In this paper, we investigate the precise behavior of orbits inside attracting basins of rational functions on ℙ1 and entire functions f in ℂ. Let R(z) be a rational function on ℙ1, \(\mathcal{A}(p)\) be the basin of attraction of an attracting fixed point p of R, and Ωi (i = 1, 2,... ) be the connected components of \(\mathcal{A}(p)\) . Assume Ω1 contains p. Let p0 ∈ Ω1 close to p. Then there exists a constant C so that for any z0 ∈ Ωi, there is a point q ∈ ∪kR−k(p0), k ≥ 0 so that the Kobayashi distance \({d_{{\Omega _i}}}({z_0},q) \le C\) . 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.
