Abstract
In this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree mge 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 _1ne {p}. Then there is a constant C so that for every point z_0 inside any Omega _i, there exists a point qin 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.
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