$ and $0<|P'(0)|, |Q'(0)|<1.$ Let $\Omega$ be the immediate attracting basin of $F(z, w)$. Then there is a constant $C$ such that for every point $(z_0, w_0)\in \Omega$, there exists a point $(\tilde{z}, \tilde{w})\in \cup_k F^{-k}(0, 0), k\geq0$ so that $d_\Omega\big((z_0, w_0), (\tilde{z}, \tilde{w})\big)\leq C, d_\Omega$ is the Kobayashi distance on $\Omega$. However, for many other cases, this result is invalid.