Interior dynamics of fatou sets

Abstract

In this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree \(m\ge 2\) 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 \(\mathcal {A}(p)\). Assume \(\Omega _1\) contains p and \(\{f^{-1}(p)\}\cap \Omega _1\ne \{p\}\). Then 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 \(d_{\Omega _i}(z_0, q)\le C\), where \(d_{\Omega _i}\) is the Kobayashi distance on \(\Omega _i.\) In paper Hu (Dynamics inside parabolic basins, 2022), we proved that this result is not valid for parabolic basins.